PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 100(114) (2016), 17–48 DOI: 10.2298/PIM1614017M

A SEQUENTIAL APPROACH TO ULTRADISTRIBUTION SPACES

Snježana Maksimović, Svetlana Mincheva-Kamińska, Stevan Pilipović, and Petar Sokoloski

Abstract. We introduce and investigate two types of the space U^{∗} of s-ultra-distributions meant as equivalence classes of suitably defined fundamental sequences of smooth functions; we prove the existence of an isomorphism between U^{∗} and the respective space D^{′∗} of ultradistributions: of Beurling type if ∗ = (p!^{t}) and of Roumieu type if ∗ = {p!^{t}}. We also study the spaces T^{∗} and T̃^{∗} of t-ultradistributions and t̃-ultradistributions, respectively, and show that these spaces are isomorphic with the space S^{′∗} of tempered ultradistributions both in the Beurling and the Roumieu case.

1. Introduction

That distributions based by Sobolev [29] and Schwartz [28] on functional analysis can be founded on a more elementary sequential approach was remarked by Mikusiński already in [18] and [19]. This idea was accomplished by him in cooperation with Sikorski in [20,21] and then, in the extended form, together with the third author Antosik in [1].

Roumieu and Beurling in [27] and [2] introduced two types of ultradistribution spaces, substantially larger than the space of distributions. However only the famous papers [12–14] of Komatsu which substantially extended the knowledge on the structure of these spaces gave impulse to an intensive development of the theory of ultradistributions of both types in various directions. In particular, the theory became an important tool of microlocal analysis.

Similarly as in the case of distributions one can expect that an ultradistribution can also be viewed as, in a sense, a limit of a sequence of functions or, more precisely, as an equivalent class of sequences of smooth functions, suitably approximating it. Our aim in this paper is to provide a sequential approach to the theory of non-quasi-analytic ultradistributions of both Beurling and Roumieu types [12]. Analogously to the sequential theory of distributions [1], we introduce sequential ultadistributions, called shortly s-ultadistributions, as equivalence classes of fundamental sequences of smooth functions which, however, are defined by means of ultradifferential operators instead of differential operators. The difference is intrinsic and requires distinct techniques: instead of polynomials we have to use functions of sub-exponential growth and apply their specific properties. In showing that the sequential approach is equivalent to the classical approach to ultradistribution theory (see [12–14]) one needs to know intrinsic structures of ultradistributions as well as of tempered ultradistributions (see [5–10, 17–25]); the equivalence of the two approaches will be proved through Hermite expansions and certain structural properties of tempered ultradistributions.

2010 Mathematics Subject Classification: Primary 46F05; Secondary 46F10.

Key words and phrases: fundamental sequences, Hermite expansions.

Communicated by Gradimir Milovanović.

In order to simplify our exposition we will consider only the Gevrey sequence of functions of the form M_{p} = p!^{t} (p ∈ ℤ_{+} ) for t > 1; they satisfy all conditions usually assumed for a general sequence (M_{p})_{p}. Therefore we use the simplified symbols D^{(t)}(Ω), D^{{t}}(Ω) for the spaces D^{(M_{p})} (Ω), D^{{M_{p} }} (Ω) of test functions on an open set Ω ⊆ ℝ^{d} and S^{(t)}(ℝ^{d}), S^{{t}}(ℝ^{d}) for the spaces S^{(M_{p})} (ℝ^{d}), S^{{M_{p} }} (ℝ^{d}) of test functions on ℝ^{d}, respectively. This concerns also their duals, i.e., D^{′(t)}(Ω), D^{′{t}}(Ω) are the spaces of ultradistributions of Beurling and Roumieu type on the set Ω and S′(ℝ^{(t)})d, S^{′{t}}(ℝ^{d}) are the spaces of tempered ultradistributions of Beurling and Roumieu type on ℝ^{d}, respectively. We traditionally use the upper index ^{∗} for a common notation of the considered spaces both in the Beurling and Roumieu cases, i.e., D^{∗}(Ω), S^{∗}(ℝ^{d}), D^{′∗}(Ω), S^{′∗}(ℝ^{d}) are common symbols for the pairs of spaces listed above. The mentioned spaces were investigated in [5, 10, 12, 16, 25] and in many other papers. It should be noted that another approach to the theory of ultradistributions was developed by D. Vogt, R. Meise and their collaborators. There exists an extensive literature in this direction with many applications; we refer here just to a few of them (and references therein): [3, 4, 23, 30].

Our approach to ultradistributions is similar to that presented in [1] for distributions. We begin with the definition of a special kind of fundamental sequences of smooth functions and the corresponding equivalence classes called sultradistributions which are elements of the space that we denote by U^{∗}(Ω). This is done in sections 2 and 3 together with an analysis of operations on s-ultradis-tributions, the convergence structure in U^{∗}(Ω) and actions of s-ultradistributions on test functions belonging to D^{∗}(Ω). In section 4 we introduce the spaces T^{∗} and T̃^{∗} of t-and t̃-ultradistributions, respectively. Again we discuss their structure, the convergence in them and actions of considered tempered ultradistributions on elements of the respective spaces of test functions. It is well-known (see e.g. [8]) that there exists a topological isomorphism between the space S^{∗}(ℝ^{d}) and the Köthe echelon space s^{∗} of sequences of sub-exponential growth. Using this fact we prove in section 5 that the spaces T^{∗} and T̃^{∗} are topologically isomorphic with the space S^{′∗}(ℝ^{d}). Applying the results of section 5, we prove in section 6 the existence of a sequential topological isomorphism between the spaces U^{∗}(Ω) and D^{′∗}(Ω).

1.1. Preliminaries. The sets of all positive integers, nonnegative integers, real and complex numbers are denoted by ℕ, ℕ_{0}, ℝ and ℂ, respectively.

For x = (x_{1}, …, x_{d}) ∈ ℝ^{d}, y = (y_{1}, …, y_{d}) ∈ ℝ^{d}, α = (α_{1}, …, α_{d}) ∈ ℕ_{0}^{d},

β = (β_{1}, …, β_{d}) ∈ ℕ_{0}^{d} and λ ∈ ℝ, we use the following notation

x + y := (x_{1} + y_{1}, …, x_{d} + y_{d}) ∈ ℝ^{d}; x + λ := (x_{1} + λ, …, x_{d} + λ) ∈ ℝ^{d};

λx := (λx_{1}, …, λx_{d}) ∈ ℝ^{d}; x ⩽ y if x_{j} ⩽ y_{j} for j = 1, …, d;

x^{α} := ∏_{j=1}^{d} x_{j}^{α_{j}} ; α! := α_{1}! …α_{d}!; \binom{α}{β} := ∏_{j=1}^{d} \binom{α_{j}}{β_{j}} for α ⩽ β;

|x| := (x_{1}^{2} + ⋯+ x_{d}^{2})^{1/2} ; |α| := α_{1} + ⋯+ α_{d};

D^{α} = D_{x}^{α} := D_{1}^{α_{1}} …D_{d}^{α_{d}} , where D_{j}^{α_{j}} := (−i∂/∂x_{j} )^{α_{j}} (j = 1, …, d)

and the following summation notation

∑_{|α|=0}^{∞} := ∑_{α∈ℕ_{0}^{d}} ;  ∑_{0⩽α⩽β} := ∑_{α_{1} =0}^{β_{1}} ⋯ ∑_{α_{d} =0}^{β_{d}} .

The symbol X^{∘} for X ⊂ ℝ^{d} means the interior of X and the symbol K ⋐ V for an open V ⊂ ℝ^{d} means that K is a compact subset of V. By Ω we denote a fixed open subset of ℝ^{d}. By C(ℝ^{d}) and C(K) for K ⋐ Ω we denote the sets

of all continuous functions on ℝ^{d} and K, respectively, and by −−−→^{C(ℝ^{d})} and −−−→^{C(K)} the uniform convergences on ℝ^{d} and K of sequences of functions in C(ℝ^{d}) and C(K), respectively; the latter is the convergence in the Banach space C(K) with the supremum norm ∥ · ∥_{∞} . We denote a sequence (α_{n})_{n∈ℕ} of numbers (functions, distributions, ultradistributions) shorter by (α_{n}) or (α_{n})_{n} and the mapping Ω ∋ x ↦ F (x) by F or F (x). The norm in L^{2}(ℝ^{d}) is denoted by ∥·∥_{2} and the convergence in L^{2}(ℝ^{d}) by ⟶^{2} . The support of a function (distribution, ultradistribution) f by suppf. A function f is called compactly supported if there is a K ⋐ ℝ^{d} such that suppf ⊂ K. For the Fourier transform of φ ∈ S(ℝ^{d}) we use the two symbols: F(φ) = φ̂ := ∫_{ℝ^{d}} φ(x)e^{−i⟨x,·⟩} dx; clearly, F(D^{α} φ) = ξ^{α} φ̂ (ξ ∈ ℝ^{d}). For the properties of the spaces of test functions D(Ω), S(ℝ^{d}) and their duals D′(Ω), S′(ℝ^{d}) we refer to [28].

We recall some notions from [12]. By the associated function, corresponding to the Gevrey sequence (p!^{t})_{p} for a fixed t > 1, we mean the following function: M(ρ) := sup_{p∈ℕ_{0}} log_{+} ρ^{p}/p!^{t} = e^{kρ^{1/t}} for ρ > 0, where k > 0 is an appropriate constant. Denote by R the set of all sequences (r_{p}) of positive numbers strictly increasing to infinity. By the (r_{p})-associated function, corresponding to (r_{p}) ∈ R, we mean the function: N_{(r_{p})} (ρ) := sup_{p∈ℕ_{0}} log_{+} ρ^{p}/N_{p} for ρ > 0, where (N_{p})_{p∈ℕ_{0}} is defined by means of (r_{p}) as follows

(1.1)  N_{0} := 1; N_{p} := p!^{t}R_{p}, where R_{p} := ∏_{j=1}^{p} r_{j} , for p ∈ ℕ.

Note that if (r_{p}) ∈ R, then for every k > 0 there is a ρ_{0} > 0 such that N_{(r_{p})} (ρ) ⩽ e^{kρ^{1/t}} for ρ > ρ_{0} (see [12]).

Let K ⋐ Ω and h > 0. We recall the definitions of some spaces of test functions [12]

E^{t,h} (K):= \{φ ∈ C^{∞} (Ω): P_{h,K} (φ) := sup_{x∈K,α∈ℕ_{0}^{d}}  \frac{|D^{α} φ(x)|}{h^{|α|} α!^{t}} < ∞\};
D_{K}^{t,h} := E^{t,h} (K) ∩ {φ ∈ C^{∞} (Ω) : suppφ ⊂ K};
D_{K}^{(t)} := \underleftarrow{lim}_{h→0} D_{K}^{t,h}; D^{(t)}(Ω) := \underrightarrow{lim}_{K⋐Ω} D_{K}^{(t)};
D_{K}^{\{t\}} := \underrightarrow{lim}_{h→∞} D_{K}^{t,h}; D^{\{t\}}(Ω) := \underrightarrow{lim}_{K⋐Ω} D_{K}^{\{t\}}.

As already said, we use the common symbol D^{∗}(Ω) for the spaces D^{(t)}(Ω) and D^{{t}}(Ω) and D^{′∗}(Ω) for their duals. Recall now (see [5]) the definitions of the spaces S^{(t)}(ℝ^{d}) and S^{{t}}(ℝ^{d}), invariant under the Fourier transform

S_{h}^{t}(ℝ^{d}) := \{f ∈ S(ℝ^{d}): ∃C > 0 ∀α, β ∈ ℕ_{0}^{d} \frac{∥x^{α} ∂^{β}f∥_{2}}{h^{|α+β|} α!^{t}β!^{t}} ⩽ C\};
S^{(t)}(ℝ^{d}) := \underleftarrow{lim}_{h→0} S_{h}^{t}(ℝ^{d});  S^{\{t\}}(ℝ^{d}) := \underrightarrow{lim}_{h→∞} S_{h}^{t}(ℝ^{d}).

Notice that the space S^{(t)}(ℝ^{d}) is nontrivial if t > 1/2, while S^{{t}}(ℝ^{d}) is nontrivial if t ⩾ 1/2. The spaces S^{(t)}(ℝ^{d}) and S^{{t}}(ℝ^{d}) are denoted commonly by S^{∗}(ℝ^{d}) and their duals by S^{′∗}(ℝ^{d}).

The Hermite polynomials H_{n} and the corresponding Hermite functions h_{n} are defined on ℝ by

H_{n}(x) := (−1)^{n}e^{x^{2}} ( \frac{d}{dx})^{n}(e^{−x^{2}} ), h_{n}(x) := (2^{n}n!\sqrt{π})^{− \frac{1}{2}} e^{−x^{2} /2} H_{n}(x)

for x ∈ ℝ, n ∈ ℕ_{0}. The d-dimensional Hermite functions h_{n} are defined by

h_{n}(x) := h_{n_{1}} (x_{1}) …h_{n_{d}} (x_{d}), x = (x_{1}, …, x_{d}) ∈ ℝ^{d}, n ∈ ℕ_{0}^{d}.

They form an orthonormal basis for L^{2}(ℝ^{d}) and are the eigenfunctions of the product H = ∏ _{i=1}^{d} (−∂^{2}/∂x_{i}^{2} +x_{i}^{2}) of the one-dimensional Hermite harmonic oscillators, so that H^{α} = ∏ _{i=1}^{d} (−∂^{2}/∂x_{i}^{2} + x_{i}^{2})^{α_{i}} and

H^{α} h_{k}(x) = (2k + 1)^{α} h_{k}(x) = ∏_{i=1}^{d} (2k_{i} + 1)^{α_{i}} h_{k}(x), x ∈ ℝ^{d}, α, k ∈ ℕ_{0}^{d}.

Note that H is a self-adjoint operator. For f ∈ S(ℝ^{d}), the Hermite coefficients are c_{k} = ∫_{ℝ^{d}} fh_{k} = (f, h_{k})_{L^{2}} for k ∈ ℕ_{0}^{d}.

We have

D^{∗}(ℝ^{d}) ↪ S^{∗}(ℝ^{d}) ↪ S(ℝ^{d}) ↪ L^{2}(ℝ^{d}) ↪ S′(ℝ^{d}) ↪ S^{′∗}(ℝ^{d}) ↪ D^{′∗}(ℝ^{d}),

where the symbol ↪ means that the identity mapping is a continuous and dense embedding.

A sequence (δ_{n}) of the form δ_{n} := n^{d}φ(n·), n ∈ ℕ, where φ ∈ D^{∗}(ℝ^{d}), φ = 1 in B(0, 1/2) and φ = 0 out of B(0, 1) (B(x_{0}, r) denotes the closed ball with the center at x_{0} and radius r) is called a delta sequence in D^{′∗}(ℝ^{d}).

1.2. Ultradifferential operators. We recall the definitions and some results related to ultradifferential operators from [12–16]. A formal expression P (D) = ∑ _{|α|=0}^{∞} a_{α} D^{α} (a_{α} ∈ ℂ), corresponding to the function P (z) = ∑ _{|α|=0}^{∞} a_{α} z^{α} (z ∈ ℂ^{d}), is called an ultradifferential operator of the Beurling class (p!^{t}) resp. of the Roumieu class {p!^{t}}) if it satisfies the condition

∃h > 0 ∃C > 0 (resp. ∀h > 0 ∃C > 0) ∀α ∈ ℕ_{0}^{d} |a_{α} | ⩽ \frac{Ch^{|α|}}{(α!)^{t}} ;

in the Roumieu case, the condition can be expressed in the equivalent form

∃(r_{p}) ∈ R ∃C > 0 ∀α ∈ ℕ_{0}^{d} |a_{α} | ⩽ \frac{C}{(α!)^{t}R_{|α|}} ,

where R_{|α|} is defined in (1.1) and t > 1 will be fixed throughout the paper. We use the common term ultradifferential operator of the class ∗ in both cases of Beurling and Roumieu.

If P (D) is an ultradifferential operator of the Beurling class (resp. of the Roumieu class), then the function P (z) satisfies, by [12, Proposition 4.5], the estimate

∃h > 0 ∃C > 0 (resp. ∀h > 0, ∃C > 0) ∀z ∈ ℂ^{d} |P (z)| ⩽ Ce^{h|z|^{1/t}} ;

in the Roumieu case, the estimate can be written in the equivalent form

∃(r_{p}) ∈ R ∃C > 0 ∀ξ ∈ ℝ^{d} |P (ξ)| ⩽ Ce^{c(|ξ|)^{1/t}} ,

where c is the subordinate function of (r_{p}), i.e., an increasing function on [0, ∞) such that c(0) = 0 and c(ρ)/ρ → 0 as ρ → ∞ corresponding to (r_{p}) by means of the identity: M(c(ρ)) = N_{(r_{p})} (ρ) (ρ > 0), where N_{(r_{p})} is the (r_{p})-associated function (see [12]).

We denote by P^{(t)} (resp. P^{{t}}) the class of ultradifferential operators P_{r}(D) of Beurling type (resp. P_{(r_{p})} (D) of Roumieu type) of the form

P_{r}(D) = ( 1 + ∑_{j=1}^{d} D_{j}^{2}) ^{l} ∏_{p=1}^{∞} [1 + ( ∑_{j=1}^{d} D_{j}^{2}) /r^{2}p^{2t}] ( = ∑_{p=0}^{∞} a_{p}D^{p}) , (1.2)

P_{(r_{p})} (D) = ( 1 + ∑_{j=1}^{d} D_{j}^{2}) ^{l} ∏_{p=1}^{∞} [1 + ( ∑_{j=1}^{d} D_{j}^{2}) /r_{p}^{2} p^{2t}] ( = ∑_{p=0}^{∞} b_{p}D^{p}) , (1.3)

where r > 0, (r_{p}) ∈ R and l ⩾ 0. Replacing D_{j} by ξ_{j} in (1.2) and (1.3) we get the ultra-polynomials P_{r}(ξ) and P_{(r_{p})} (ξ) of the Beurling and Roumieu type corresponding to the ultradifferential operators P_{r}(D) and P_{(r_{p})} (D), respectively. They can be described in the following way (see [12]):

Ultra-polynomials of Beurling type are of sub-exponential growth, i.e., there are constants C_{1}, C_{2}, C > 0 and h_{1}, h_{2}, h > 0 such that

(1.4)  C_{1}e^{h_{1} |ξ|^{1/t}} ⩽ |P_{r}(ξ)| ⩽ C_{2}e^{h_{2} |ξ|^{1/t}} , ξ ∈ ℝ^{d},

and

|a_{p}| ⩽ Ch^{p}/p!^{t}, p ∈ ℕ_{0}.

The description of ultra-polynomials in the Roumieu case is more difficult. One can prove, similarly to the Beurling case, that for a given (r_{p}) ∈ R and its subordinate function c there exists a constant C > 0 such that

(1.5)  Ce^{c(|ξ|)^{1/t}} ⩽ |P_{(r_{p})} (ξ)|, ξ ∈ ℝ^{d}.

To get a suitable upper estimate we have to find a sequence (r_{0,p}) ∈ R and its subordinate function c_{0} such that the inequality

(1.6)  (1 + |ξ|^{2})^{l}|P_{(r_{p})} (ξ)| ⩽ C_{0}e^{c_{0} (|ξ|)^{1/t}} , ξ ∈ ℝ^{d}.

holds for some C_{0} > 0 and all l ⩾ 0. For this aim we use the following property of a subordinate function which is a consequence of Lemma 3.12 (see also Lemma 3.10) in [12].

Let c be an arbitrary subordinate function and put c̃ := 2c. There exists a sequence (r_{p}^{0}) ∈ R such that the (r_{p}^{0})-associated function N_{(r_{p}^{0} )} and the subordinate function c_{0} corresponding to (r_{p}^{0}) satisfy the inequality

M(c̃(ρ)) ⩽ N_{(r_{p}^{0} )} (ρ) = M(c_{0}(ρ)), ρ > 0,

Consequently,

c_{0}(ρ) ⩾ c̃(ρ) = 2c(ρ), ρ > 0.

The above remarks can be formulated as follows:

Lemma 1.1. For an arbitrary subordinate function c, corresponding to some sequence (r_{p}) ∈ R, and h > 0 there exist a sequence (r_{p}^{0}) ∈ R and its subordinate function c_{0} such that

c_{0}(ρ) ⩾ hc(ρ), ρ > 0.

In particular, for a given subordinate function c there exists another subordinate function c_{0} (both corresponding to appropriate sequences from R) such that c_{0}(ρ) ⩾ c(2ρ) for all ρ > 0.

In Subsection 4.2 we will need also the following lemma.

Lemma 1.2. (a) If P_{r} ∈ P^{(t)} (resp. P_{(r_{p})} ∈ P^{{t}}), then there exist r_{0} > 0 (resp. (r_{p}^{0}) ∈ R), C > 0 and ε > 0 such that

\begin{matrix}|D^{α} P_{r}(x)| & ⩽ \frac{Cα!}{ε^{|α|}} e^{r_{0} |x|^{1/t}} , & x ∈ ℝ^{d}, α ∈ ℕ_{0}^{d} \\ ( resp. |D^{α} P_{(r_{p})} (x)| & ⩽ \frac{Cα!}{ε^{|α|}} e^{c_{(r^{0}_{p} )} (|x|)^{1/t}} , & x ∈ ℝ^{d}, α ∈ ℕ_{0}^{d}) , \end{matrix}

where c_{(r_{p}^{0} )} is the subordinate function corresponding to the sequence (r_{p}^{0}).

(b) If Pr̃ ∈ P^{(t)} (resp. P_{(r̃_{p} )} ∈ P^{{t}}), then there exist r̃_{0} > 0 (resp. (r̃_{p}^{0}) ∈ R), C > 0 and ε > 0 such that

\begin{matrix}|D^{α} (1/P_{r}(x))| & ⩽ \frac{Cα!}{ε^{|α|}} e^{−r̃_{0} |x|^{1/t}} , & x ∈ ℝ^{d}, α ∈ ℕ_{0}^{d} \\ ( resp. |D^{α} (1/P_{(r̃_{p} )}(x))| & ⩽ \frac{Cα!}{ε^{|α|}} e^{−c_{(r̃^{0}_{p} )} (|x|)^{1/t}} , & x ∈ ℝ^{d}, α ∈ ℕ_{0}^{d}) , \end{matrix}

where c_{(r̃_{p}^{0} )} is the subordinate function corresponding to the sequence (r̃_{p}^{0}).

Proof. In the proof of both parts we use Lemma 1.1 and the Cauchy integral formula for ultra-polynomials on the circles K(x, ε) around x ∈ ℝ^{d}. The proof of (b) follows from the estimate

|D^{α} (1/P_{(r̃_{p} )}(x))| ⩽ \frac{Cα!}{ε^{|α|}} e^{−N_{(r̃p )} (|x|)} , x ∈ ℝ^{d}, α ∈ ℕ_{0}^{d},

shown in [26, Lemma 2.1], because we can construct c_{(r̃_{p}^{0} )} such that

N_{(r̃_{p} )}(|x|) = M(c_{(r̃_{p} )}(|x|)) = Cc_{(r̃_{p} )}(|x|)^{1/t} = c_{(r̃_{p}^{0} )}(|x|)^{1/t}

for some C > 0 and |x| > 0, in view of Lemma 3.10 in [12] and Lemma 1.1. The proof of (a) follows from Proposition 4.5 in [12]; see also the last part of the proof of Theorem 10.2 in [12]. □

The symbol P^{∗} will be common for the classes P^{(t)} and P^{{t}} of ultradifferential operators of Beurling and Roumieu types and P^{2∗} will mean the space of t-ultradistributions of both types. The corresponding spaces of ultradifferentiabile functions will be denoted by P_{u}^{∗} and P_{u}^{2∗}, respectively. This notation looks complicated but it helps to distinct the different use of P: P (D), P (x), P (ξ). To simplify the exposition we will usually consider ultradifferential operators of the form (1.2) and (1.3), but in some proofs we need their general form.

Denote by μ_{β} the operator acting on measurable functions G as follows: (μ_{β}G)(ξ) := (iξ)^{β}G(ξ) for ξ ∈ ℝ^{d} and β ∈ ℕ_{0}^{d}; in particular μ_{0}G = G. We will use in the sequel the following assertion: for arbitrary β ∈ ℕ_{0}^{d}, q ∈ [1, ∞] and P_{r}(D) ∈ P^{(t)} with r > 0 (resp. P_{(r_{p})} (D) ∈ P^{{t}} with (r_{p}) ∈ R) there exists Pr̃(D) ∈ P^{(t)} with r̃ > r (resp. P_{(r̃_{p})} ∈ P^{{t}} with (r̃_{p}) ∈ R, r_{p}/r̃_{p} ↓ 0 as p → ∞) such that

(1.7)

μ_{β} \frac{P_{r}}{Pr̃} ∈ L^{q}(ℝ^{d}) and F^{−1} ( μ_{β} \frac{P_{r}}{Pr̃}) ∈ L^{q}(ℝ^{d})
( resp. μ_{β} \frac{P_{(r_{p})}}{P_{(r̃_{p} )}} ∈ L^{q}(ℝ^{d}) and F^{−1} ( μ_{β} \frac{P_{(r_{p})}}{P_{(r̃_{p} )}} ) ∈ L^{q}(ℝ^{d}))

and, analogously, with P_{u}^{2∗} instead of P_{u}^{∗}, the following one

(1.8)  ( \frac{P_{r}(2α + 1)}{Pr̃(2α + 1)} )_{α∈ℤ^{d}} ∈ l^{q} ( resp. ( \frac{P_{(r_{p})} (2α + 1)}{P_{(r̃_{p} )}(2α + 1)} )_{α∈ℤ^{d}} ∈ l^{q}) ,

where P (2α + 1) := ∑ _{|k|=0}^{∞} a_{k} ∏ _{i=1}^{d} (2α_{i} + 1)^{k_{i}} for α ∈ ℕ_{0}^{d}.

The following well-known assertions will also be used in the sequel; their proofs can be found e.g. in [5, 25].

Lemma 1.3. (a) A smooth function φ on ℝ^{d} belongs to S^{∗}(ℝ^{d}) iff for arbitrary P ∈ P^{∗} and P_{1} ∈ P_{u}^{∗} we have ∥P_{1}P (−D)φ∥_{2} < ∞.

(b) If φ_{n} ∈ S^{∗}(ℝ^{d}) (n ∈ ℕ_{0}) and φ_{n} ⟶^{S^{∗}} φ_{0} as n → ∞, then for arbitrary

P ∈ P^{∗} and P_{1} ∈ P_{u}^{∗} we have P_{1}P (−D)φ_{n} ⟶^{S^{∗}} P_{1}P (−D)φ_{0} as n → ∞.

(c) If φ_{n} ∈ S^{∗}(ℝ^{d}) (n ∈ ℕ_{0}) and φ_{n} ⟶^{S^{∗}} φ_{0} as n → ∞, then for every P ∈ P^{∗}

we have P (H)φ_{n} ⟶^{S^{∗}} P (H)φ_{0} as n → ∞.

2. Fundamental sequences

Let us recall that Schwartz distributions in the sequential approach presented in [1] are equivalence classes of fundamental sequences of smooth functions defined with the use of derivatives of finite order. We introduce s-ultradistributions of Beurling and Roumieu type in a similar way, but our fundamental sequences are defined by means of the ultradifferential operators P_{r}(D) and P_{(r_{p})} (D), respectively, instead of finite order differential operators.

If P ∈ P^{∗} and F is an integrable function compactly supported, then P (z)F(F)(z) (z ∈ ℂ^{d}) is an entire function of sub-exponential growth on ℝ^{d}. If the inverse Fourier transform F^{−1} (P F̂) is a locally integrable function, then we define

(2.1)  P (D)F (x) := F^{−1} (P F̂)(x), x ∈ ℝ^{d}

to give the meaning for the formal acting of the ultradifferential operator P (D) on a compactly supported smooth function. If F is a compactly supported smooth function such that suppF ⊂ K_{1} ⋐ Ω and P ∈ P^{∗} is of the form P (D) := ∑ _{|α|=0}^{∞} a_{α} D^{α}, then the left hand side of (2.1) is meant as follows

(2.2)  P (D)F (x) := lim_{k→∞} P_{k}(D)F (x), x ∈ K,

where P_{k}(D) := ∑ _{|α|=0}^{k} a_{α} D^{α} and the limit in (2.2) is assumed to exist for every x ∈ K and to be a smooth function on K. In this case the limit defines f(x) = P (D)F (x) for x ∈ K and gives the meaning of (2.3) below.

Definition 2.1. A sequence (f_{n}) of smooth functions defined on an open set Ω ⊂ ℝ^{d} is called s-fundamental (of type ∗, i.e., of Beurling or Roumieu type, respectively) in Ω if for arbitrary K_{1} ⋐ Ω and K ⋐ K_{1}^{∘} there exist an ultradifferential operator P (D) ∈ P^{∗}, a sequence (F_{n}) of smooth functions on Ω and a continuous function F_{0} on Ω such that

f_{n} = P (D)F_{n} on K (n ∈ ℕ), suppF_{n} ⊂ K_{1} (n ∈ ℕ_{0}) (2.3)

and F_{n} −−−→^{C(K)} F_{0} as n → ∞.

The equality in (2.3) is meant in the sense of (2.2). In the sequel, for a given K ⋐ Ω we will always take a set K_{1} ⋐ Ω with K ⋐ K_{1}^{∘} which is sufficiently close to K, not referring explicitly about it (one can show, taking an appropriate cut-of function, that the definition does not depend on the choice of K_{1}).

Remark 2.1. 1^{∘} One can consider in (2.3) all ultradifferential operators of class ∗, not only belonging to P^{∗}, which gives a more general form of the definition. Since both formulations are equivalent, we will use the above one for simplicity.

2^{∘} Let Ω_{1}, Ω be open sets in ℝ^{d} such that Ω_{1} ⊂ Ω. If a sequence (f_{n}) is s-fundamental in Ω, then it is s-fundamental in Ω_{1}.

3^{∘} Let (f_{n}) be a sequence of smooth functions in Ω. If for every open set Ω_{0} ⊂ Ω the sequence (f_{n}|Ω_{0}) is s-fundamental in Ω_{0}, then (f_{n}) is s-fundamental in Ω.

Definition 2.2. Let (f_{n}) and (g_{n}) be s-fundamental sequences in an open set Ω. We write (f_{n}) ∼ (g_{n}) if for arbitrary K_{1} ⋐ Ω and K ⋐ K_{1}^{∘}, there exist an ultradifferential operator P ∈ P^{∗} and sequences (F_{n}), (G_{n}) of smooth functions on Ω such that

f_{n} = P (D)F_{n}, g_{n} = P (D)G_{n} on K (n ∈ ℕ),

suppF_{n}, suppG_{n} ⊂ K_{1} (n ∈ ℕ), and F_{n} →←^{C(K)} G_{n} as n → ∞,

where the symbol F_{n} →←^{C(K)} G_{n} means that (F_{n}) and (G_{n}) converge in C(K) to a common continuous function H on Ω.

Remark 2.2. It is clear that if (f_{n}) and (g_{n}) are s-fundamental sequences in Ω, then (f_{n}) ∼ (g_{n}) iff the sequence f_{1}, g_{1}, f_{2}, g_{2}, f_{3}, g_{3}, … is s-fundamental in Ω.

Obviously, the relation ∼ is reflexive and symmetric. To prove its transitivity we need some auxiliary statements.

Proposition 2.1. Fix K_{1} ⋐ Ω and K ⋐ K_{1}^{∘}. Assume that (f_{n}) satisfies Definition 2.1 in the Roumieu case, i.e., f_{n} = P_{(r_{p})} (D)F_{n} on K, suppF_{n} ⊂ K_{1} for

n ∈ ℕ_{0} and F_{n} −−−→^{C(K)} F_{0} as n → ∞ for a sequence (F_{n}) of smooth functions and a continuous function F_{0} on Ω. Then there are a P_{(r̃_{p} )} ∈ P^{{t}}, where (r̃_{p}) ∈ R with r_{p}/r̃_{p} ↓ 0 as p → ∞ and F^{−1} (P_{(r_{p})} /P_{(r̃_{p} )}) ∈ L^{1}(ℝ^{d}), smooth functions F̃_{n} (n ∈ ℕ) and a continuous function F̃_{0} on Ω such that

f_{n} = P_{(r̃_{p} )}(D)F̃_{n} on K (n ∈ ℕ), supp F̃_{n} ⊂ K_{1} (n ∈ ℕ_{0})

and F̃_{n} −−−→^{C(K)} F̃_{0} as n → ∞.

The same assertion holds in the Beurling case (with the corresponding notation).

Proof. By (1.7), for arbitrary β ∈ ℕ_{0}^{d} (in particular, for β = 0) we can find (r̃_{p}) ∈ R with r_{p}/r̃_{p} ↓ 0 as p → ∞ such that

μ_{β} \frac{P_{(r_{p})}}{P_{(r̃_{p} )}} ∈ L^{1}(ℝ^{d}) and F^{−1} ( μ_{β} \frac{P_{(r_{p})}}{P_{(r̃_{p} )}} ) ∈ L^{1}(ℝ^{d}).

Define

(2.4)  F̃_{n} := κ_{K} [F^{−1} ( \frac{P_{(r_{p})}}{P_{(r̃_{p} )}} ) ∗ (κ_{K} F_{n})], n ∈ ℕ_{0},

where κ_{K} is a smooth function in D^{{t}}(Ω) such that

(2.5)  κ_{K} (x) = \{ \begin{matrix}1, & if x ∈ K, \\ 0, & if x ∈ K_{1}^{c}. \end{matrix}

Then supp F̃_{n} ⊂ K_{1} for n ∈ ℕ_{0} and

P_{(r̃_{p} )}(D)F̃_{n}(x) = P_{(r̃_{p} )}(D)[F^{−1} ( \frac{P_{(r_{p})}}{P_{(r̃_{p} )}} ) ∗ (κ_{K} F_{n})](x)
= [F^{−1} (P_{(r_{p})} ) ∗ (κ_{K} F_{n})](x) = P_{(r_{p})} (D)(κ_{K} F_{n})(x) = f_{n}(x)

for x ∈ K and n ∈ ℕ. Since

F^{−1} (P_{(r_{p})}/P_{(r̃_{p} )}) ∈ L^{1}(ℝ^{d}) and κ_{K} F_{n} −−−→^{C(K)} κ_{K} F_{0} as n → ∞,

it follows that F̃_{n} −−−→^{C(K)} F̃_{0} as n → ∞. □

Proposition 2.2. If for arbitrary K_{1} ⋐ Ω and K ⋐ K_{1}^{∘} there are a P ∈ P^{∗}, smooth functions F_{n} (n ∈ ℕ) and a continuous function F_{0} on Ω with suppF_{n} ⊂ K_{1}

(n ∈ ℕ_{0}) such that F_{n} −−−→^{C(K)} F_{0} and P (D)F_{n}(x) → 0 for x ∈ K as n → ∞, then

F_{0} = 0 on Ω. In particular, if F is a smooth function on Ω and P (D)F (x) = 0 for x ∈ Ω, then F = 0 on Ω.

Proof. Fix K_{1} ⋐ Ω, K ⋐ K_{1}^{∘} and κ_{K} as in (2.5). By the assumption,
(2.6)  P (D)(κ_{K} F_{n})(x) → 0 (x ∈ K), κ_{K} F_{n} −−−→^{C(K)} κ_{K} F_{0} as n → ∞.

Since lim_{m→∞} [P (D) − P_{m}(D)](κ_{K} F_{n}) = 0 in ℝ^{d}, it follows from (2.6) that

lim_{n→∞} lim_{m→∞} P_{m}(ξ){ \atop κ_{K} F_{n}}(ξ) = lim_{n→∞} P (ξ){ \atop κ_{K} F_{n}}(ξ) = 0, ξ ∈ ℝ^{d}.

and

lim_{n→∞} P (ξ)[{ \atop κ_{K} F_{n}}(ξ) − { \atop κ_{K} F_{0}}(ξ)] = 0, ξ ∈ ℝ^{d}.

This implies P (ξ){ \atop κ_{K} F_{0}}(ξ) = 0 for ξ ∈ ℝ^{d}, so κ_{K} (x)F_{0}(x) = 0 for x ∈ K. Hence F_{0} = 0 in K and thus F_{0} = 0 in Ω, since K ⋐ Ω was arbitrarily chosen. The particular case is clear if we take F_{n} = F for n ∈ ℕ. □

Proposition 2.3. Fix K, K_{1} ⋐ Ω such that K ⋐ K_{1}^{∘}. Assume that, for sequences (r_{p}), ( r̃_{p}) ∈ R, smooth functions F_{n}, F̃_{n} on Ω (n ∈ ℕ) and continuous functions F_{0}, F̃_{0} on Ω, we have

f_{n} = P_{(r_{p})} (D)F_{n} on K (n ∈ ℕ), suppF_{n} ⊂ K_{1} (n ∈ ℕ_{0}),

F_{n} −−−→^{C(K)} F_{0} as n → ∞

and

f_{n}(x) = P_{(r̃_{p} )}(D) F̃_{n} on K (n ∈ ℕ), supp F̃_{n} ⊂ K_{1} (n ∈ ℕ_{0}),

F̃_{n} −−−→^{C(K)} F̃_{0} as n → ∞.

Then there are a sequence (r̄_{p}) ∈ R, r_{p}/r̄_{p} ↓ 0, r̃_{p}/r̄_{p} ↓ 0 as p → ∞ such that

(2.7)  F^{−1} ( \frac{P_{(r_{p})}}{P_{(r̄_{p} )}} ) , F^{−1} ( \frac{P_{(r̃_{p} )}}{P_{(r̄_{p} )}} ) ∈ L^{1}(ℝ^{d}),

and sequences (F_{n,1} ), (F_{n,2} ) of smooth functions and continuous functions F_{0,1}, F_{0,2} on Ω, with suppF_{n,1} , suppF_{n,2} ⊂ K_{1} (n ∈ ℕ_{0}), such that

(2.8)  f_{n} = P_{(r̄_{p} )}(D)F_{n,j} on K (n ∈ ℕ), F_{n,j} −−−→^{C(K)} F_{0,j} as n → ∞

for j = 1, 2. Moreover F_{0,1} = F_{0,2} in Ω.

The same assertion also holds in the Beurling case (with the corresponding notation).

Proof. The existence of (r̄_{p}) ∈ R satisfying (2.7) follows from (1.7). Define

F_{n,1} := κ_{K} [F^{−1} ( \frac{P_{(r_{p})}}{P_{(r̄_{p} )}} ) ∗ (κ_{K} F_{n})], F_{n,2} := κ_{K} [F^{−1} ( \frac{P_{(r̃_{p} )}}{P_{(r̄_{p} )}} ) ∗ (κ_{K} F̃_{n})],

where κ_{K} is a smooth function in D^{{t}}(Ω) which satisfies (2.5). Using Proposition 2.1, one can deduce (2.8). Finally, by Proposition 2.2, we conclude that F_{0,1} = F_{0,2} in Ω. □

Proposition 2.4. Relation ∼ introduced in Definition 2.2 is transitive.

Proof. We will prove the assertion only in the Roumieu case; the proof in the Beurling case is similar. Suppose that (f_{n}) ∼ (g_{n}) and (g_{n}) ∼ (h_{n}) and fix K, K_{1} ⋐ Ω so that K ⋐ K_{1}^{∘}. Now select K̃ ⋐ Ω such that K ⋐ K̃^{∘} and K̃ ⋐ K_{1}^{∘}.

By the assumption and Definition 2.2, there exist (r_{p}), (r̃_{p}) ∈ R and sequences (F_{n}), (G_{n}), (G̃_{n}), (H_{n}) of smooth functions on Ω such that

f_{n} = P_{(r_{p})} (D)F_{n}, g_{n} = P_{(r_{p})} (D)G_{n} on K (n ∈ ℕ),

suppF_{n}, suppG_{n} ⊂ K̃ (n ∈ ℕ), F_{n} →←^{C(K)} G_{n} as n → ∞

and

g_{n} = P_{(r̃_{p} )}(D)G̃_{n}, h_{n} = P_{(r̃_{p} )}(D)H_{n} on K̃ (n ∈ ℕ),

supp G̃_{n}, suppH_{n} ⊂ K_{1} (n ∈ ℕ), G̃_{n} −→⟵^{C(K̃ )} H_{n} as n → ∞.

In view of Proposition 2.3, there exist an appropriate (r̄_{p}) ∈ R and convergent sequences (F_{n,1} ), (G_{n,1} ), (G̃_{n,1} ), (H_{n,1} ) of smooth functions, all having supports contained in K_{1}, such that

f_{n} = P_{(r̄_{p} )}(D)F_{n,1} , g_{n} = P_{(r̄_{p} )}(D)G_{n,1} = P_{(r̄_{p} )}(D)G̃_{n,1} , h_{n} = P_{(r̄_{p} )}(D)H_{n,1}

on K. Moreover F_{n,1} →←^{C(K)} G_{n,1} and G̃_{n,1} →←^{C(K)} H_{n,1} as n → ∞.

If we put now

H̃_{n,1} (x) := G_{n,1} (x) − G̃_{n,1} (x) + H_{n,1} (x), x ∈ K,

then h_{n} = P_{(r̄_{p} )}(D)H̃_{n,1} in K and F_{n,1} →←^{C(K)} H̃_{n,1} as n → ∞, which means that

(f_{n}) ∼ (h_{n}). This completes the proof. □

2.1. Sequential ultradistributions.

Definition 2.3. Let (f_{n}) be a s-fundamental sequence (of type ∗) in an open set Ω ⊂ ℝ^{d}. The class of all s-fundamental sequences equivalent to (f_{n}) with respect to the relation ∼ is called a sequential ultradistributon or, shortly, s-ultradistribution (of type ∗) and denoted by f = [f_{n}]. The set of all s-ultradistributions (of type ∗) on Ω is denoted by U^{∗}(Ω).

Remark 2.3. 1^{∘} By Proposition 2.2, f = [f_{n}] = 0 on Ω for f ∈ U^{∗}(Ω) if for arbitrary K_{1} ⋐ Ω and K ⋐ K_{1}^{∘} there exist a sequence (F_{n}) of smooth functions on Ω and an ultradifferential operator P ∈ P^{∗} such that

f_{n} = P (D)F_{n} on K (n ∈ ℕ), suppF_{n} ⊂ K_{1} (n ∈ ℕ)

and F_{n} −−−→^{C(K)} 0 as n → ∞.

2^{∘} Let f ∈ U^{∗}(Ω). If f = 0 on Ω_{1} for every Ω_{1} ⊂ Ω, then f = 0 on Ω.

Definition 2.4. By the support of an s-ultradistribution f ∈ U^{∗}(Ω) we mean the complement of the union of all open sets where f = 0. We say that an sultradistribution f = [f_{n}] or a s-fundamental sequence (f_{n}) is compactly supported if there exists K ⋐ ℝ^{d} such that suppf_{n} ⊂ K for n ∈ ℕ. Then we write suppf ⊂ K or supp(f_{n}) ⊂ K. In this case there exist a sequence of smooth functions (F_{n}), a continuous function F_{0}, an ultradifferential operator P ∈ P^{∗} and K_{1} ⋐ Ω so that

f_{n}(x) = P (D)F_{n}(x) on ℝ^{d} (n ∈ ℕ), suppF_{n} ⊂ K_{1} (n ∈ ℕ_{0})

and F_{n} −−−→^{C(K)} F_{0} as n → ∞.

Example 2.1. Let F be a compactly supported continuous function in ℝ^{d}, (δ_{n}) be a delta sequence in D^{′∗}(ℝ^{d}) and let F_{n} := F ∗ δ_{n} for n ∈ ℕ. Then (F_{n}) is a s-fundamental sequence on ℝ^{d}.

Example 2.2. Let (f_{n}) be a s-fundamental sequence on Ω ⊆ ℝ^{d} and (K_{n}) be an increasing sequence of compact sets such that K_{n} ⊂ K_{n+1}^{∘} for n ∈ ℕ. Put Ω := ⋃ _{n∈ℕ} K_{n} and consider the open sets

Ω_{n} := {x ∈ Ω: d(x, ∂Ω) > 1/n}, Ω_{n,n} := {x ∈ Ω_{n} : d(x, ∂Ω_{n}) > 1/n}

and functions κ_{n} ∈ C_{0}^{∞} (Ω) such that

κ_{n}(x) = \{ \begin{matrix}1, & if x ∈ Ω_{n,n} , \\ 0, & if x ∈ Ω_{n}^{c}, \end{matrix}

for n ∈ ℕ. Then the sequence (f̃_{n} ), where

f̃_{n} (x) = \{ \begin{matrix}((κ_{n}f_{n}) ∗ δ_{n})(x), & if x ∈ Ω_{n}, \\ 0, & if x ∈ Ω_{n}^{c}, \end{matrix}

is s-fundamental on Ω and (f_{n}) ∼ (f̃_{n} ).

Example 2.3. Let f ∈ D′(Ω) for an open Ω ⊆ ℝ^{d}, let Ω_{n}, Ω_{n,n} and κ_{n} be as

in Example 2.2 and let P_{(r_{p})} ∈ P^{{t}}. Put

(2.9)  F_{n}(x) := \{ \begin{matrix}F^{−1} (f̂_{n} /P_{(r_{p})} )(x), & if x ∈ Ω, \\ 0, & if x ∈ Ω_{n}^{c}, \end{matrix}

for n ∈ ℕ, where f_{n} := (κ_{n}f) ∗ δ_{n} = P_{(r_{p})} (D)F_{n} and P_{(r_{p})} means in (2.9) the function corresponding to an ultra-differential operator P_{(r_{p})} . Since κ̂_{n}f , for every n ∈ ℕ, is bounded by a polynomial, it follows from (1.6) that (f_{n}) is a s-fundamental sequence of the Roumieu type on Ω, that is [f_{n}] ∈ U^{{t}}(Ω). In a similar way, we can also represent f as an element of U^{(t)}(Ω) by the use P_{r}(D) instead of P_{(r_{p})} (D).

Remark 2.4. If (f_{n}) is a s-fundamental sequence in the sense of Definition

2.1, then F_{n} −−−→^{C(K)} F_{0} as n → ∞ for every K ⋐ Ω. Examples 2.1 and 2.3 show that every s-fundamental sequence (f_{n}) for which (2.3) holds can be identified with the formal representation f = P (D)F_{0} on K, since from the general theory of ultradistributions we know that for arbitrary K_{1} ⋐ Ω and K ⋐ K_{1}^{∘} there exist P ∈ P^{∗} and F_{0} ∈ C(Ω) such that

f = P (D)F_{0} on K and suppF_{0} ⊂ K_{1}.

This will be justified by (3.3) and the last section.

2.2. Operations on s-ultradistributions. We start from the operations of addition and multiplication by a constant. Let f, g ∈ U^{∗}(Ω) and λ ∈ ℂ, where f = [f_{n}], g = [g_{n}] for some s-fundamental sequences (f_{n}), (g_{n}). Using Proposition 2.3, one can prove that (f_{n} + g_{n}) and (λf_{n}) are s-fundamental sequences on Ω, so we may define s-ultradistributions f + g := [f_{n} + g_{n}] and λf := [λf_{n}]. By Remark 2.2, the definitions are consistent. Consequently, U^{∗}(Ω) is a vector space.

Next consider the operation of differentiation. If f = [f_{n}] ∈ U^{{t}}(Ω), i.e., (f_{n}) is a s-fundamental sequence of the Roumieu type, and let β ∈ ℕ_{0}^{d}. We will show that the sequence (f_{n}^{(β)} ) is s-fundamental of the Roumieu type. For arbitrary K_{1} ⋐ Ω and K ⋐ K_{1}^{∘} take (F_{n}) and P_{(r_{p})} (D) according to Definition 2.1. Since f_{n}^{(β)} = P_{(r_{p})} (D)F_{n}^{(β)} on K, it follows from (1.7) as in the proof of Proposition 2.1 for β = 0 that there exist P_{(r̃_{p} )} ∈ P^{{t}}, a sequence (F̃_{n}) of smooth functions and a continuous function F̃_{0} on Ω defined by (cf. (2.4))

F̃_{n} := κ_{K} [F^{−1} ( μ_{β} \frac{P_{(r_{p})}}{P_{(r̃_{p} )}} ) ∗ (κ_{K} F_{n})], n ∈ ℕ_{0},

with supp F̃_{n} ⊂ K_{1} (n ∈ ℕ_{0}), such that P_{(r_{p})} (D)F_{n}^{(β)} = P_{(r̃_{p} )}(D)F̃_{n} on K and

F̃_{n} −−−→^{C(K)} F̃_{0} as n → ∞. Consequently, (f_{n}^{(β)} ) is s-fundamental and we define f^{(β)} = [f_{n}^{(β)} ]. By Remark 2.2, the definition is consistent and f^{(β)} ∈ U^{{t}}(Ω). An analogous assertion holds in the Beurling case.

Let us discuss now the operations of multiplication and convolution by a function from E^{∗}(Ω). We consider only the Roumieu case. The Beurling case is similar.

Fix ω ∈ E^{{t}}(Ω) and let f = [f_{n}] ∈ U^{{t}}(Ω). We will show that the sequence (ωf_{n}) is s-fundamental, so one can define ωf := [ωf_{n}] ∈ E^{{t}}(Ω) and the definition is consistent, by Remark 2.2. For every K ⋐ Ω there exist P_{(r_{p})} ∈ P^{{t}}, a sequence (F_{n}) of smooth functions and a continuous function F_{0} in Ω with suppF_{n} ⊂ K_{1}

(n ∈ ℕ_{0}) such that ωf_{n} = ωP_{(r_{p})} (D)F_{n} in K and F_{n} −−−→^{C(K)} F_{0}, n → ∞. We can

assume that ω is compactly supported multiplying it by a cut-of function equal to 1 on K. We have

ω̂(ξ) ⩽ Ce^{−h|ξ|^{1/t}} and |P_{(r_{p})} (ξ)F̂_{n}| ⩽ C_{1}e^{c(|ξ|)^{1/t}} , ξ ∈ ℝ^{d}

for some constants C > 0, C_{1} > 0, h > 0 and a subordinate function c, in view of (1.6). By (1.7), there exists a P_{(r̃_{p} )} ∈ P^{{t}}, where (r̃_{p}) ∈ R with r_{p}/r̃_{p} ↓ 0 as p → ∞, such that (ω̂ ∗ (P_{(r_{p})} F̂_{n}))/P_{(r̃_{p} )} ∈ L^{1}(ℝ^{d}). Now, defining

G_{n} := κ_{K} F^{−1} ( \frac{ω̂ ∗ (P_{(r_{p})} F̂_{n})}{P_{(r̃_{p} )}}) , n ∈ ℕ_{0},

we have ωf_{n} = P_{(r̃_{p} )}(D)G_{n} on K (n ∈ ℕ), suppG_{n} ⊂ K_{1} (n ∈ ℕ_{0}) and G_{n} −−−→^{C(K)}

G_{0} as n → ∞. Hence, (ωf_{n}) is a s-fundamental sequence on Ω.

Moreover, if f = [f_{n}] and ω ∈ D^{{t}}(Ω), then ω∗f = [ω∗f_{n}], since ω∗P (D)F_{n} = P (D)(ω ∗ F_{n}) on every compact set K ⊂ ℝ^{d} for n ∈ ℕ.

More generally, we have the following assertion: if (f_{n}) and (g_{n}) are s-fun-damental sequences on ℝ^{d} and supp(g_{n}) ⊂ K_{0} ⋐ ℝ^{d}, then (f_{n} ∗ g_{n}) is a sfundamental sequence on ℝ^{d}.

3. Sequences of s-ultradistributions

Definition 3.1. Let f^{m} ∈ U^{∗}(Ω) for m ∈ ℕ_{0}, i.e., f^{m} = [(f_{n}^{m})_{n}], where

(f_{n}^{m})_{n} means a s-fundamental sequence representing f^{m} for m ∈ ℕ_{0}. We say that the sequence (f^{m}) converges to f^{0} in U^{∗}(Ω) and write f^{m} ⟶^{s} f^{0} as m → ∞ or s-lim_{m→∞} f^{m} = f^{0} if for arbitrary K_{1} ⋐ Ω and K ⋐ K_{1}^{∘} there exist a P ∈ P^{∗}, smooth functions F_{n}^{m} on Ω (m ∈ ℕ_{0}, n ∈ ℕ) and continuous functions F^{m} on Ω (m ∈ ℕ_{0}), all supported by K_{1}, such that

f_{n}^{m} = P (D)F_{n}^{m} on K (n ∈ ℕ, m ∈ ℕ_{0});

F_{n}^{m} −−−→^{C(K)} F_{n}^{0} as m → ∞ uniformly in n ∈ ℕ;

F_{n}^{m} −−−→^{C(K)} F^{m} as n → ∞ (m ∈ ℕ_{0}) and F^{m} −−−→^{C(K)} F^{0} as m → ∞.

We know that the above assumptions imply that

lim_{m→∞} lim_{n→∞} F_{n}^{m} = lim_{n→∞} lim_{m→∞} F_{n}^{m} in C(K).

Theorem 3.1. If the limit s-lim_{m→∞} f^{m} exists, then it is unique.

Proof. Assume that f^{m} ⟶^{s} f and f^{m} ⟶^{s} g, where f = [f_{n}], g = [g_{n}] ∈ U^{∗}(Ω) and f^{m} = [(f_{n}^{m})_{n}] ∈ U^{∗}(Ω) for m ∈ ℕ. We will prove that f = g.

Fix arbitrary K_{1} ⋐ Ω and K ⋐ K_{1}^{∘}. According to Definition 3.1, there exist P, P̃ ∈ P^{∗}, smooth functions F_{n}^{m}, F_{n}, G_{n}^{m}, G_{n}, on Ω and continuous functions F^{m}, F, G^{m}, G on Ω (n, m ∈ ℕ), all supported by K_{1}, such that

f_{n}^{m} = P (D)F_{n}^{m}, f_{n} = P (D)F_{n} on K (n, m ∈ ℕ);

F_{n}^{m} −−−→^{C(K)} F_{n} as m → ∞ uniformly in n ∈ ℕ; F_{n} −−−→^{C(K)} F as n → ∞;

F_{n}^{m} −−−→^{C(K)} F^{m} as n → ∞ (m ∈ ℕ); F^{m} −−−→^{C(K)} F as m → ∞;

and, on the other hand,

f_{n}^{m} = P̃(D)G_{n}^{m}, g_{n} = P̃(D)G_{n} on K (n, m ∈ ℕ);

G_{n}^{m} −−−→^{C(K)} G_{n} as m → ∞ uniformly in n ∈ ℕ; G_{n} −−−→^{C(K)} G as n → ∞;

G_{n}^{m} −−−→^{C(K)} G^{m} as n → ∞ (m ∈ ℕ); G^{m} −−−→^{C(K)} G as m → ∞.

By Proposition 2.3, there exist an ultradifferential operator P̄ ∈ P^{∗}, smooth functions F̄_{n}^{m}, Ḡ_{n}^{m}, F̄_{n}, Ḡ_{n} and H_{n}^{m} := F̄_{n}^{m} − Ḡ_{n}^{m}, H_{n} := F̄_{n} − Ḡ_{n} on Ω as well as continuous functions F̄^{m}, Ḡ^{m}, F̄, Ḡ and H^{m} := F̄^{m} − Ḡ^{m}, H := F̄ − Ḡ on Ω (n, m ∈ ℕ), all supported by K_{1}, such that

0 = P̄(D)H_{n}^{m}, f_{n} − g_{n} = P̄(D)H_{n} on K (n, m ∈ ℕ);

H_{n}^{m} −−−→^{C(K)} H_{n} as m → ∞ uniformly in n ∈ ℕ; H_{n} −−−→^{C(K)} G as n → ∞;

H_{n}^{m} −−−→^{C(K)} H^{m} as n → ∞ (m ∈ ℕ) and H^{m} −−−→^{C(K)} H as m → ∞.

Hence H_{n}^{m} = 0 on K^{∘} for n, m ∈ ℕ, by Proposition 2.2. This implies that H = 0, i.e., F̄ = Ḡ on K^{∘}. Since K ⋐ Ω was fixed arbitrarily, we conclude that f = g on Ω. □

3.1. Action on test functions from D^{∗}(Ω) . Let f = [f_{n}] ∈ U^{∗}(Ω), where (f_{n}) is a s-fundamental sequence satisfying Definition 2.1, i.e., for arbitrary K, K_{1} ⋐ Ω with K ⋐ K_{1}^{∘} there exist P (D) ∈ P^{∗}, smooth functions F_{n} (n ∈ ℕ) and a continuous function F_{0} on Ω such that

f_{n} = P (D)F_{n} on K (n ∈ ℕ), suppF_{n} ⊂ K_{1} (n ∈ ℕ_{0}) (3.1)

and F_{n} −−−→^{C(K)} F_{0} as n → ∞.

By the action of f on the test functions from D^{∗}(Ω) we mean the mapping

(3.2)  D^{∗}(Ω) ∋ φ ↦ (f, φ)_{U^{∗} (Ω)} ∈ ℝ,

where

(3.3)  (f, φ)_{U^{∗} (Ω)} := lim_{n→∞} ∫_{K} f_{n}(x)φ(x)dx = ∫_{K} F_{0}(x)[P (−D)φ](x)dx.

If, beside (3.1), we have

f_{n} = P̃(D)F̃_{n} on K (n ∈ ℕ), supp F̃_{n} ⊂ K_{1} (n ∈ ℕ_{0})

and F̃_{n} −−−→^{C(K)} F̃_{0} as n → ∞

for some P̃(D) ∈ P^{∗}, smooth functions F̃_{n} (n ∈ ℕ) and a continuous function F̃_{0} on Ω, then

lim_{n→∞} ∫_{K} f_{n}(x)φ(x)dx = ∫_{K} F̃_{0}(x)[P̃(−D)φ](x)dx,

i.e., definition in (3.3) is consistent.

Clearly, 3.2 is a linear mapping. To prove that mapping (3.2) is sequentially continuous we need the following result from [12]: if φ_{n} → φ_{0} in D^{∗}(Ω), then P (D)φ_{n} → P (D)φ_{0} in D^{∗}(Ω) for every P (D) ∈ P^{∗}.

Theorem 3.2. (a) Let f ∈ U^{∗}(Ω) and φ_{m} → φ_{0} as m → ∞ in D^{∗}(Ω). Then

(f, φ_{m})_{U^{∗} (Ω)} → (f, φ_{0})_{U^{∗} (Ω)} as m → ∞.

(b) Let f^{m} → f^{0} as m → ∞ in U^{∗}(Ω). Then (f^{m}, φ)_{U^{∗} (Ω)} → (f^{0}, φ)_{U^{∗} (Ω)} as m → ∞ for every φ ∈ D^{∗}(Ω).

Proof. (a) Supose that f = [f_{n}], where (f_{n}) is a s-fundamental sequence, i.e., (3.1) holds for given K_{1} ⋐ Ω and K ⋐ K_{1}^{∘} and suitable P (D) ∈ P^{∗} and F_{n} (n ∈ ℕ_{0}). By (3.3), we have

lim_{m→∞} [(f, (φ_{m} − φ_{0})_{U^{∗} (Ω)} ] = lim_{m→∞} ∫_{K} F_{0}(x)[P (−D)(φ_{n} − φ_{0})](x)dx = 0.

(b) Let φ ∈ D_{K}^{∗}. Using the notation of Definition 3.1 we obtain

lim_{m→∞} (f^{m} − f^{0}, φ)_{U^{∗} (Ω)} = lim_{m→∞} lim_{n→∞} (f_{n}^{m} − f_{n}, φ)_{U^{∗} (Ω)}

= lim_{m→∞} lim_{n→∞} ∫_{K} (F_{n}^{m} − F_{n})(x)[P (−D)φ](x)dx.

Since F_{n}^{m} −−−→^{C(K)} F_{n} as m → ∞ uniformly in n ∈ ℕ and F_{n} −−−→^{C(K)} F^{0} as n → ∞, we

have

lim_{n→∞} lim_{m→∞} ∫_{K} (F_{n}^{m} − F_{n})(x)[P (−D)φ](x)dx = 0.  □

4. Tempered sequential ultradistributions

4.1. t-Tempered sequential ultradistributions. Recall that

H^{α} = ∏_{i=1}^{d} (−∂^{2}/∂x_{i}^{2} + x_{i}^{2})^{α_{i}} , α = (α_{1}, …, α_{d}) ∈ ℕ_{0}^{d}

and that the symbol P^{2∗} means either P^{(2t)} or P^{{2t}}. Let P ∈ P^{2∗}. We will use ultradifferential operators of the form P (H) = ∑ _{|α|=0}^{∞} a_{α} H^{α} of Beurling class (p!^{2t}) (resp. of Roumieu class {p!^{2t}}) such that

∃h > 0 ∃C > 0 (resp.∀h > 0 ∃C > 0) ∀α ∈ ℕ_{0}^{d} |a_{α} | ⩽ \frac{Ch^{|α|}}{(α!)^{2t}} ;

in the Roumieu case the given condition is equivalent to

∃(r_{p}) ∈ R ∃C > 0 ∀α ∈ ℕ_{0}^{d} |a_{α} | ⩽ \frac{C}{(α!)^{2t}R_{|α|}} ,

where R_{|α|} is defined in (1.1).

Definition 4.1. A sequence (f_{n}) of functions from L^{2}(ℝ^{d}) ∩ C^{∞} (ℝ^{d}) is called t-fundamental (of type ∗) in ℝ^{d} if there exist an ultradifferential operator P ∈ P^{2∗}, functions F_{n} ∈ L^{2}(ℝ^{d})∩C^{∞} (ℝ^{d}) of the form F_{n} = ∑ _{|α|=0}^{∞} c_{α,n} h_{α} with (c_{α,n} )_{α∈ℕ_{0}^{d}} ∈ l^{2} for n ∈ ℕ and a function F_{0} ∈ L^{2}(ℝ^{d}) such that

(4.1)  f_{n} = P (H)F_{n} on ℝ^{d} and F_{n} ⟶^{2} F_{0} as n → ∞,

where P (H)F_{n} in (4.1) is meant in L^{2}(ℝ^{d}) ∩ C^{∞} (ℝ^{d}) as follows

lim_{k→∞} P_{k}(H)F_{n} = lim_{k→∞} ∑_{|α|=0}^{k} a_{α} H^{α} F_{n} = ∑_{|α|=0}^{∞} c_{α,n} P (2α + 1)h_{α} , (n ∈ ℕ).

We will need later the following assertion which enables us to change in representations of t-fundamental sequences an L^{2}-convergent sequence with a sequence which converges both in L^{2}(ℝ^{d}) and uniformly on ℝ^{d}.

Lemma 4.1. Assume that F_{n} ∈ L^{2}(ℝ^{d}) ∩ C^{∞} (ℝ^{d}) for n ∈ ℕ_{0} and F_{n} ⟶^{2} F_{0} as n → ∞. Denote F̃_{n} := F^{−1} (G_{n}), where G_{n}(ξ) := (1+|ξ|)^{−d/2} F̂_{n}(ξ) for ξ ∈ ℝ^{d} and n ∈ ℕ_{0}. Then (F̃_{n}) is a bounded sequence of smooth functions such that F̃_{n} ⟶^{2} F̃_{0}

as n → ∞ and F̃_{n} −−−→^{C(ℝ^{d})} F̃_{0} as n → ∞.

Proof. It is clear that the sequence (F̃_{n}) is bounded and F̃_{n} ⟶^{2} F̃_{0}, due to the Parseval identity. By the Schwarz inequality, we have

∥F̃_{n} − F̃_{0}∥_{∞} = ∥F^{−1} (G_{n} − G_{0})∥_{∞} ⩽ ( ∫_{ℝ^{d}} (1 + |ξ|^{2})^{−d} dξ)^{1/2} ∥F̂_{n} − F̂_{0}∥_{2},

which proves the uniform convergence. □

Definition 4.2. Let (f_{n}) and (g_{n}) be t-fundamental sequences. We write (f_{n}) ∼_{1} (g_{n}) if there exist sequences (F_{n}), (G_{n}) of functions F_{n}, G_{n} ∈ L^{2}(ℝ^{d}) ∩ C^{∞} (ℝ^{d}) (n ∈ ℕ), both convergent in L^{2}(ℝ^{d}), and an operator P ∈ P^{2∗} such that

f_{n} = P (H)F_{n}, g_{n} = P (H)G_{n} on ℝ^{d} and F_{n} − G_{n} ⟶^{2} 0 as n → ∞.

The following two assertions will be used in the sequel.

Proposition 4.1. If there exist P ∈ P^{2∗}, functions F_{n} ∈ L^{2}(ℝ^{d}) ∩ C^{∞} (ℝ^{d}) for n ∈ ℕ and F_{0} ∈ L^{2}(ℝ^{d}) such that F_{n} = ∑ _{|α|=0}^{∞} c_{α,n} h_{α} with (c_{α,n} )_{α∈ℕ_{0}^{d}} ∈ l^{2} for

n ∈ ℕ_{0} and, moreover, F_{n} ⟶^{2} F_{0} and P (H)F_{n} ⟶^{2} 0 as n → ∞, then F_{0} = 0 on ℝ^{d}. In particular, if F ∈ L^{2}(ℝ^{d}) ∩ C^{∞} (ℝ^{d}) and P (H)F = 0 in L^{2}(ℝ^{d}), then F = 0 on ℝ^{d}.

Proof. By the assumption, we have (c_{α,n} )_{α∈ℕ_{0}^{d}} → (c_{α,0} )_{α∈ℕ_{0}^{d}} in l^{2} and

P (H)F_{n} = ∑_{|α|=0}^{∞} P (2α + 1)c_{α,n} h_{α} ⟶^{2} 0

as n → ∞. Consequently, (c_{α,n} )_{α∈ℕ_{0}^{d}} → 0 in l^{2} as n → ∞. Hence F_{0} = 0. The particular case is clear. □

Proposition 4.2. Assume that f_{n} = P_{(r_{p})} (H)F_{n} = P_{(r̃_{p} )}(H)F̃_{n} on ℝ^{d} (n ∈ ℕ)

and F_{n} ⟶^{2} F_{0}, F̃_{n} ⟶^{2} F̃_{0} as n → ∞ for some P_{(r_{p})} , P_{(r̃_{p} )} ∈ P^{{2t}} and functions

F_{n}, F̃_{n} ∈ L^{2}(ℝ^{d}) ∩ C^{∞} (ℝ^{d}) and F_{0}, F̃_{0} ∈ L^{2}(ℝ^{d}) of the form

F_{n} = ∑_{|α|=0}^{∞} c_{α,n} h_{α} , F̃_{n} = ∑_{|α|=0}^{∞} c̃_{α,n} h_{α} (n ∈ ℕ_{0}),

where (c_{α,n} )_{α∈ℕ_{0}^{d}} , (c̃_{α,n} )_{α∈ℕ_{0}^{d}} ∈ l^{2} for n ∈ ℕ_{0}. Then there are a P_{(r̄_{p} )} ∈ P^{{2t}}, where (r̄_{p}) ∈ R with r_{p}/r̄_{p} ↓ 0 and r̃_{p}/r̄_{p} ↓ 0 as p → ∞, such that

(4.2)  ( \frac{P_{(r_{p})} (2α + 1)}{P_{(r̄_{p} )}(2α + 1)} ) _{α∈ℕ_{0}^{d}} ∈ l^{∞} , ( \frac{P_{(r̃_{p} )}(2α + 1)}{P_{(r̄_{p} )}(2α + 1)} ) _{α∈ℕ_{0}^{d}} ∈ l^{∞} ,

and functions G_{n}, G̃_{n} ∈ L^{2}(ℝ^{d}) ∩ C^{∞} (ℝ^{d}) and G_{0}, G̃_{0} ∈ L^{2}(ℝ^{d}) of the form

G_{n} = ∑_{|α|=0}^{∞} \frac{P_{(r_{p})} (2α + 1)}{P_{(r̄_{p} )}(2α + 1)} c_{α,n} h_{α} , G̃_{n} = ∑_{|α|=0}^{∞} \frac{P_{(r̃_{p} )}(2α + 1)}{P_{(r̄_{p} )}(2α + 1)} c̃_{α,n} h_{α}

for n ∈ ℕ_{0}, satisfying the conditions

f_{n} = P_{(r̄_{p} )}(H)G_{n} = P_{(r̄_{p} )}(H)G̃_{n} on ℝ^{d} (n ∈ ℕ)

and

(4.3)  G_{n} ⟶^{2} G_{0}, G̃_{n} ⟶^{2} G̃_{0} as n → ∞.

Moreover, G_{n} = G̃_{n} on ℝ^{d} for n ∈ ℕ_{0}.

The same holds in the Beurling case with an appropriate notation.

Proof. The existence of P_{(r̄_{p} )}(H) follows from (1.8). It is clear that H^{β}h_{α} = (2α + 1)^{β}h_{α} for α, β ∈ ℕ_{0}^{d}. By 4.2, we have

G_{n} = ∑_{|α|=0}^{∞} \frac{P_{(r_{p})} (2α + 1)}{P_{(r̄_{p} )}(2α + 1)} c_{α,n} h_{α} = P_{(r_{p})} (H) ∑_{|α|=0}^{∞} \frac{c_{α,n} h_{α}}{P_{(r̄_{p} )}(2α + 1)}

for n ∈ ℕ_{0} and a similar representation holds for G̃_{n} (n ∈ ℕ_{0}). Hence

P_{(r̄_{p} )}(H)G_{n} = P_{(r̄_{p} )}(H)P_{(r_{p})} (H) ∑_{|α|=0}^{∞} \frac{c_{α,n} h_{α}}{P_{(r̄_{p} )}(2α + 1)}

= P_{(r_{p})} (H)P_{(r̄_{p} )}(H) ∑_{|α|=0}^{∞} \frac{c_{α,n} h_{α}}{P_{(r̄_{p} )}(2α + 1)} = P_{(r_{p})} (H)F_{n} = f_{n}

and, similarly, P_{(r̄_{p} )}(H)G̃_{n} = f_{n} on ℝ^{d} forn ∈ ℕ. We deduce from (4.2) that G_{n} and G̃_{n} are smooth L^{2} functions and (4.3) holds. By Proposition 4.1, we conclude that G_{n} = G̃_{n} on ℝ^{d} for n ∈ ℕ and, consequently, G_{0} = G̃_{0}. □

It is clear that the relation ∼_{1} is reflexive and symmetric. We shall prove that ∼_{1} is transitive.

Proposition 4.3. Relation ∼_{1} is transitive.

Proof. We prove the assertion in the Roumieu case; the proof in the Beurling case is similar. Let (f_{n}) ∼_{1} (g_{n}) and (g_{n}) ∼_{1} (h_{n}). Then there exist P_{(r_{p})} , P_{(r̃_{p} )} ∈ P^{{2t}} and sequences (F_{n}), (G_{n}), (G_{n}^{1}), (H_{n}) of functions in L^{2}(ℝ^{d}) ∩ C^{∞} (ℝ^{d}), all convergent in L^{2}(ℝ^{d}), such that

f_{n} = P_{(r_{p})} (H)F_{n}, g_{n} = P_{(r_{p})} (H)G_{n} = P_{(r̃_{p} )}(H)G_{n}^{1}, h_{n} = P_{(r̃_{p} )}(H)H_{n}

on ℝ^{d} for n ∈ ℕ and F_{n} − G_{n} ⟶^{2} 0, G_{n}^{1} − H_{n} ⟶^{2} 0 as n → ∞.

By Proposition 4.2, there is a P_{(r̄_{p} )} ∈ P^{{2t}}, where (r̄_{p}) ∈ R with r_{p}/r̄_{p} ↓ 0 and r̃_{p}/r̄_{p} ↓ 0 as p → ∞, and there are suitable functions F̄_{n}, Ḡ_{n}, Ḡ_{n}^{1}, H̄_{n} on ℝ^{d} such that f_{n} = P_{(r̄_{p} )}(H)F̄_{n}, g_{n} = P_{(r̄_{p} )}(H)Ḡ_{n} = P_{(r̄_{p} )}(H)Ḡ_{n}^{1}, h_{n} = P_{(r̄_{p} )}(H)H̄_{n} on ℝ^{d}

for n ∈ ℕ and F̄_{n} −Ḡ_{n} ⟶^{2} 0, Ḡ_{n}^{1} −H̄_{n} ⟶^{2} 0 as n → ∞. Putting Φ_{n} := Ḡ_{n} −Ḡ_{n}^{1} +H̄_{n},

we have h_{n} = P_{(r̄_{p} )}(H)Φ_{n} on ℝ^{d} and, moreover, F̄_{n} − Φ_{n} ⟶^{2} 0 as n → ∞. Hence (f_{n}) ∼_{1} (h_{n}), i.e., ∼_{1} is transitive. □

Definition 4.3. Let (f_{n}) be a t-fundamental sequence (of type ∗) in an open set Ω ⊂ ℝ^{d}. The class of all t-fundamental sequences equivalent to (f_{n}) with respect to the relation ∼_{1} is called a t-tempered sequential ultradistributon or, shortly, t-ultradistribution (of type ∗) and denoted by f = [f_{n}]. The set of all t-ultradistributions (of type ∗) in Ω ⊂ ℝ^{d} is denoted by T^{∗} = T^{∗}(ℝ^{d}).

Remark 4.1. As in the space U^{∗}(Ω) (see Section 2.2) we can consider appropriate operations in T^{∗}. But we do not go into details, remarking only that the operations of addition and multiplication by a constant are well defined in this set, i.e., T^{∗} is a vector space.

Example 4.1. Let F_{0} ∈ L^{2}(ℝ^{d}) and let (δ_{n}) be a delta-sequence in D^{∗}(ℝ^{d}). Define F_{n} := F_{0} ∗ δ_{n} for n ∈ ℕ. Then (F_{n}) is a sequence of smooth functions in L^{2}(ℝ^{d}) which is t-fundamental in both the Beurling and Roumieu cases.

Example 4.2. Let f = [f_{n}] ∈ T^{∗} be of the form f_{n} = P (H)F_{n}, where P ∈ P^{2∗}

and F_{n} ∈ L^{2}(ℝ^{d}) ∩ C^{∞} (ℝ^{d}) for n ∈ ℕ. Moreover, assume that F_{n} ⟶^{2} F_{0} as n → ∞ for some function F_{0} ∈ L^{2}(ℝ^{d}). Define f̃_{n} := P (H)(F_{n} ∗ δ_{n}) for n ∈ ℕ. Then (f̃_{n} ) is a t-fundamental sequence in ℝ^{d} and (f_{n}) ∼_{1} (f̃_{n} ).

Definition 4.4. Let f^{m} ∈ T^{∗} for m ∈ ℕ_{0}, i.e., f^{m} = [(f_{n}^{m})_{n}], where (f_{n}^{m})_{n}

means a t-fundamental sequence representing f^{m} for m ∈ ℕ_{0}. We say that the sequence (f^{m}) converges to f^{0} in T^{∗} and write f^{m} ⟶^{t} f^{0} as m → ∞ or tlim_{m→∞} f^{m} = f^{0} if there exist a P ∈ P^{2∗}, smooth functions F_{n}^{m} ∈ L^{2}(ℝ^{d}) (m ∈ ℕ_{0}, n ∈ ℕ) and functions F^{m} ∈ L^{2}(ℝ^{d}) (m ∈ ℕ_{0}) such that

f_{n}^{m} = P (H)F_{n}^{m} on ℝ^{d} (n ∈ ℕ, m ∈ ℕ_{0});

F_{n}^{m} ⟶^{2} F_{n}^{0} as m → ∞, uniformly in n ∈ ℕ;

F_{n}^{m} ⟶^{2} F^{m} as n → ∞ (m ∈ ℕ_{0}) and F^{m} ⟶^{2} F^{0} as m → ∞.

Assumptions in the definition imply that

lim_{m→∞} lim_{n→∞} F_{n}^{m} = lim_{n→∞} lim_{m→∞} F_{n}^{m} in L^{2}(ℝ^{d}).

36 MAKSIMOVIĆ, MINCHEVA-KAMIŃSKA, PILIPOVIĆ, AND SOKOLOSKI

Theorem 4.1. If the limit t-lim_{m→∞} f^{m} exists, then it is unique.

Proof. In the Roumieu case, let f^{m} ⟶^{s} f and f^{m} ⟶^{s} g, where f^{m} = [(f_{n}^{m})_{n}] ∈ T^{{t}} for m ∈ ℕ and f = [f_{n}], g = [g_{n}] ∈ T^{{t}}. We will show that f = g.

By Definition 4.4, there exist ultradifferential operators P_{(r_{p})} , P_{(r̃_{p} )} ∈ P^{{2t}} with (r_{p}), (r̃_{p}) ∈ R, smooth functions F_{n}^{m}, F_{n}, G_{n}^{m}, G_{n} ∈ L^{2}(ℝ^{d}) and functions F^{m}, F, G^{m}, G ∈ L^{2}(ℝ^{d}) (n, m ∈ ℕ) such that

f_{n}^{m} = P_{(r_{p})} (H)F_{n}^{m}, f_{n} = P_{(r_{p})} (H)F_{n} on ℝ^{d} (n, m ∈ ℕ);

F_{n}^{m} ⟶^{2} F_{n} as m → ∞ uniformly in n ∈ ℕ; F_{n} ⟶^{2} F as n → ∞;

F_{n}^{m} ⟶^{2} F^{m} as n → ∞ (m ∈ ℕ); F^{m} ⟶^{2} F as m → ∞

and, on the other hand, f_{n}^{m} = P_{(r̃_{p} )}(H)G_{n}^{m}, g_{n} = P_{(r̃_{p} )}(H)G_{n} on ℝ^{d} (n, m ∈ ℕ);

G_{n}^{m} ⟶^{2} G_{n} as m → ∞ uniformly in n ∈ ℕ; G_{n} ⟶^{2} G as n → ∞;

G_{n}^{m} ⟶^{2} G^{m} as n → ∞ (m ∈ ℕ); G^{m} ⟶^{2} G as m → ∞.

By Proposition 4.2, there exist a P_{(r̄_{p} )} ∈ P^{{2t}}, where (r̄_{p}) ∈ R, with r_{p}/r̄_{p} ↓ 0 and r̃_{p}/r̄_{p} ↓ 0 as p → ∞, smooth functions F̄_{n}^{m}, Ḡ_{n}^{m}, F̄_{n}, Ḡ_{n} in L^{2}(ℝ^{d}) and functions F̄^{m}, Ḡ^{m}, F̄, Ḡ in L^{2}(ℝ^{d}) for n, m ∈ ℕ such that the following conditions are satisfied

0 = P_{(r̄_{p} )}(H)Φ_{n}^{m}, f_{n} − g_{n} = P̄(D)(H)Φ_{n} on ℝ^{d} (n, m ∈ ℕ); Φ_{n}^{m} ⟶^{2} Φ_{n} as m → ∞ uniformly in n ∈ ℕ; Φ_{n} ⟶^{2} Φ as n → ∞;

Φ_{n}^{m} ⟶^{2} Φ^{m} as n → ∞ (m ∈ ℕ) and Φ^{m} ⟶^{2} Φ as m → ∞,

where Φ_{n}^{m} := F̄_{n}^{m} − Ḡ_{n}^{m}, Φ_{n} := F̄_{n} − Ḡ_{n}, Φ^{m} := F̄^{m} − Ḡ^{m} for n, m ∈ ℕ and Φ := F̄ − Ḡ. Hence P_{(r̄_{p} )}(H)Φ_{n} = lim_{m→∞} P_{(r̄_{p} )}(H)(Φ_{n}^{m} − Φ_{n}) = 0 on ℝ^{d} (n ∈ ℕ). As in Proposition 4.1, we conclude that F̄ = Ḡ on ℝ^{d} and thus f = g.

The proof in the Beurling case is similar. □ We need the following assertion: Lemma 4.2. Let φ be a function in D^{∗}(Ω) equal to 1 on B(0, 1/2) and let

φ_{m}(x) := φ(x/m) for x ∈ ℝ^{d} and m ∈ ℕ. If f = [f_{n}] is a t-ultradistribution, then

fφ_{m} ⟶^{t} f as m → ∞.

Proof. Let P ∈ P^{2∗} (F_{n}) be a sequence of functions corresponding to (f_{n}) according to Definition 4.1. Put F_{n}^{m} := φ_{m}F_{n} for m, n ∈ ℕ. Then f^{m} = P (H)φ_{m}F_{n} ∈ T^{∗}, because φ_{m}F_{n} ⟶^{2} φ_{m}F_{0} as n → ∞ for every fixed m. Since

∥F_{n}^{m} − F_{n}∥_{2}^{2} ⩽ ∫_{|x|> \frac{m}{2}} |φ^{2}(x/m) − 1| |F_{n}(x)|^{2}dx, n, m ∈ ℕ,

we have F_{n}^{m} ⟶^{2} F_{n} as m → ∞ uniformly in n. This completes the proof. □

Corollary 4.1. For every t-ultradistribution f^{0} there exists a sequence (f^{m}) of t-ultradistributions such that f^{m} ⟶^{t} f^{0} as m → ∞.

Remark 4.2. As in the case of s-fundamental sequences commented in Remark 2.4, every t-fundamental sequence (f_{n}) for which (4.1) holds can be identified with the formal representation f = P (H)F_{0}, since any other representation defines the same element of T^{∗} (see (4.15) of Subsection 4.3).

4.2. t̃-Tempered sequential ultradistributions. In this subsection we develop a sequential theory of tempered ultradistributions closely related to the sequential approach of sections 2 and 3.

Definition 4.5. A sequence (f_{n}) of smooth functions is called t̃-fundamental (of type ∗) in ℝ^{d} if there exist an ultradifferential operator P ∈ P^{∗}, a function P_{1} ∈ P_{u}^{∗} and functions F_{0} ∈ L^{2}(ℝ^{d}) and F_{n} ∈ L^{2}(ℝ^{d}) ∩ C^{∞} (ℝ^{d}) for n ∈ ℕ such that

(4.4)  f_{n} = P (D)(P_{1}F_{n}) on ℝ^{d} and F_{n} ⟶^{2} F_{0} as n → ∞.

The action of P (D) on P_{1}F_{n} is understood as in Section 2; it is the limit of ∑ _{|α|=0}^{m} a_{α} D^{α} (P_{1}F_{n})(x) as m → ∞ for x ∈ ℝ^{d}. The following assertion will enable us to transfer one form of a fundamental sequence into another one.

For a given h > 0 and a subordinate function c denote for simplicity

E_{±h} (u) := e^{±hu^{1/t}} and E^{±c} (u) := e^{±c(u)^{1/t}} for u ⩾ 0.

Lemma 4.3. Let P_{1}, F_{n} and F_{0} be as in (4.4). For P_{1} assume (1.4) in the Beurling case (resp. (1.6) in the Roumieu case) in the form

|P_{1}(x)| ⩽ CE_{h_{1}} (|x|) (resp. |P_{1}(x)| ⩽ CE^{c_{1}} (|x|)), x ∈ ℝ^{d},

where h_{1} > 0 is a constant (resp. c_{1} is a subordinate function). Then

(a) for a given h > 0 (resp. for a given subordinate function c) there exists r > 0 (resp. (r_{p}) ∈ R) such that

|[E_{h}F^{−1} (P_{1}/P_{r})](x)| < ∞ (resp. |[E^{c}F^{−1} (P_{1}/P_{(r_{p})} )](x)| < ∞)

for every x ∈ ℝ^{d}.

(b) there exists r > 0 (resp. (r_{p}) ∈ R) such that

E_{−2h_{1}} [(P_{1}F_{n} − P_{1}F_{0}) ∗ F^{−1} (P_{1}/P_{r})] ⟶^{2} 0

(resp. E^{−c} [(P_{1}F_{n} − P_{1}F_{0}) ∗ F^{−1} (P_{1}/P_{(r_{p})} )] ⟶^{2} 0)

as n → ∞, where c is a subordinate function such that 2c_{1}^{1/t} ⩽ c^{1/t}.

Proof. We will prove the assertions only in the Roumieu case; the proof in the Beurling case is similar.

To prove (a) choose (r_{p}^{0}) ∈ R such that E^{c}(|x|) ⩽ P_{(r_{p}^{0} )} (x) for x ∈ ℝ^{d}. The proof will be completed if we show that there exists (r_{p}) ∈ R such that P_{(r_{p}^{0} )} (D)(P_{1}/P_{(r_{p})} ) ∈ L^{1}(ℝ^{d}), since then the function P_{(r_{p}^{0} )} F^{−1} (P_{1}/P_{(r_{p})} ) belongs to L^{∞} (ℝ^{d}). For all x ∈ ℝ^{d}, we have

(4.5) P_{(r_{p}^{0} )} (D)(P_{1}/P_{(r_{p})} )(x) = ∑_{|α|=0}^{∞} a_{α} ∑_{0⩽γ⩽α} \binom{α}{γ} [D^{α−γ} P_{1}D^{γ}(1/P_{(r_{p})} )](x),

where

(4.6)  |a_{α} | ⩽ \frac{C}{(α!)^{t}R_{|α|}^{0}} ( R_{|α|}^{0} := ∏_{i⩽|α|} r_{i}^{0}) ,  α ∈ ℕ_{0}^{d}

for some C > 0. By Lemma 1.2, there exist a subordinate function c̃_{1} (related to P_{1}) and a subordinate function c̃_{(r_{p})} (suitably chosen to fulfill the inequality c̃_{1}(|x|) ⩽ c̃_{(r_{p})} (|x|) for x ∈ ℝ^{d}) such that

(4.7)  |D^{α−γ} P_{1}(x)| ⩽ \frac{C(α − γ)!}{ε^{|α−γ|}} Ec̃_{1} (|x|),  x ∈ ℝ^{d}, α, γ ∈ ℕ_{0}^{d}, γ ⩽ α;
(4.8)  |D^{γ}(1/P_{(r_{p})} )(x)| ⩽ \frac{Cγ!}{ε^{|γ|}}E^{−c̃_{(rp )}} ,  x ∈ ℝ^{d}, γ ∈ ℕ_{0}^{d},

By (4.5), (4.6), (4.7) and (4.8), we get

|P_{(r_{p}^{0} )} (D)(P_{1}/P_{(r_{p})} )(x)| ⩽ C^{2}( ∑_{|α|=0}^{∞} \frac{(2/ε)^{α}}{(α!)^{t−1} R_{|α|}^{0}} ) Ec̃_{1} (|x|)E^{−c̃_{(rp )}} (|x|).

This proves (a), since the sum on the right-hand side is finite.

To prove (b) note that, by Lemma 4.1, we can assume that (F_{n}) is a bounded sequence of smooth functions in L^{2}(ℝ^{d}). By the assumption and (1.5), there exists a suitable subordinate function c_{0} (depending on (r_{p}) ∈ R) satisfying

(4.9)  C_{0} := ∫_{ℝ^{d}} E^{4c_{1}} (|s|)E^{−2c_{0}} (|s|)ds < ∞

and there is a constant C > 0 such that

|[(P_{1}F_{n} − P_{1}F_{0})∗F^{−1} (P_{1}/P_{(r_{p})} )](x)|
⩽ C∥F_{n} − F_{0}∥_{2}( ∫_{ℝ^{d}} E^{2c_{1}} (|x − s|)E^{−2c_{0}} (|s|)ds) ^{1/2}
⩽ CC_{0}∥F_{n} − F_{0}∥_{2}E^{2c_{1}} (|x|)

for all x ∈ ℝ^{d}, due to (4.9). Hence assertion (b) easily follows. □

Definition 4.6. Let (f_{n}) and (g_{n}) be t̃-fundamental sequences. We write (f_{n}) ∼_{2} (g_{n}), if there exist sequences (F_{n}), (G_{n}) of functions F_{n}, G_{n} ∈ L^{2}(ℝ^{d}) ∩ C^{∞} (ℝ^{d}) (n ∈ ℕ), both convergent in L^{2}(ℝ^{d}), an operator P ∈ P^{∗} and a function P_{1} ∈ P_{u}^{∗} such that

f_{n} = P (D)(P_{1}F_{n}), g_{n} = P (D)(P_{1}G_{n}) on ℝ^{d}

and F_{n} − G_{n} ⟶^{2} 0 as n → ∞.

Proposition 4.4. If the assumptions of Definition 4.6 are satisfied for (f_{n}) and if P (D)(P_{1}F_{n}) ⟶^{2} 0, then F_{n} ⟶^{2} 0 as n → ∞. In particular, if P (D)(P_{1}F) = 0, then F = 0.

Proof. Using the Fourier transform, we have P { \atop P_{1}F_{n}} ⟶^{2} 0. The same is true for { \atop P_{1}F_{n}}, then for P_{1}F_{n} and, finally, for F_{n}. The particular case is clear. □

The key assertion in this subsection is related to the change of representative of some t̃-ultradistribution (see Definition 4.7 below). We consider the Roumieu case.

Assume that P_{(r_{p})} , P_{(r̃_{p} )} ∈ P^{{t}}, P_{(r_{p})}^{1} , P_{(r̃_{p} )}^{2} ∈ P_{u}^{{t}} and (F_{n}), (F̃_{n}) are sequences of functions in L^{2}(ℝ^{d}) ∩ C^{∞} (ℝ^{d}), both convergent in L^{2}(ℝ^{d}), satisfying Definition 4.5, i.e.,

(4.10)  f_{n} = P_{(r_{p})} (D)(P_{(r_{p})}^{1} F_{n}) = P_{(r̃_{p} )}(D)(P_{(r̃_{p} )}^{2} F̃_{n}) on ℝ^{d} (n ∈ ℕ),

so that

(4.11)  max{|P_{(r_{p})} (x)|, |P_{(r̃_{p} )}(x)|, |P_{(r_{p})}^{1} (x)|, |P_{(r̃_{p} )}^{2}(x)|} ⩽ Ce^{c(|x|)^{1/t}}

for a suitable subordinate function c and x ∈ ℝ^{d}. We may assume, without loosing generality, that P_{(r_{p})}^{1} = P_{(r̃_{p} )}^{2} = P_{(r̄_{p} )}. Actually one can use in (4.10) instead of P_{(r_{p})}^{1} (x)F_{n}(x) and P_{(r̃_{p} )}^{2}(x)F̃_{n}(x), the following expressions

P_{(r̄_{p} )}(x)\frac{P_{(r_{p})}^{1} (x)F_{n}(x)}{P_{(r̄_{p} )}(x)} and P_{(r̄_{p} )}(x)\frac{P_{(r̃_{p} )}^{2}(x)F̃_{n}(x)}{P_{(r̄_{p} )}(x)}, x ∈ ℝ^{d},

respectively, where the sequence (r̄_{p}) ∈ R is increasing slowly enough to guarantee L^{2}-convergence of the sequences

( \frac{P_{(r_{p})}^{1} (x)F_{n}(x)}{P_{(r̄_{p} )}(x)}) , ( \frac{P_{(r̃_{p} )}^{2}(x)F̃_{n}(x)}{P_{(r̄_{p} )}(x)}) .

The above remarks concern the following proposition.

Proposition 4.5. Assume that P_{(r_{p})} , P_{(r̃_{p} )} ∈ P^{{t}}, P_{(r̄_{p} )} ∈ P_{u}^{{t}} and functions F_{n}, F̃_{n} ∈ L^{2}(ℝ^{d}) ∩ C^{∞} (ℝ^{d}) for n ∈ ℕ, and F_{0}, F̃_{0} ∈ L^{2}(ℝ^{d}) satisfy Definition 4.5,

i.e., f_{n} = P_{(r_{p})} (D)(P_{(r̄_{p} )}F_{n}) = P_{(r̃_{p} )}(D)(P_{(r̄_{p} )} F̃_{n}) on ℝ^{d} (n ∈ ℕ) and F_{n} ⟶^{2} F_{0},

F̃_{n} ⟶^{2} F̃_{0} as n → ∞, so that (4.11) holds. Then there exist P_{(r_{p}^{0} )} ∈ P^{{t}} with (r_{p}^{0}) ∈ R and functions F̄_{n} ∈ L^{2}(ℝ^{d}) ∩ C^{∞} (ℝ^{d}) for n ∈ ℕ and F̄_{0} ∈ L^{2}(ℝ^{d}) such that

f_{n} = P_{(r_{p}^{0} )} (D)(P_{(r̄_{p} )} F̄_{n}) on ℝ^{d} and F̄_{n} ⟶^{2} F̄_{0} as n → ∞.

Proof. We know that there exists (r_{p}^{0}) ∈ R such that if we put

G_{n} := F^{−1} (F_{n}/P_{(r_{p}^{0} )} ) and G̃_{n} := F^{−1} (F̃_{n}/P_{(r_{p}^{0} )} ) on ℝ^{d} (n ∈ ℕ_{0}),

then f_{n} = P_{(r_{p}^{0} )} (D)(P_{(r̄_{p} )}G_{n}) = P_{(r_{p}^{0} )} (D)(P_{(r̄_{p} )} G̃_{n}) on ℝ^{d} (n ∈ ℕ) and, more-

over, G_{n} ⟶^{2} G_{0} and G̃_{n} ⟶^{2} G̃_{0} as n → ∞.

By Proposition 4.4, G_{n} = G̃_{n} on ℝ^{d} for n ∈ ℕ_{0}, so the assertion follows for the functions F̄_{n} := G_{n} = G̃_{n} (n ∈ ℕ_{0}). □

To prove that ∼_{2}, introduced in Definition 4.6, is an equivalence relation, it suffices to show that it is transitive.

Proposition 4.6. The relation ∼_{2} is transitive.

We omit the proof of the proposition, because it is similar to the proofs of Propositions 2.4 and 4.3. One has to use appropriate representations as it was demonstrated in those proofs.

Definition 4.7. Let (f_{n}) be a t̃-fundamental sequence (of type ∗) in ℝ^{d}. The class of all t̃-fundamental sequences equivalent to (f_{n}) with respect to the relation ∼_{2} is called a t̃-tempered sequential ultradistributon or, shortly, t̃-ultradistribution (of type ∗) and denoted by f = [f_{n}]. The set of all t̃-ultradistributions (of type ∗) in ℝ^{d} is denoted by T̃^{∗} = T̃^{∗}(ℝ^{d}).

We give the convergence structure in T̃^{∗}.

Definition 4.8. Let f^{m} ∈ T̃^{∗} for m ∈ ℕ_{0}, i.e., f^{m} = [(f_{n}^{m})_{n}], where (f_{n}^{m})_{n} means a t̃-fundamental sequence representing f^{m} for m ∈ ℕ_{0}. We say that the sequence (f^{m}) converges to f^{0} in T̃^{∗} and write f^{m} ⟶^{t̃} f^{0} as m → ∞ or t̃- lim_{m→∞} f^{m} = f^{0} if there exist P ∈ P^{∗} and P_{1} ∈ P_{u}^{∗}, smooth functions F_{n}^{m} ∈ L^{2}(ℝ^{d}) (m ∈ ℕ_{0}, n ∈ ℕ) and functions F^{m} ∈ L^{2}(ℝ^{d}) (m ∈ ℕ_{0}) such that

f_{n}^{m} = P (D)(P_{1}F_{n}^{m}) on ℝ^{d} (n ∈ ℕ, m ∈ ℕ_{0});

F_{n}^{m} ⟶^{2} F_{n}^{0} as m → ∞, uniformly in n ∈ ℕ;

F_{n}^{m} ⟶^{2} F^{m} as n → ∞ (m ∈ ℕ_{0}) and F^{m} ⟶^{2} F^{0} as m → ∞.

The assumptions of the definition imply that

lim_{m→∞} lim_{n→∞} F_{n}^{m} = lim_{n→∞} lim_{m→∞} F_{n}^{m} in L^{2}(ℝ^{d}).

By suitable modifications of the proofs of Proposition 4.4 and Theorem 4.1, one can prove the following theorem

Theorem 4.2. If the limit t̃-lim_{n→∞} f^{n} exists, then it is unique.

By [25] we know that every f ∈ S^{′∗}(ℝ^{d}) can be identified with the formal representation f = P (D)(P_{1}F_{0}), where P ∈ P^{∗}, P_{1} ∈ P_{u}^{∗} and F_{0} ∈ L^{2}(ℝ^{d}) is a function of the form F_{0} = ∑ _{|α|=0}^{∞} c_{α,0} h_{α} ∈ L^{2}(ℝ^{d}) with (c_{α,0} )_{α∈ℕ_{0}^{d}} ∈ l^{2}.

Actually, we need the following assertion:

Lemma 4.4. An f is an element of the space S^{′∗}(ℝ^{d}) if and only if

(4.12)  f = P (D)(P_{1}F_{0})

for some P ∈ P^{∗}, P_{1} ∈ P_{u}^{∗} and F_{0} ∈ L^{2}(ℝ^{d}) of the form

(4.13)  F_{0} = ∑_{|α|=0}^{∞} c_{α,0} h_{α} with (c_{α,0} ) ∈ l^{2},

that is

(f, φ)_{S^{′∗}} = ∫_{ℝ^{d}} F_{0}(x)P_{1}(x)P (−D)φ(x)dx, φ ∈ S^{∗}(ℝ^{d}).

The proof is a consequence of the well known representation theorem based on the Hahn–Banach theorem and assertions (a) and (b) of Lemma 1.3.

We may formulate the above assertion in the form of the proposition which will be needed in Section 5.

Proposition 4.7. Let f ∈ S^{′∗}(ℝ^{d}) be of the form (4.12)–(4.13). Then the sequence (f_{n}), where f_{n} := (P (D)(P_{1}F_{n}) and F_{n} := ∑ _{|α|=0}^{n} c_{α,0} h_{α} for n ∈ ℕ, is t̃-fundamental and determines f̃ = [f_{n}] ∈ T̃^{∗}.

Conversely, if f̃ = [f_{n}] ∈ T̃^{∗}, where (f_{n}) is a t̃-fundamental sequence of the form (4.4) in Definition 4.5, then the corresponding f = P (D)(P_{1}F_{0}), where F_{0} is the L^{2}-limit of the sequence (F_{n}), is an element of S^{′∗}(ℝ^{d}).

The above correspondence between S^{′∗}(ℝ^{d}) and T̃^{∗} defines a linear bijection between these spaces.

4.3. Tempered ultradistributions as functionals. Let f = [f_{n}] be an element of T^{∗}, where the functions f_{n} are of the form f_{n} = P (H)F_{n} on ℝ^{d} with P ∈ P^{2∗} and F_{n} ∈ L^{2}(ℝ^{d}) for n ∈ ℕ such that F_{n} ⟶^{2} F_{0} as n → ∞ for some F_{0} ∈ L^{2}(ℝ^{d}).

We define the action of f = [f_{n}] on S^{∗}(ℝ^{d}) as the mapping

(4.14)  S^{∗}(ℝ^{d}) ∋ φ ↦ f(φ) := (f, φ)_{T ^{∗}} ∈ ℝ,

where

(4.15)  (f, φ)_{T ^{∗}} := lim_{n→∞} (f_{n}, φ) = ∫_{ℝ^{d}}(F_{0}P (H)φ)(x)dx = (F_{0}, P (H)φ)_{L^{2}} .

As in the case of s-ultradistributions, if there is another representation of f_{n} in the form f_{n} = P̃(H)F̃_{n} on ℝ^{d} for n ∈ ℕ, where P̃ ∈ P^{2∗} and F̃_{n} ⟶^{2} F̃_{0} as n → ∞, then we have

lim_{n→∞} (f_{n}, φ) = ∫_{ℝ^{d}}(F̃_{0}P̃(H)φ)(x)dx = ∫_{ℝ^{d}}(F_{0}P (H)φ)(x)dx,

i.e., the definition of (f, φ)_{T ^{∗}} in (4.15) is consistent. Lemma 1.3 implies that the mapping in (4.14) is linear.

We prove now the same for f = [f_{n}] ∈ T̃^{∗}, where the functions f_{n} are of the

form f_{n} = P (D)(P_{1}F_{n}) for P ∈ P^{∗}, P_{1} ∈ P_{u}^{∗}, F_{n} ∈ L^{2}(ℝ^{d}) (n ∈ ℕ) and F_{n} ⟶^{2} F

as n → ∞ for some F ∈ L^{2}(ℝ^{d}).

The action of f = [f_{n}] ∈ T̃^{∗} on φ ∈ S^{∗}(ℝ^{d}) is defined as the mapping

(4.16)  S^{∗}(ℝ^{d}) ∋ φ ↦ (f, φ)T̃ ^{∗} ∈ ℝ,

where

(4.17)  (f, φ)T̃ ^{∗} := lim_{n→∞} (F_{n}, P_{1}P (−D)φ) = ∫_{ℝ^{d}}(F_{0}P_{1}P (−D)φ)(x)dx.

Note that the limit in (4.16) exists, because P_{1}P (−D)φ ∈ S^{∗}(ℝ^{d}), in view of part (b) of Lemma 1.3.

If f_{n} is represented in another form: f_{n} = P̃(D)(P̃_{1}F̃_{n}) on ℝ^{d} with P̃(D) ∈ P^{∗},

P̃_{1} ∈ P_{u}^{∗}, F̃_{n} ∈ L^{2}(ℝ^{d}) for n ∈ ℕ and F̃_{n} ⟶^{2} F̃_{0} as n → ∞ for some F̃_{0} ∈ L^{2}(ℝ^{d}),

then

lim_{n→∞} (F̃_{n}, P̃_{1}P̃(−D)φ) = lim_{n→∞} (F_{n}, P_{1}P (−D)φ), φ ∈ S^{∗}(ℝ^{d}),

i.e., the definition in (4.17) is consistent. The linearity of the mapping (4.16) follows by Lemma 1.3.

The continuity of the mappings (4.14) and (4.16) follows from the following assertion.

Proposition 4.8. Let f ∈ T^{∗} (resp. f ∈ T̃^{∗}) and let φ_{n} ∈ S^{∗}(ℝ^{d}) for n ∈ ℕ_{0}

be functions such that φ_{n} ⟶^{S^{∗}} φ_{0} as n → ∞. Then

(f, φ_{n})_{T ^{∗}} → (f, φ_{0})_{T ^{∗}} (resp. (f, φ_{n})T̃ ^{∗} → (f, φ_{0})T̃ ^{∗}) as n → ∞.

Proof. If f ∈ T^{∗}, then we have (f, φ_{n})_{T ^{∗}} = (F_{0}, P (H)φ_{n})_{L^{2}} for n ∈ ℕ_{0}, according to (4.1) and (4.15). Hence, by the Schwarz inequality, we get

|(f, φ_{n})_{T ^{∗}} − (f, φ_{0})_{T ^{∗}}| = |(F_{0}, P (H)(φ_{n} − φ_{0}))_{L^{2}} |
⩽ ∥F_{0}∥_{2} · ∥P (H)(φ_{n} − φ_{0})∥_{2}

and the assertion follows, in view of part (c) of Lemma 1.3. The proof in the case f ∈ T̃^{∗} is analogous. □

The above result can be generalized in the following way:

Proposition 4.9. Let f^{m} ∈ T^{∗} (resp. f^{m} ∈ T̃^{∗}) and φ_{m} ∈ S^{∗}(ℝ^{d}) for

m ∈ ℕ_{0}. If f^{m} ⟶^{t} f^{0} (resp. f^{m} ⟶^{t̃} f^{0}) and φ_{m} ⟶^{S^{∗}} φ_{0} as m → ∞, then

(f^{m}, φ_{m})_{T ^{∗}} → (f^{0}, φ_{0})_{T ^{∗}} (resp. (f^{m}, φ_{m})T̃ ^{∗} → (f^{0}, φ_{0})T̃ ^{∗}) as m → ∞.

Proof. We give the proof only in the T^{∗} case. By Definition 4.4, we have

lim_{m→∞} (f^{m}, φ_{m})_{T ^{∗}} = lim_{m→∞} lim_{n→∞} (F_{n}^{m}, P (H)φ_{m})_{L^{2}} ,

where F_{n}^{m} ⟶^{2} F^{m} as n → ∞ for every m ∈ ℕ and F^{m} ⟶^{2} F^{0} as m → ∞. Hence, using the Schwarz inequality, we have

|(F^{m}, P (H)φ_{m})_{L^{2}} − (F^{0}, P (H)φ_{0})_{L^{2}} |
⩽ |(F^{m}, P (H)(φ_{m} − φ_{0}))_{L^{2}} | + |(F^{m} − F^{0}, P (H)φ_{0})_{L^{2}} |

⩽ ∥F^{m}∥_{2} · ∥P (H)(φ_{m} − φ_{0})∥_{2} + ∥F^{m} − F^{0}∥_{2} · ∥P (H)φ_{0}∥_{2}.

It suffices now to use again part (c) of Lemma 1.3 to complete the proof. □

5. Relations between spaces of tempered ultradistributions

In connection with the spaces S^{∗}(ℝ^{d}) and S^{′∗}(ℝ^{d}), where ∗ = (t) in the Beurling case (resp. ∗ = {t} in the Roumieu case) consider the following spaces of numerical sequences

s^{∗} = { (a_{α})_{α∈ℕ_{0}^{d}} : ∀h > 0 (resp. ∃h > 0) ∑_{|α|=0}^{∞} |a_{α} |^{2}e^{h|α|^{1/(2t)}} < ∞}

s^{′∗} = { (b_{α})_{α∈ℕ_{0}^{d}} : ∃k > 0 (resp. ∀k > 0) ∑_{|α|=0}^{∞} |b_{α} |^{2}e^{−k|α|^{1/(2t)}} < ∞} .

By the Köthe theory of echelon and co-echelon spaces (see [11]) the spaces s^{∗} and s^{′∗} with their natural convergence structure constitute a dual pair.

It is well known that the mapping

(5.1)  s^{∗} ∋ (a_{α})_{α∈ℕ_{0}^{d}} ↦ ∑_{|α|=0}^{∞} a_{α} h_{α} ∈ S^{∗}(ℝ^{d})

is a bijective isomorphism between the spaces s^{∗} and S^{∗}(ℝ^{d}).

On the other hand, to every f ∈ T^{∗} we can assign a unique (b_{α})_{α∈ℕ_{0}^{d}} ∈ s^{′∗}. In fact, assume that f = [f_{n}] ∈ T^{∗} satisfies (4.1), i.e.,

(5.2)  f_{n} = P (H)F_{n} on ℝ^{d} (n ∈ ℕ) and F_{n} ⟶^{2} F_{0} as n → ∞,

where

(5.3)  F_{n} = ∑_{|α|=0}^{∞} c_{α,n} h_{α} with (c_{α,n} )_{α∈ℕ_{0}^{d}} ∈ l^{2}  (n ∈ ℕ_{0}).

Morepover, let φ ∈ S^{∗}(ℝ^{d}) be of the form

(5.4)  φ = ∑_{|α|=0}^{∞} r_{α} h_{α} .

We know that (r_{α})_{α∈ℕ_{0}^{d}} ∈ s^{∗} (see 5.1).

By (4.15), (5.2), (5.3) and (5.4), we have

(5.5)  (f, φ)_{T ^{∗}} = lim_{n→∞} ∑_{|α|=0}^{∞} P (2α + 1)c_{α,n} r_{α} = ∑_{|α|=0}^{∞} b_{α} r_{α} ,

where

(5.6)  b_{α} := P (2α + 1)c_{α,0} ,  α ∈ ℕ_{0}^{d},

because c_{α,n} → c_{α,0} in l^{2} as n → ∞ for α ∈ ℕ_{0}^{d}.

Assign to f = [f_{n}] ∈ T^{∗} the sequence (b_{α})_{α∈ℕ_{0}^{d}} ∈ s^{′∗} defined in (5.6) which does not depend on a representation (f_{n}) of f, by Proposition 4.2.

The described mapping is a bijective isomorphism between T^{∗} and s^{′∗}.

Let us recall the following well known assertion (see [5]- [10]):

Proposition 5.1. The bijective isomorphism (5.1) induces the isomorphism of s^{′∗} onto S^{′∗}(ℝ^{d}) given by

s^{′∗} ∋ (b_{α})_{α∈ℕ_{0}^{d}} ↦ ∑_{|α|=0}^{∞} b_{α} h_{α} ∈ S^{′∗}(ℝ^{d}).

According to the preceding remarks and Proposition 5.1 to each f = [f_{n}] ∈ T^{∗} one can uniquely assign an element of S^{′∗}(ℝ^{d}) of the form ∑ _{|α|=0}^{∞} b_{α} h_{α}. On the other hand, we can assign to f = [f_{n}] ∈ T^{∗} the functional T on S^{∗}(ℝ^{d}) given by T (φ) := (f, φ)_{T ^{∗}}, φ ∈ S^{∗}(ℝ^{d}), where (f, φ)_{T ^{∗}} is defined in (5.5). Clearly, the functional T is linear and continuous on S^{∗}(ℝ^{d}), i.e., T ∈ S^{′∗}(ℝ^{d}).

Conversely, if T ∈ S^{′∗}(ℝ^{d}) is of the form T = ∑ _{|α|=0}^{∞} b_{α} h_{α}, then the sequence (f_{n}) of functions f_{n} given by (5.2), where

F_{n} = ∑_{|α|=0}^{n} \frac{b_{α}h_{α}}{P (2α + 1)} (n ∈ ℕ) and F_{0} = ∑_{|α|=0}^{∞} \frac{b_{α}h_{α}}{P (2α + 1)} ,

is t-fundamental and f = [f_{n}] is the element of T^{∗} corresponding to T. Thus we have

Proposition 5.2. The mapping B : T^{∗} → S^{′∗}(ℝ^{d}) given by

T^{∗} ∋ f ↦ T = B(f) ∈ S^{′∗}(ℝ^{d}),

where T (φ) := (f, φ)_{T ^{∗}} for φ ∈ S^{∗}(ℝ^{d}), is a linear and sequentially continuous bijection.

We formulate now the concluding theorem of this section.

Theorem 5.1. (i) For each continuous linear functional T on S^{∗}(ℝ^{d}) there exists a unique t-ultradistribution f ∈ T^{∗} such that

(5.7)  T (φ) = (f, φ)_{T ^{∗}}, φ ∈ S^{∗}(ℝ^{d}).

Conversely, for each t-ultradistribution f, formula (5.7) defines a sequentially continuous linear functional on S^{∗}(ℝ^{d}).

The correspondence between continuous linear functionals on S^{∗}(ℝ^{d}) and tultradistributions in T^{∗}, described by (5.7), is bijective.

(ii) A sequence (f^{m}) of t-ultradistributions f^{m} ∈ T^{∗}, represented by t-funda-mental sequences (f_{n}^{m})_{n} for m ∈ ℕ, converges to f^{0} ∈ T^{∗} if and only if

(5.8)  lim_{m→∞} lim_{n→∞} (f_{n}^{m}, φ)_{T ^{∗}} = (f^{0}, φ)_{T ^{∗}},  φ ∈ S^{∗}(ℝ^{d}).

Proof. Assertion (i) is already proved above.

In order to show (ii) it suffices to prove that (5.8) implies f^{m} ⟶^{t} f^{0} as m → ∞. We apply the notation from Definition 4.4. Assume that the Hermite expansions of the functions F_{n}^{m} ∈ L^{2}(ℝ^{d}) (m ∈ ℕ_{0}, n ∈ ℕ), F^{m} ∈ L^{2}(ℝ^{d}) (m ∈ ℕ_{0}) in Definition 4.4, and of a given function φ ∈ S^{∗}(ℝ^{d}) are of the form

F_{n}^{m} = ∑_{|α|=0}^{∞} a_{α,n}^{m} h_{α} , F^{m} = ∑_{|α|=0}^{∞} a_{α}^{m}h_{α} (m ∈ ℕ_{0}, n ∈ ℕ), φ = ∑_{|α|=0}^{∞} r_{α} h_{α} .

By the duality of s^{∗} and s^{′∗}, we have

∑_{|α|=0}^{∞} a_{α,n}^{m} r_{α} → ∑_{|α|=0}^{∞} a_{α,n}^{0} r_{α} as m → ∞, uniformly in n ∈ ℕ.

Moreover, A_{n} → A_{0} as n → ∞ in s^{′∗}, where A_{n} := (a_{α,n}^{0} )_{α∈ℕ_{0}^{d}} ∈ s^{′∗} for n ∈ ℕ_{0}. This implies the assertion. □

Remark 5.1. We have shown in Propositions 4.7 and 4.9 that there exists a linear continuous bijection between the spaces S^{′∗}(ℝ^{d}) and T̃^{∗}, i.e., the spaces are isomorphic: S^{′∗}(ℝ^{d}) ≅ T̃^{∗}.

Remark 5.1, together with Theorem 5.1, leads to the following conclusion.

Theorem 5.2. The spaces T^{∗} and T̃^{∗} are isomorphic: T^{∗} ≅ T̃^{∗}. This means that every f ∈ T^{∗} can be represented as an equivalence class of t̃-fundamental sequences in the sense of Definition 4.5. Conversely, every f ∈ T̃^{∗} can be represented as an equivalence class of t-fundamental sequence in the sense of Definition 4.1. The convergence structures described in Definitions 4.4 and 4.8 are equivalent.

6. s-Ultradistributions as continuous linear functionals

We recall that a linear functional f on the corresponding space of test functions is an ultradistiribution or tempered ultradistribution if it is sequentially continuous.

Using our approach to the t̃-ultradistributions we prove:

Theorem 6.1. If f ∈ T^{∗} ≅ T̃^{∗}, then f ∈ U^{∗}(Ω).

Proof. We know that there exist P ∈ P^{∗} and P_{1} ∈ P_{u}^{∗} and there are functions

F_{n} ∈ L^{2}(ℝ^{d}) ∩ C^{∞} (ℝ^{d}) for n ∈ ℕ and F_{0} ∈ L^{2}(ℝ^{d}) such that F_{n} ⟶^{2} F_{0} as n → ∞

and f_{n} = P (D)(P_{1}F_{n}) on ℝ^{d} for n ∈ ℕ.

Fix K, K_{1} ⋐ Ω such that K ⋐ K_{1}^{∘} and a function κ_{K} ∈ D^{∗}(Ω) as in (2.5). We have P_{1}F_{n} = κ_{K} P_{1}F_{n} on K for n ∈ ℕ and the inequality

( ∫_{ℝ^{d}} |[κ_{K} P_{1}(F_{n} − F_{0})](x)|^{2}dx) ^{1/2} ⩽ sup_{x∈K} |P_{1}(x)| · ∥F_{n} − F_{0}∥_{2}

implies κ_{K} P_{1}F_{n} ⟶^{2} κ_{K} P_{1}F_{0} as n → ∞, so

F(κ_{K} P_{1}F_{n}) ⟶^{2} F(κ_{K} P_{1}F_{0}) as n → ∞.

Moreover, by Proposition 4.1, we have

F̃_{n} −−−→^{C(K)} F̃_{0} as n → ∞ and supp F̃_{n} ⊂ K_{1} (n ∈ ℕ_{0}),

where

F̃_{n} := κ_{K} F^{−1} (⟨·⟩^{−d} { \atop κ_{K} P_{1}F_{n}}) for n ∈ ℕ_{0}

with ⟨·⟩^{−d} meaning the function: ⟨ξ⟩^{−d} = (1 + |ξ|^{2})^{−d/2} for ξ ∈ ℝ^{d}.

If P̃(D) := (1 + D_{1}^{2} + …+ D_{d}^{2})^{d/2}P (D), then P̃ ∈ P^{∗} and f_{n} = P̃(D)F̃_{n} on K, so (f_{n}) is an s-fundamental sequence in the sense od Definition 2.1. □

The following assertion follows from the proof of Theorem 6.1.

Corollary 6.1. Let f ∈ T^{∗} ≅ T̃^{∗}, let Ω be an open set in ℝ^{d} and let θ ∈ D^{∗}(ℝ^{d}) be a function such that suppθ ⊂ Ω. Then θf = [f_{n}] ∈ U^{∗}(ℝ^{d}) for some s-fundamental sequence (f_{n}) having the following properties: for every K ⋐ Ω there exist P ∈ P^{∗} and functions F_{n} such that

f_{n} = P (D)F_{n} on K (n ∈ ℕ) and F_{n} −−−→^{C(ℝ^{d})} F_{0} as n → ∞.

Moreover

suppF_{n} ⊂ Ω for n ∈ ℕ_{0}.

Now, we are able to prove the main result of this section.

Main Theorem 6.2. (i) For every continuous linear functional T on D^{∗}(Ω) there exists a unique s-ultradistribution f ∈ U^{∗}(Ω) such that

(6.1)  T (φ) = (f, φ)_{U^{∗} (Ω)} , φ ∈ D^{∗}(Ω),

where (f, φ)_{U^{∗} (Ω)} is defined by (3.3).

Conversely, for every s-ultradistribution f ∈ U^{∗}(Ω), formula (6.1) defines a continuous linear functional T on D^{∗}(Ω).

The correspondence between continuous linear functionals on D^{∗}(Ω) and sultradistributions in U^{∗}(Ω), described by (6.1), is bijective.

(ii) A sequence of s-ultradistributions f^{m} ∈ U^{∗}(Ω) converges to an s-ultradistri-bution f^{0} ∈ U^{∗}(Ω) if and only if

lim_{m→∞} (f^{m}, φ)_{U^{∗} (Ω)} = (f^{0}, φ)_{U^{∗} (Ω)} for all φ ∈ D^{∗}(Ω).

Proof. It suffices to prove only the first part of assertion (i).

Consider a locally finite covering of Ω consisting of bounded open subsets Ω_{i} and Ω̃_{i} of Ω such that \overline{Ω}_{i} ⋐ Ω̃_{i} and let functions φ_{i} ∈ D^{∗}(Ω) form a partition of unity for i ∈ ℕ, i.e., φ_{i}(x) = 1 if x ∈ Ω_{i} and suppφ_{i} ⊂ Ω̃_{i} for i ∈ ℕ. If T is a continuous linear functional on D^{∗}(Ω), then T_{i}, defined by T_{i}(ψ) = T (φ_{i}ψ) for ψ ∈ S^{∗}(ℝ^{d}), is a continuous linear functional on S^{∗}(ℝ^{d}).

By Lemma 4.4 and Theorem 5.1, there is a sequence of f^{i} ∈ T^{∗} ≅ T̃^{∗} such that T_{i}(ψ) = (f^{i}, ψ)_{T ^{∗}} in the sense of (4.15) and T_{i}(ψ) = (f^{i}, ψ)T̃ ^{∗} in the sense of (4.16) for ψ ∈ S^{∗}(ℝ^{d}) and i ∈ ℕ. By Corollary 6.1, each f^{i} can be represented in the form f^{i} = [(f_{n}^{i})_{n}] ∈ U^{∗}(ℝ^{d}), where

f_{n}^{i} = P_{i}(D)F_{n}^{i} (n ∈ ℕ), F_{n}^{i} −−−→^{C(ℝ^{d})} F^{i} as n → ∞, suppF_{n}^{i} ⊂ Ω̃_{i} (n ∈ ℕ_{0})

for some P_{i} ∈ P^{∗} and functions F_{n}^{i} (n ∈ ℕ_{0}) with suitable properties.

Fix a pair of different i, j such that Ω_{i} ∩ Ω_{j} ≠ ∅. We have

T (φ) = (f^{i}, φ)_{U^{∗} (Ω)} = (f^{j} , φ)_{U^{∗} (Ω)} for φ ∈ D^{∗}(Ω_{i} ∩ Ω_{j}).

Consider f^{i} − f^{j} ∈ U^{∗}(ℝ^{d}) as an s-ultradistribution restricted to Ω_{i} ∩ Ω_{j}. There exists an s-fundamental sequence of r_{n}^{i,j} such that f^{i}−f^{j} = [(r_{n}^{i,j} )_{n}] on Ω_{i}∩Ω_{j}. This means f^{i} − f^{j} = [(r_{n}^{i,j} )_{n}], where r_{n}^{i,j} are smooth functions with suppr_{n}^{i,j} ⊂ Ω̃_{i} ∩ Ω̃_{j} such that for every K ⋐ Ω_{i} ∩Ω_{j} we have r_{n}^{i,j} = P_{i,j} (D)R_{n}^{i,j} on K for some P_{i,j} ∈ P^{∗} and functions R_{n}^{i,j} for n ∈ ℕ; moreover, R_{n}^{i,j} → 0 as n → ∞ uniformly on K. By Proposition 2.2, we have f^{i} = f^{j} on Ω_{i} ∩ Ω_{j}. Since our covering of Ω is locally finite, we may define

(6.2)  f := ∑_{i∈ℕ} f^{i}φ_{i} ∈ U^{∗}(Ω) with f|Ω_{i} = f^{i} (i ∈ ℕ).

Consequently, T (φ) = (f, φ)_{U^{∗} (Ω)} for φ ∈ D^{∗}(Ω) (see (3.3)).

Suppose T (φ) = (f^{1}, φ)_{U^{∗} (Ω)} and T (φ) = (f^{2}, φ)_{U^{∗} (Ω)} for f^{1}, f^{2} ∈ U^{∗}(Ω) and for φ ∈ D^{∗}(Ω). Let f^{1} − f^{2} = [g_{n}], where (g_{n}) is an s-fundamental sequence of the form g_{n} = P (D)(G_{n}) on K ⋐ Ω for suitable P and G_{n}. By (3.3), lim_{n→∞} g_{n} = 0 on Ω. Hence, by Proposition 2.2, f^{1} − f^{2} = [g_{n}] = 0, so the s-ultradistribution f defined in (6.2) is unique. □

Acknowledgements. The first, third and fourth authors were supported by the Project 174024 of the Serbian Ministry of Science. The second author was partly supported by the Center for Innovation and Transfer of Natural Sciences and Engineering Knowledge.

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Faculty of Electrical Engineering (Received 18 02 2016) University of Banja Luka (Revised 19 04 2016) Banja Luka Bosnia and Herzegovina

snjezana.maksimovic@etfbl.net

Faculty of Mathematics and Natural Sciences University of Rzeszow Rzeszow Poland

svetlana.kaminska@gmail.com

Faculty of Sciences and Mathematics University of Novi Sad Novi Sad Serbia

pilipovic@dmi.uns.ac.rs

Faculty of Mathematics and Natural Sciences University of Sts. Cyril and Methodius Skopje Macedonia

petar@pmf.ukim.mk