PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 99(113) (2016), 59–65 DOI: 10.2298/PIM1613059J

EXTENSION THEOREM OF WHITNEY TYPE FOR S(ℝ_{+}^{d}) BY USE OF THE KERNEL THEOREM

Smiljana Jakšić and Bojan Prangoski

Abstract. We study the expansions of the elements in S(ℝ_{+}^{d} ) and S′(ℝ_{+}^{d} ) with respect to the Laguerre orthonormal basis, extending the result of M. Guillemot-Teissier in the one dimensional case. As a consequence, we obtain Kernel theorem for S(ℝ_{+}^{d} ) and S′(ℝ_{+}^{d} ) and an extension theorem of Whitney type for S(ℝ_{+}^{d} ).

1. Introduction

We denote by ℝ_{+}^{d} the set (0, ∞)^{d} and by \overline{ℝ_{+}^{d}} its closure, i.e., [0, ∞)^{d}. We will consider the space S(ℝ_{+}^{d} ) which consists of all f ∈ C^{∞} (ℝ_{+}^{d} ) such that all derivatives D^{p}f, p ∈ ℕ_{0}^{d}, extend to continuous functions on \overline{ℝ_{+}^{d}} and

sup_{x∈ℝ_{+}^{d}} x^{k}|D^{p}f(x)| < ∞, for all k, p ∈ ℕ_{0}^{d}.

With this system of seminorms, S(ℝ_{+}^{d} ) becomes an (F)-space.

The results concerning the extension of a smooth function or a function of class C^{k} out of some region and various reformulation of such problems are called extension theorems of Whitney type. One can see Whitney [11], Seeley [8] and Hörmander [3, Theorem 2.3.6, p. 48]. Here we deal with a problem of extension of a function from S(ℝ_{+}^{d} ) onto S(ℝ^{d}). Theorem 4.2 is the main result of the paper. For the purpose of this theorem we prove the Schwartz kernel theorem for S(ℝ_{+}^{d} ) and S′(ℝ_{+}^{d} ), Theorem 4.1.

Recall, for n = 0, 1, 2 … the functions

L_{n}(x) = \frac{e^{x}}{n!} ( \frac{d}{dx})^{n}(e^{−x} x^{n}), x > 0

2010 Mathematics Subject Classification: Primary 46F05.

Key words and phrases: tempered distributions on ℝ_{+}^{d} ; kernel theorem for tempered distributions on ℝ_{+}^{d} ; smooth extensions of smooth rapidly decreasing functions.

Partially supported by the Ministry of Science, Republic of Serbia, project 174024.

Communicated by Stevan Pilipović.

are the Laguerre polynomials and L_{n}(x) = L_{n}(x)e^{− \frac{x}{2}} are Laguerre functions; {L_{n}(x), n = 0, 1, …} is an orthonormal basis for L^{2}(0, ∞) [10, p. 108].

The problem of expanding the elements of S′(ℝ_{+} ) with respect to the Laguerre orthonormal basis has been treated by Guillemont-Teissier in [4] and Duran in [1]: If T ∈ S′(ℝ_{+} ) and a_{n} = ⟨T, L_{n}(x)⟩, then T = ∑ _{n=0}^{∞} a_{n}L_{n}(x) and {a_{n}}_{n=0}^{∞} decreases slowly. Conversely, if {a_{n}}_{n=0}^{∞} decreases slowly, then there exists T ∈ S′(ℝ_{+} ) such that T = ∑ _{n=0}^{∞} a_{n}L_{n}(x).

The papers [7, 12, 13] contain expansions of the same kind as in [4, 1]. The novelty of this paper is an extension of the results of [4] for the d-dimensional case. This leads to the Schwartz kernel theorem (Theorem 4.1) which states that there is one-to-one correspondence between elements from S′(ℝ_{+}^{m+n} ) in two sets of variables x and y, and the continuous linear mappings of (S(ℝ_{+}^{m}))_{y} into (S′(ℝ_{+}^{m}))_{x}. As a consequence of Theorem 4.2, we explain the convolution in S′(ℝ_{+}^{d} ) in the last remark.

The structure of the paper is as follows. We recall in Section 3 some properties of Laguerre series and prove the convergence of the Laguerre series in S(ℝ_{+}^{d} ) and S′(ℝ_{+}^{d} ). In Section 4, we state Schwartz’s kernel theorem for S(ℝ_{+}^{d} ) and prove an extension theorem of Whitney type for S(ℝ_{+}^{d} ).

2. Notation

We use the standard multi-index notation. Given α = (α_{1}, …, α_{d}) ∈ ℕ_{0}^{d}, we write |α| = ∑ _{i=1}^{d} α_{i}, x^{α} = (x_{1}, …, x_{d})^{(α_{1} ,...,α_{d})} = ∏ _{i=1}^{d} x_{i}^{α_{i}} , D^{α} = ∏ _{i=1}^{d} \frac{∂^{α_{i}}}{∂x_{i} ^{α_{i}}} for the partial derivative and X^{α} f(x) = x^{α} f(x) for the multiplication operator. For x ∈ ℝ^{d}, |x| stands for the standard Euclidean norm in ℝ^{d}.

Let s be the space of rapidly decreasing sequences, i.e.,

{a_{n}}_{n∈ℕ_{0}^{d}} ∈ s ⇔ ∑_{n∈ℕ_{0}^{d}} |a_{n}|^{2}n^{2k} < ∞, for all k ∈ ℕ.

Then s′ stands for the strong dual of s, the space of slowly increasing sequences:

{a_{n}}_{n∈ℕ_{0}^{d}} ∈ s′ ⇔ ∑_{n∈ℕ_{0}^{d}} |a_{n}|^{2}n^{−2k} < ∞, for a k ∈ ℕ.

3. Laguerre series

The d-dimensional Laguerre functions

L_{n}(x) = L_{n_{1}} (x_{1}) ⋯ L_{n_{d}} (x_{d}) = ∏_{i=1}^{d} L_{n_{i}} (x_{i})

form an orthonormal basis for L^{2}(ℝ_{+}^{d} ) and are the eigenfunctions of the Laguerre operator E = (D_{1}(x_{1}D_{1}) − \frac{x_{1}}{4} ) ⋯ (D_{d}(x_{d}D_{d}) − \frac{x_{d}}{4} ) , E : S(ℝ_{+}^{d} ) → S(ℝ_{+}^{d} )

L_{n}(x) → E(L_{n}(x)) = ∏_{i=1}^{d} −( n_{i} + \frac{1}{2}) L_{n}(x).

EXTENSION THEOREM OF WHITNEY TYPE FOR S(ℝ_{+}^{d} ) 61

Note that E is a self-adjoint operator, i.e.,

⟨Ef, g⟩ = ⟨f, Eg⟩, f, g ∈ dom(E) = {f ∈ L^{2}(ℝ_{+}^{d} ); Ef ∈ L^{2}(ℝ_{+}^{d} )}.

For f ∈ S(ℝ_{+}^{d} ) we define the n-th Laguerre coefficient by a_{n} = ∫_{ℝ_{+}^{d}} f(x)L_{n}(x)dx.

The Laguerre series of the function f ∈ S(ℝ_{+}^{d} ) is ∑ _{n∈ℕ_{0}^{d}} a_{n}L_{n}(x).

In [4, p.547], the following bound on the one-dimensional Laguerre functions is obtained:

|x^{k}( \frac{d}{dx})^{p}L_{n}(x)| ⩽ C_{p,k}(n + 1)^{p+k} , x ⩾ 0, n, p, k ⩾ 0.

Finding the bound on the d-dimensional Laguerre functions involves not complicated calculation. Hence

(3.1)  |x^{k}D^{p}L_{n}(x)| ⩽ C_{p,k} ∏_{i=1}^{d} (n_{i} + 1)^{p_{i} +k_{i}} , x ∈ ℝ_{+}^{d} , n, p, k ∈ ℕ_{0}^{d}.

3.1. Convergence of the Laguerre series in S(ℝ_{+}^{d} ).

Theorem 3.1. For f ∈ S(ℝ_{+}^{d} ), let a_{n}(f) = ∫_{ℝ_{+}^{d}} f(x)L_{n}(x) dx. Then f =

∑ _{n∈ℕ_{0}^{d}} a_{n}(f)L_{n} and the series converges absolutely in S(ℝ_{+}^{d} ). Moreover the mapping ι : S(ℝ_{+}^{d} ) → s, ι(f) = {a_{n}(f)}_{n∈ℕ_{0}^{d}} , is a topological isomorphism.

Proof. For f ∈ S(ℝ_{+}^{d} ) we have

a_{n}(Ef) = ⟨Ef, L_{n}⟩ = ⟨f, E(L_{n})⟩ = a_{n}(f)(−1)^{d} ∏_{i=1}^{d} ( n_{i} + \frac{1}{2}) .

Moreover,

a_{n}(E^{p}f) = a_{n}(f) ∏_{i=1}^{d} (−1)^{p_{i}} ( n_{i} + \frac{1}{2})^{p_{i}}

for any p ∈ ℕ^{d}. As E^{p}f ∈ S(ℝ_{+}^{d} ) ⊂ L^{2}(ℝ_{+}^{d} ), we have

∑_{n∈ℕ_{0}^{d}} |a_{n}(f)|^{2} ∏_{i=1}^{d} ( n_{i} + \frac{1}{2})^{2p_{i}} < ∞, for every p ∈ ℕ_{0}^{d},

i.e., {a_{n}(f)}_{n∈ℕ_{0}^{d}} ∈ s. Clearly f = ∑ _{n∈ℕ_{0}^{d}} a_{n}(f)L_{n} as elements of L^{2}(ℝ_{+}^{d} ). By (3.1), we obtain

(3.2)  ∑_{n∈ℕ_{0}^{d}} |x^{k}D^{p}(a_{n}(f)L_{n}(x))| ⩽ C_{p,k} ∑_{n∈ℕ_{0}^{d}} |a_{n}(f)| ∏_{i=1}^{d} (n_{i} + 1)^{p_{i} +k_{i}} < ∞

which yields the absolute convergence of the series in S(ℝ_{+}^{d} ).

To prove that ι is a topological isomorphism, first observe that by the above consideration it is well defined and it is clearly an injection. Let {a_{n}}_{n∈ℕ_{0}^{d}} ∈ s. Define f = ∑ _{n∈ℕ_{0}^{d}} a_{n}L_{n} ∈ L^{2}(ℝ_{+}^{d} ). Now (3.2) proves that this series converges in S(ℝ_{+}^{d} ), hence f ∈ S(ℝ_{+}^{d} ). Thus ι is bijective. Observe that, (3.2) proves that ι^{−1} is continuous. Since S(ℝ_{+}^{d} ) and s are (F)-spaces, the open mapping theorem proves that ι is topological isomorphism. □

3.2. Convergence of the Laguerre series in S′(ℝ_{+}^{d} ).

Theorem 3.2. For T ∈ S′(ℝ_{+}^{d} ), let b_{n}(T) = ⟨T, L_{n}⟩. Then {b_{n}(T)}_{n∈ℕ_{0}^{d}} ∈ s′ and T = ∑ _{n∈ℕ_{0}^{d}} b_{n}(T)L_{n}. The series converges absolutely in S′(ℝ_{+}^{d} ). Conversely, if {b_{n}}_{n∈ℕ_{0}^{d}} ∈ s′, then there exists a T ∈ S′(ℝ_{+}^{d} ) such that T = ∑ _{n∈ℕ_{0}^{d}} b_{n}L_{n}. As a consequence, S′(ℝ_{+}^{d} ) is topologically isomorphic to s′.

Proof. Assume that {b_{n}}_{n∈ℕ_{0}^{d}} ∈ s′. Then there exists a k ∈ ℕ such that ∑ _{n∈ℕ_{0}^{d}} |b_{n}|^{2}(|n| + 1)^{−2k} < ∞. For a bounded subset B of S(ℝ_{+}^{d} ), Theorem 3.1 implies that there exists C > 0 such that ∑ _{n∈ℕ_{0}^{d}} |a_{n}(f)|^{2}(|n| + 1)^{2k} ⩽ C, for all f ∈ B, where we denote {a_{n}(f)}_{n∈ℕ_{0}^{d}} = ι(f). Observe that for an arbitrary q ∈ ℕ we have

∑_{|n|⩽q} sup_{f∈B} |⟨b_{n}L_{n}, f⟩|⩽ sup_{f∈B} ∑_{n∈ℕ_{0}^{d}} ∑_{m∈ℕ_{0}^{d}} |⟨b_{n}L_{n}, a_{m}(f)L_{m}⟩|

= sup_{f∈B} ∑_{n∈ℕ_{0}^{d}} |b_{n}||a_{n}(f)| ⩽ C′,

i.e.,

∑_{n∈ℕ_{0}^{d}} sup_{f∈B} |⟨b_{n}L_{n}, f⟩| < ∞,

hence ∑ _{n∈ℕ_{0}^{d}} b_{n}L_{n} converges absolutely in S′(ℝ_{+}^{d} ).

Let T ∈ S′(ℝ_{+}^{d} ). Theorem 3.1 implies that ^{t}ι : s′ → S′(ℝ_{+}^{d} ) is an isomorphism (^{t}ι denotes the transpose of ι). Now, one easily verifies that (^{t}ι)^{−1} T = {b_{n}}_{n∈ℕ_{0}^{d}} , where b_{n}(T) = ⟨T, L_{n}⟩. Observe that for f ∈ S(ℝ_{+}^{d} )

⟨T, f⟩ = ∑_{n∈ℕ_{0}^{d}} a_{n}(f)⟨T, L_{n}⟩ = ∑_{n∈ℕ_{0}^{d}} a_{n}(f)b_{n}(T) = ⟨ ∑_{n∈ℕ_{0}^{d}} b_{n}(T)L_{n}, f⟩ ,

i.e., T = ∑ _{n∈ℕ_{0}^{d}} b_{n}(T)L_{n}. □

4. Kernel theorem

The completions of the tensor product are denoted by ˆ⊗_{ϵ} and ˆ⊗_{π} with respect to ϵ and π topologies. If they are equal, we drop the subindex.

Proposition 4.1. The spaces S(ℝ_{+}^{d} ) and S′(ℝ_{+}^{d} ) are nuclear.

Proof. Since s is nuclear, Theorem 3.1 implies that S(ℝ_{+}^{d} ) is also nuclear. Now S′(ℝ_{+}^{d} ) is nuclear as the strong dual of a nuclear (F)-space. □

Theorem 4.1. The following canonical isomorphisms hold:

S(ℝ_{+}^{m}) ˆ⊗S(ℝ_{+}^{n} ) ≅ S(ℝ_{+}^{m+n} ), S′(ℝ_{+}^{m}) ˆ⊗S′(ℝ_{+}^{n} ) ≅ S′(ℝ_{+}^{m+n} ).

Proof. The second isomorphism follows from the first one since S(ℝ_{+}^{d} ) is a nuclear (F)-space. Thus, it is enough to prove the first isomorphism.

Step 1: From Theorem 3.1 it follows that S(ℝ_{+}^{m})⊗S(ℝ_{+}^{n} ) is dense in S(ℝ_{+}^{m+n} ). It suffices to show that the latter induces on the former the topology π = ϵ (the π and the ϵ topologies are the same because S(ℝ_{+}^{d} ) is nuclear). Since the bilinear mapping (f, g) ↦ f ⊗ g of S(ℝ_{+}^{m}) × S(ℝ_{+}^{n} ) into S(ℝ_{+}^{m+n} ) is separately continuous, it follows that it is continuous (S(ℝ_{+}^{m}) and S(ℝ_{+}^{n} ) are (F)-spaces). The continuity of this bilinear mapping proves that the inclusion S(ℝ_{+}^{m}) ⊗_{π} S(ℝ_{+}^{n} ) → S(ℝ_{+}^{m+n} ) is continuous, hence the topology π is stronger than the induced one from S(ℝ_{+}^{m+n} ) onto S(ℝ_{+}^{m}) ⊗ S(ℝ_{+}^{n} ).

Step 2: Let A′ and B′ be equicontinuous subsets of S′(ℝ_{+}^{m}) and S′(ℝ_{+}^{n} ), respectively. There exist C > 0 and j, l ∈ ℕ such that

sup_{T ∈A′} |⟨T, φ⟩| ⩽ C∥φ∥_{j,l} and sup_{F ∈B′} |⟨F, ψ⟩| ⩽ C∥ψ∥_{j,l} ,

where

∥f∥_{j,l} = sup_{\begin{matrix}|k|⩽j \\ |p|⩽l \end{matrix}} sup_{x∈ℝ_{+}^{d}} |x^{k}D^{p}f(x)| < ∞.

For all T ∈ A′ and F ∈ B′ we have

|⟨T_{x} ⊗ F_{y}, χ(x, y)⟩| = |⟨F_{y}, ⟨T_{x}, χ(x, y)⟩⟩| ⩽ C sup_{\begin{matrix}|k|⩽j \\ |p|⩽l \end{matrix}} sup_{y∈ℝ_{+}^{n}} |y^{k}⟨T_{x}, D_{y}^{p}χ(x, y)⟩|

⩽ C^{2} sup_{\begin{matrix}|k|⩽j \\ |p|⩽l \end{matrix}} sup_{\begin{matrix}|k′|⩽j \\ |p′|⩽l \end{matrix}} sup_{\begin{matrix}x∈ℝ_{+}^{m} \\ y∈ℝ_{+}^{n} \end{matrix}} |x^{k′}y^{k}D_{x}^{p′}D_{y}^{p}χ(x, y)|

⩽ C^{2}∥χ(x, y)∥_{(k′,k),(p′,p)} , for all χ ∈ S(ℝ_{+}^{m}) ⊗ S(ℝ_{+}^{n} ).

It follows that the ϵ topology on S(ℝ_{+}^{m}) ⊗ S(ℝ_{+}^{n} ) is weaker than the induced one from S(ℝ_{+}^{m+n} ). □

As a consequence of this theorem we have the following important

Theorem 4.2. The restriction mapping f ↦ f|_{ℝ_{+}^{d}} , S(ℝ^{d}) → S(ℝ_{+}^{d} ) is a topological homomorphism onto.

The space S(ℝ_{+}^{d} ) is topologically isomorphic to the quotient space S(ℝ^{d})/N, where N = {f ∈ S(ℝ^{d}) | suppf ⊆ ℝ^{d} ∖ ℝ_{+}^{d} }. Consequently, S′(ℝ_{+}^{d} ) can be identified with the closed subspace of S′(ℝ^{d}) which consists of all tempered distributions with support in \overline{ℝ_{+}^{d}} .

Proof. Obviously, the restriction mapping f ↦ f|_{ℝ_{+}^{d}} , S(ℝ^{d}) → S(ℝ_{+}^{d} ) is continuous. We prove its surjectivity by induction on d. For clarity, denote the d-dimensional restriction by R_{d}. For d = 1, the surjectivity of R_{1} is proved in [1, p. 168]. Assume that R_{d} is surjective. By the open mapping theorem, R_{d} and R_{1} are topological homomorphisms onto since all the underlying spaces are (F)-spaces. By the above theorem, R_{d} ˆ⊗_{π}R_{1} is continuous mapping from S(ℝ^{d+1} ) to S(ℝ_{+}^{d+1} ) (S(ℝ^{d}) ˆ⊗S(ℝ) ≅ S(ℝ^{d+1} ) by the Schwartz kernel theorem). Clearly R_{d} ˆ⊗_{π}R_{1} = R_{d+1} . As S(ℝ^{d+1} ) and S(ℝ_{+}^{d+1} ) are (F)-spaces, [6, Theorem 7, p. 189] implies that R_{d+1} is also surjective.

The surjectivity of the restriction mapping together with the open mapping theorem implies that it is homomorphism. Clearly N is a closed subspace of S(ℝ^{d}) and kerR_{d} = N. Thus R_{d} induces natural topological isomorphism between S(ℝ^{d})/N and S(ℝ_{+}^{d} ). Hence (S(ℝ^{d})/N) ′_{b} is topologically isomorphic to S′(ℝ_{+}^{d} ) (the index b stands for the strong dual topology). Since S(ℝ^{d}) is an (F S)-space, [5, Theorem A.6.5, p. 255] implies that (S(ℝ^{d})/N) ′_{b} is topologically isomorphic to the closed subspace N^{⊥} = {T ∈ S′(ℝ^{d}) | ⟨T, f⟩ = 0, for all f ∈ N} of S′(ℝ^{d}) which is exactly the subspace of all tempered distributions with support in \overline{ℝ_{+}^{d}} . □

Given f, g ∈ S′(ℝ_{+}^{d} ), Theorem 4.2 implies that we can consider them as elements of S′(ℝ^{d}) with support in \overline{ℝ_{+}^{d}} . Now, one easily verifies that for each φ ∈ S(ℝ^{d}), we have (f(x)⊗g(y))φ(x+y) ∈ D′_{L^{1}} (ℝ^{2d}), hence the S′-convolution of f and g exists (see [9, p. 26]). Also, if suppφ∩\overline{ℝ_{+}^{d}} = ∅, then (f(x)⊗g(y))φ(x+y) = 0, hence suppf ∗ g ⊆ \overline{ℝ_{+}^{d}} , i.e., f ∗ g ∈ S′(ℝ_{+}^{d} ). Thus

⟨f ∗ g, φ⟩ = ⟨f(x) ⊗ g(y), φ(x + y)⟩, φ ∈ S(ℝ_{+}^{d} )

(observe that the function φ^{Δ} (x, y) = φ(x + y) is an element of S(ℝ_{+}^{2d})).

Remark 4.1. [1, Remark 3.7 for d = 1] Let us show that S′(ℝ_{+}^{d} ) is a convolution algebra. Given f, g ∈ S′(ℝ_{+}^{d} ), we compute the n-th Laguerre coefficient of f ∗ g. If a_{n} = ⟨f, L_{n}⟩ and b_{n} = ⟨g, L_{n}⟩, then

⟨f ∗ g, L_{n}(t)⟩ = ⟨f(x) ⊗ g(y), L_{n}(x + y)⟩.

Now, L_{n}^{1}(x+y) = ∑ _{k=0}^{n} L_{n−k} (x)L_{k}(y) and L_{n}(t) = L_{n}^{1}(t)−L_{n−1}^{1} (t) (see [2, p. 192]) where L_{n}^{1}(x) = ∑ _{k=0}^{n} \binom{n+1}{n−k}((−x)^{k}/k!). In order to simplify the proof, we consider the case d = 2. Then

⟨f ∗ g, L_{n}(t)⟩ = ⟨ f(x) ⊗ g(y), ∏_{i=1}^{2} (L_{n_{i}}^{1} (x_{i} + y_{i}) − L_{n_{i} −1}^{1} (x_{i} + y_{i}))⟩

= ⟨ f(x) ⊗ g(y), ∏_{i=1}^{2} ( ∑_{k_{i} =0}^{n_{i}} L_{n_{i} −k_{i}} (x_{i})L_{k_{i}} (y_{i}) − ∑_{k_{i} =0}^{n_{i} −1} L_{n_{i} −k_{i} −1} (x_{i})L_{k_{i}} (y_{i})) ⟩

= ⟨ f(x)g(y), ∑_{k⩽(n_{1} ,n_{2} )} L_{(n_{1} ,n_{2} )−k} (x)L_{k}(y) − ∑_{k⩽(n_{1} −1,n_{2} )} L_{(n_{1} −1,n_{2} )−k} (x)L_{k}(y)

− ∑_{k⩽(n_{1} ,n_{2} −1)} L_{(n_{1} ,n_{2} −1)−k} (x)L_{k}(y) + ∑_{k⩽(n_{1} −1,n_{2} −1)} L_{(n_{1} −1,n_{2} −1)−k} (x)L_{k}(y)⟩

= ∑_{k⩽(n_{1} ,n_{2} )} a_{(n_{1} ,n_{2} )−k} b_{k} − ∑_{k⩽(n_{1} −1,n_{2} )} a_{(n_{1} −1,n_{2} )−k} b_{k}

− ∑_{k⩽(n_{1} ,n_{2} −1)} a_{(n_{1} ,n_{2} −1)−k} b_{k} + ∑_{k⩽(n_{1} −1,n_{2} −1)} a_{(n_{1} −1,n_{2} −1)−k} b_{k},

where a_{n} or b_{n} equals zero if some component of the subindex n is less than zero. It is easy to verify that if (a_{n})_{n∈ℕ^{2}} ∈ s′ and (b_{n})_{n∈ℕ^{2}} ∈ s′, then ⟨f ∗ g, L_{n}(t)⟩ ∈ s′.

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Faculty of Forestry (Received 26 08 2015) Belgrade University Belgrade, Serbia

   smiljana.jaksic@sfb.bg.ac.rs

Faculty of Mechanical Engineering University Ss. Cyril and Methodius Skopje, Macedonia

   bprangoski@yahoo.com