Georgian Mathematical Journal 1(1994), No. 2, 213-227

ON INTEGRAL BERNSTEIN OPERATORS IN SOME CLASSES OF MEASURABLE BIVARIATE FUNCTIONS

ROMAN TABERSKI

Abstract. The two main theorems are cencerned with the approximation of (complex-valued) functions on the real plane by sums of Bernstein pseudoentire functions. They are formulated and proved in Section 4, after prior determination of the suitable integral operators. Analogous results for pseudopolynomial approximations were obtained by Brudnyǐ, Gonska, and Jetter ([2],[3]).

1. Preliminaries. Let L_{loc}(R) [resp. AC_{loc}(R)] be the set of all univariate (complex-valued) functions Lebesgue-integrable (absolutely continuous) on every compact subinterval of R := (−∞, ∞). Denote by L_{loc}^{p}(R^{2}), 1 ≤ p ≤ ∞, the set of all measurable bivariate (complex-valued) functions Lebesgue-integrable with pth power (essentially bounded when p = ∞) on every finite two-dimensional integral lying on the plane R^{2} = R × R; write L_{loc}(R^{2}) instead of L_{loc}^{1}(R^{2}). Denote by C(R) [resp. C(R^{2})] the set of all (complex-valued) functions continuous on R [R^{2}].

Given any bivariate (complex-valued) function f ≡ f(·, ·) measurable on R^{2}, the quantity

∥f∥_{p} := \begin{cases}( ∫∫_{R^{2}} |f(u, v)|^{p}du dv)^{1/p} & if 1 ≤ p < ∞, \\ ess sup_{(u,v)∈R^{2}} |f(u, v)| & if p = ∞ \end{cases}  (1)

is finite or infinite. In the case ∥f∥_{p} < ∞ the function f is said to be of class L^{p} ≡ L^{p}(R^{2}); in symbols, f(·, ·) ∈ L^{p}. The notations f(·, ·) ∈ L_{loc}^{p}(R^{2}), g(·, v) ∈ L_{loc}(R), etc. have a similar meaning.

Let f ≡ f(·, ·) be a (complex-valued) function defined on R^{2} and let k, l ∈ N_{0} ≡ {0, 1, 2, … }. Determine the partial differences of f at a point (x, y) ∈ R^{2}, i.e.,

1991 Mathematics Subject Classification. 41A25, 41A35.

∗Research supported by KBN grant 2 1079 91 01.

1Δ_{λ}^{k}f(x, y) := ∑_{μ=0}^{k}(−1)^{k−μ}\binom{k}{μ}f(x + μλ, y),

1Δ_{η}^{l}f(x, y) := ∑_{ν=0}^{k}(−1)^{l−ν} \binom{l}{ν}f(x, y + νη),

with the real increments λ, η. Introduce also the mixed difference Δ_{λ,η}^{k,l} := 1Δ_{λ}^{k}(^{2}Δ_{η}^{l}f(x, y)).

It is easy to see that

Δ_{λ,η}^{k,l} = ∑_{μ=0}^{k} ∑_{ν=0}^{l}(−1)^{k+l−μ−ν} \binom{k}{μ}\binom{l}{ν}f(x + μλ, y + νη).

In particular, Δ_{λ,η}^{0,0}f(x, y) = f(x, y) and

Δ_{λ,η}^{1,1}f(x, y) = f(x + λ, y + η) − f(x + λ, y) − f(x, y + η) + f(x, y).

Further, Δ_{λ,η}^{1,1}(Δ_{λ,η}^{k,l}f(x, y)) = Δ_{λ,η}^{k+1,l+1}f(x, y).

The first (weak) derivative of f at (x, y) is given by

f^{(1)}(x, y) := lim_{λ→0} \frac{1}{λ^{2}} Δ_{λ,λ}^{1,1}f(x, y)

whenever the right-hand side exists. The (weak) derivatives of f of higher orders are defined successively:

f^{(j)}(x, y) := (f^{(j−1)})^{(1)}(x, y) for j = 2, 3, ….

Moreover, by convention, f^{(0)}(x, y) ≡ f(x, y) and

f^{(m,n)}(x, y) ≡ \frac{∂^{m}}{∂x^{m}} (\frac{∂^{n}}{∂y^{n}} f(x, y)) for m, n ∈ N_{0}.

Considering any (complex-valued) function f measurable on R^{2}, one can define its mixed L^{p}-modulus of smoothness:

ω_{k,l}(δ_{1}, δ_{2}; f)_{p} := sup {∥Δ_{λ,η}^{k,l}f∥_{p} : 0 ≤ λ ≤ δ_{1}, 0 ≤ η ≤ δ_{2}}.

This quantity, with fixed p ≥ 1 and k, l ∈ N_{0}, may be finite or infinite for positive numbers δ_{1}, δ_{2}. If there exist three non-negative numbers M, α, β such that ω_{k,l}(δ_{1}, δ_{2}; f)_{p} ≤ Mδ_{1}^{α}δ_{2}^{β} for all δ_{1}, δ_{2} ∈ (0, 1], we say that f belongs to the Hölder class H_{α,β,p}^{(k,l)}. More generally, if

∥f∥_{φ,ψ,p}^{(k,l)} ≡ sup \{\frac{1}{φ(λ)ψ(η)}∥Δ_{λ,η}^{k,l}f∥_{p} : 0 < λ, η ≤ 1\} < ∞,  (2)

where φ, ψ mean positive non-decreasing functions on (0, 1] and φ(1) = ψ(1) = 1, the function f is said to be of class H_{φ,ψ,p}^{(k,l)}. In the case φ(δ) = δ^{α}, ψ(δ) = δ^{β} for δ ∈ (0, 1], the left-hand side of identity (2) will be denoted by ∥f∥_{α,β,p}^{(k,l)}.

Denote by E_{σ} [resp. E_{σ,τ}] the set of all univariate (bivariate) entire functions of exponential type of order σ [(σ, τ)] at most. Clearly, if F (·, ·) ∈ E_{σ,τ} (σ, τ ≥ 0), then F (·, v) ∈ E_{σ} and F (u, ·) ∈ E_{τ} for all u, v ∈ R. Moreover, F (·, ·) ∈ C(R^{2}). In the case where Φ(·, v) ∈ E_{σ} (resp. Ψ(u, ·) ∈ E_{τ}) for almost every v ∈ R [u ∈ R], Φ(z, ·) ∈ L_{loc}(R) [Ψ(·, z) ∈ L_{loc}(R^{2})] for every complex number z, and Φ(·, ·) ∈ L_{loc}(R^{2}) [Ψ(·, ·) ∈ L_{loc}(R^{2})], we call Φ [Ψ] a pseudoentire function of class W_{σ}^{1} [W_{τ}^{2}].

The aim of this paper is to present the Jackson type theorems, in L^{p}- norms (1) and seminorms (2), for some (complex-valued) functions defined and measurable on R^{2}. We begin with auxiliary results about the mixed differences and Bernstein’s singular integrals used in our approximation problems.

2. Estimates for the mixed differences and moduli of smoothness. Consider a (complex-valued) function f ≡ f(·, ·) defined and measurable on the plane R^{2}. Denote by k, l two non-negative integers. Take an arbitrary p satisfying the condition: 1 ≤ p ≤ ∞.

Proposition 1. If λ, η ∈ R and n ∈ N ≡ {1, 2, 3, … }, then

∥Δ_{mλ,nη}^{k,l}f∥_{p} ≤ m^{k}n^{l}∥Δ_{λ,η}^{k,l}f∥_{p}.  (3)

Proof. Given arbitrary x, y ∈ R, let g(x, y) := ^{2}Δ_{nη}^{l}f(x, y).

By identity (5) of [5], p.116,

g(x, y) = ∑_{ν_{1}=0}^{n−1} ⋯ ∑_{ν_{l}=0}^{n−1}^{2}Δ_{η}^{l}f(x, y + ν_{1}η + ⋯ + ν_{i}η)

and

1Δ_{mλ}^{k}g(x, y) = ∑_{μ_{1}=0}^{m−1} ⋯ ∑_{μ_{k}=0}^{m−1}^{1}Δ_{λ}^{k}g(x + μ_{1}λ + ⋯ + μ_{k}λ, y).

Hence

1Δ_{mλ,nη}^{k,l}f(x, y) = ∑_{μ_{1}=0}^{m−1} ⋯ ∑_{μ_{k}=0}^{m−1} ∑_{ν_{1}=0}^{n−1} ⋯

⋯ ∑_{ν_{l}=0}^{n−1} Δ_{λ,η}^{k,l}f(x + μ_{1}λ + ⋯ + μ_{k}λ, y + ν_{1}η + ⋯ + ν_{i}η).

Applying Minkowski’s inequality, we get at once estimate (3).

From (3) it follows that

ω_{k,l}(mδ_{1}, nδ_{2}; f)_{p} ≤ m^{k}n^{l}ω_{k,l}(δ_{1}, δ_{2}; f)_{p} (m, n ∈ N)  (4)

for all non-negative numbers δ_{1}, δ_{2}, whence , in case a, b ≥ 0

ω_{k,l}(aδ_{1}, bδ_{2}; f)_{p} ≤ ([a] + 1)^{k}([b] + 1)^{l}ω_{k,l}(δ_{1}, δ_{2}; f)_{p}.  (5)

Clearly, the estimates (3)-(5) are useful only with the finite right-hand sides.

Proposition 2. Let f have the partial derivatives

f^{(0,n)}(u, ·), …, f^{(m−1,n)}(u, ·) ∈ L_{loc}(R) (m, n ∈ N)

for every u ∈ R, and let f^{(m−1,n)}(·, v) ∈ AC_{loc}(R) for almost every v ∈ R. Further, given any c > 0, suppose the existence of positive number M_{c} such that ess sup_{−c≤u≤c} |f^{(m,n)}(u, v)| ≤ M_{c} for almost every v ∈ [−c, c]. Then, in the case k ≥ m and λ, η ∈ R,

∥Δ_{λ,η}^{k,l}f∥_{p} ≤ |λ|^{m}|η|^{n}∥Δ_{λ,η}^{k−m,l−n}f^{(m,n)}∥_{p}.  (6)

Proof. By our assumption, the (Lebesgue) integrals

∫_{0}^{η} ⋯ ∫_{0}^{η} f^{(j,n)}(u, w + t_{1} + ⋯ + t_{n})dt_{1} ⋯ dt_{n} ≡ F_{j}(u, w)
(j = 0, 1, …, m)

exist for all u, w ∈ R if j < m and for almost every u and every w ∈ R if j = m.

Given u, w ∈ R, one has

|\frac{1}{h} ∫_{0}^{h} f^{(m,n)}(u + s, w + t_{1} + ⋯ + t_{n})ds| ≤ M_{c}

uniformly in h ∈ [−1, 1]

(h ≠ 0) for almost every point (t_{1}, ⋯, t_{n}) of the n-dimensional interval [−|η|, |η|]^{n} whenever c ≥ max(|u| + 1, |w| + n|η|). Moreover,

lim_{h→0} \frac{1}{h} ∫_{0}^{h} f^{(m,n)}(u + s, w + t_{1} + ⋯ + t_{n})ds =

= f^{(m,n)}(u, w + t_{1} + ⋯ + t_{n})

for almost every u. Hence

lim_{h→0} \frac{1}{h}\{F_{m−1}(u + h, w) − F_{m−1}(u, w)\} =

= ∫_{0}^{η} ⋯ ∫_{0}^{η} f^{(m,n)}(u, w + t_{1} + ⋯ + t_{n})dt_{1} ⋯ dt_{n}, i.e., F_{m−1}^{(1,0)}(u, w) = F_{m}(u, w) for a.e. u ∈ R and every w ∈ R, by the Lebesgue dominated convergence theorem.

Next, when m ≥ 2, (u, w) ∈ R^{2} and 0 < |h| ≤ 1,

\frac{1}{h}\{F_{m−2}(u + h, w) − F_{m−2}(u, w)\} =

= ∫_{0}^{η} ⋯ ∫_{0}^{η} \{\frac{1}{h} ∫_{0}^{h} f^{(m−1,n)}(u + s, w + t_{1} + ⋯ + t_{n})ds\}dt_{1} ⋯ dt_{n},

|\frac{1}{h}f^{(m−1,n)}(u + s, w + t_{1} + ⋯ + t_{n})ds| =

= |\frac{1}{h}\{ ∫_{0}^{s} f^{(m,n)}(u + z, w + t_{1} + ⋯ + t_{n})dz +

+f^{(m−1,n)}(u, w + t_{1} + ⋯ + t_{n})}ds| ≤

≤ |\frac{1}{h}|f^{(m,n)}(u + z, w + t_{1} + ⋯ + t_{n})|dz| +

+|f^{(m−1,n)}(u, w + t_{1} + ⋯ + t_{n})|

and

lim_{h→0} \frac{1}{h} ∫_{0}^{h} f^{(m−1,n)}(u + s, w + t_{1} + ⋯ + t_{n})ds =

= f^{(m−1,n)}(u, w + t_{1} + ⋯ + t_{n})

for almost all (t_{1}, …, t_{n}) ∈ [−|η|, |η|]^{n}. Therefore, as previously, F_{m−2}^{(1,0)}(u, w) = F_{m−1}(u, w) for all (u, w) ∈ R^{2} Analogously, when m ≥ 3, F_{m−3}^{(1,0)}(u, w) = F_{m−2}(u, w) for all (u, w) ∈ R^{2}, etc. Consequently, if m ≥ 1, F_{0}^{(m−1,0)}(u, w) = F_{m−1}(u, w) for all (u, w) ∈ R^{2}, and F_{0}^{(m,0)}(u, w) = F_{m}(u, w) for a.e. u ∈ R and every w ∈ R

Further, in the case −c ≤ a ≤ b ≤ c and |w| + n|η| ≤ c,

|F_{m−1}(b, w) − F_{m−1}(a, w)| =

= | ∫_{0}^{η} ⋯ ∫_{0}^{η} { ∫_{a}^{b} f^{(m,n)}(s, w + t_{1} + ⋯ + t_{n})ds}dt_{1} ⋯ dt_{n} ≤

≤ M_{c}(b − a)|η|^{n}.

Hence F_{m−1}(·, w) ∈ AC_{loc}(R) for every w ∈ R.

By identity (4) of [5], p.116,

Δ_{λ,η}^{m,n}f(x, y) = ^{1}Δ_{λ}^{m}(^{2}Δ_{η}^{n}f(x, y)) = ^{1}Δ_{λ}^{m}F_{0}(x, y) =
= ∫_{0}^{λ} ⋯ ∫_{0}^{λ} F_{0}^{(m,0)}(x + s_{1} + ⋯ + s_{m}, y)ds_{1} ⋯ ds_{m} =
= ∫_{0}^{λ} ⋯ ∫_{0}^{λ} F_{m}(x + s_{1} + ⋯ + s_{m}, y)ds_{1} ⋯ ds_{m}

for arbitrary x, y, λ, η ∈ R. Thus

Δ_{λ,η}^{m,n}f(x, y) = ∫_{0}^{λ} ⋯ ∫_{0}^{λ} { ∫_{0}^{η} ⋯

⋯ ∫_{0}^{η} f^{(m,n)}(x+s_{1}+⋯+s_{m}, y+t_{1}+⋯+t_{n})dt_{1}...dt_{n}}ds_{1}...ds_{m}.

Observing that

Δ_{λ,η}^{k,l}f(x, y) = Δ_{λ,η}^{k−m,l−n}(Δ_{λ,η}^{m,n}f(x, y)) =
= ∫_{0}^{λ} ⋯ ∫_{0}^{λ} { ∫_{0}^{η} ⋯ ∫_{0}^{η} Δ_{λ,η}^{k−m,l−n} ×

×f^{(m,n)}(x + s_{1} + ⋯ + s_{m}, y + t_{1} + ⋯ + t_{n})dt_{1} ⋯ dt_{n}}ds_{1} ⋯ ds_{m}

and applying the generalized Minkowski’s inequality, we obtain (6).

Estimate (6) immediately implies

ω_{k,l}(δ_{1}, δ_{2}; f)_{p} ≤ δ_{1}^{m}δ_{2}^{n}ω_{k−m,l−n}(δ_{1}, δ_{2}; f^{(m,n)})_{p} for δ_{1}, δ_{2} ≥ 0. (7)

Proposition 3. Let (f_{0}, f_{1}, …, f_{ρ}) be a system of (complex-valued) functions of two real variables, such that f_{ρ} ∈ L_{loc}(R^{2}) (ρ ∈ N) and, for j = 1, …, ρ,

f_{ρ−j}(x, y) = f_{ρ−j}(x, 0) + f_{ρ−j}(0, y) − f_{ρ−j}(0, 0) +
+ ∫_{0}^{x} ∫_{0}^{y} f_{ρ−j+1}(u, v) du dv if x, y ≠ 0,

f_{ρ−j}(x, 0), f_{ρ−j}(0, y) are defined for all real x, y and, when j < ρ, f_{ρ−j}(·, 0), f_{ρ−j}(0, ·) ∈ L_{loc}(R). Suppose that the integer k is greater than or equal to ρ. Then, for all λ, η ∈ R

∥Δ_{λ,η}^{k,k}f_{0}∥_{p} ≤ |λη|^{ρ}∥Δ_{λ,η}^{k−ρ,k−ρ}f_{ρ}∥_{p}.  (8)

Proof. Given arbitrary x, y, λ, η ∈ R, we have

Δ_{λ,η}^{1,1}f_{0}(x, y) = ∫_{x}^{x+y} ∫_{y}^{y+η} f_{1}(s, t) ds dt =

= ∫_{0}^{λ} ∫_{0}^{η} f_{1}(x + s_{1}, y + t_{1}) ds_{1}dt_{1},
Δ_{λ,η}^{2,2}f_{0}(x, y) = ∫_{0}^{λ} ∫_{0}^{η} Δ_{λ,η}^{1,1}f_{1}(x + s_{1}, y + t_{1}) ds_{1}dt_{1} =
= ∫_{0}^{λ} ∫_{0}^{η} { ∫_{0}^{λ} ∫_{0}^{η} f_{2}(x + s_{1} + s_{2}, y + t_{1} + t_{2})ds_{2}dt_{2}} ds_{1}dt_{1}, etc.

Therefore

Δ_{λ,η}^{k,k}f_{0}(x, y) = Δ_{λ,η}^{k−ρ,k−ρ}(Δ_{λ,η}^{ρ,ρ}f_{0}(x, y)) =
= ∫_{0}^{λ} ⋯ ∫_{0}^{λ} ∫_{0}^{η} ⋯ ∫_{0}^{η} Δ_{λ,η}^{k−ρ,k−ρ} ×
×f_{ρ}(x + s_{1} + ⋯ + s_{ρ}, y + t_{1} + ⋯ + t_{ρ})ds_{1} ⋯ ds_{ρ}dt_{1} ⋯ dt_{ρ}.

This immediately implies estimate (8).

From (8) it follows that

ω_{k,k}(δ_{1}, δ_{2}; f_{0})_{p} ≤ (δ_{1}δ_{2})^{ρ}ω_{k−ρ,k−ρ}(δ_{1}, δ_{2}; f_{ρ})_{p}  (9)

for all non-negative numbers δ_{1}, δ_{2}.

3. Basic properties of the Bernstein singular integrals. Consider the entire functions g_{σ}, G_{σ,k} of exponential type of positive order σ, with positive integer parameters r, k given by

g_{σ}(z) := (\frac{1}{z} sin \frac{σz}{2r} )^{2r}, G_{σ,k}(ζ) := ∑_{μ=1}^{k}(−1)^{μ} \frac{1}{μ}\binom{k}{μ}g_{σ}(\frac{ζ}{μ}).

Write

γ_{σ} := ∫_{R} g_{σ}(t) dt = 2(\frac{σ}{2r} )^{2r−1} ∫_{0}^{∞} (\frac{sin v}{v} )^{2r}dv.

Suppose that f ≡ f(·, ·) is a (complex-valued) function defined and measurable on R^{2}, such that

I(f) ≡ ∫∫_{R^{2}} \frac{|f(u, v)|}{(1 + u^{2r})(1 + v^{2r})} du dv < ∞.  (10)

Take arbitrary τ > 0, l ∈ N and complex numbers z_{1} = x_{1} + iy_{1}, z_{2} = x_{2} + iy_{2} (x_{m}, y_{m} ∈ R). Introduce the singular integrals

J_{σ,τ} [f](z_{1}, z_{2}) :=(γ_{σ}γ_{τ})^{−1}∫∫_{R^{2}}f(u, v)G_{σ,k}(z_{1} − u)G_{τ,l}(z_{2} − v) du dv, (11)
J_{σ}^{1}[f](z_{1}, v) := γ_{σ}^{−1} ∫_{R} f(u, v)G_{σ,k}(z_{1} − u)) du (v ∈ R),  (12)
J_{τ}^{2}[f](u, z_{2}) := γ_{τ}^{−1} ∫_{R} f(u, v)G_{τ,l}(z_{2} − v) dv (u ∈ R),  (13)

which are due to S.Bernstein ([1], pp.421-432).

The double integral (11) exists in the Lebesgue sense for all complex z_{1}, z_{2}; the single (Lebesgue) integrals (12) and (13) exist for every complex z_{1} [resp. z_{2}] and almost every real v [resp. u]. More precise assertions will be presented below.

Proposition 4. The relation

J_{σ,τ} [f](z_{1}, z_{2}) = O((1 + x_{1}^{2r})(1 + x_{2}^{2r})e^{σ|y_{1}|+τ|y_{2}|}I(f))  (14)

holds uniformly in x_{1}, y_{1}, x_{2}, y_{2} ∈ R; moreover, J_{σ,τ} [f] coincides with some bivariate entire function.

Proof. Putting

S(z_{1}, z_{2}) := ∫∫_{R^{2}} f(u, v)g_{σ}(z_{1} − u)g_{τ} (z_{2} − v) du dv,

we can, formally, write

S(z_{1}, z_{2}) = ( ∫_{x_{1}−1}^{x_{1}+1} ∫_{x_{2}−1}^{x_{2}+1} + ∫_{x_{1}+1}^{∞} ∫_{x_{2}+1}^{∞} +
+ ∫_{x_{1}+1}^{∞} ∫_{x_{2}−1}^{x_{2}+1} + ⋯)f(u, v)g_{σ}(z_{1} − u)g_{τ} (z_{2} − v) du dv =
= T_{0}(z_{1}, z_{2}) + T_{1}(z_{1}, z_{2}) + T_{2}(z_{1}, z_{2}) + · · · .

In view of (10), f ∈ L_{loc})R^{2}). Hence the term T_{0}(z_{1}, z_{2}) exists because g_{σ} ∈ E_{σ}, g_{τ} ∈ E_{τ}. Furthermore,

|T_{0}(z_{1}, z_{2})| ≤ ∫_{x_{1}−1}^{x_{1}+1} ∫_{x_{2}−1}^{x_{2}+1} |f(u, v)|(\frac{σ}{2r} )^{2r}e^{σ|y_{1}|}(\frac{τ}{2r} )^{2r}e^{τ|y_{2}|}du dv ≤

≤ (\frac{στ}{4r^{2}} )^{2r}e^{σ|y_{1}|+τ|y_{2}|} ∫_{x_{1}−1}^{x_{1}+1} ∫_{x_{2}−1}^{x_{2}+1} \frac{|f(u, v)|}{(1 + u^{2r})(1 + v^{2r})}du dv ·

·{1 + (|x_{1}| + 1)^{2r}}{1 + (|x_{2}| + 1)^{2r}} ≤

≤ 4(\frac{στ}{4r^{2}} )^{2r}e^{σ|y_{1}|+τ|y_{2}|}I(f)(|x_{1}| + 1)^{2r}(|x_{2}| + 1)^{2r}.

Next

|T_{1}(z_{1}, z_{2})|≤∫_{x_{1}+1}^{∞}∫_{x_{2}+1}^{∞}|f(u, v)|e^{σ|y_{1}|}(x_{1}−u)^{−2r}e^{τ|y_{2}|}(x_{2}−v)^{−2r} du dv ≤

≤ e^{σ|y_{1}|+τ|y_{2}|} ∫_{x_{1}+1}^{∞} ∫_{x_{2}+1}^{∞} \frac{|f(u, v)|}{(1 + u^{2r})(1 + v^{2r})}du dv ·

·{1 + (|x_{1}| + 1)^{2r}}{1 + (|x_{2}| + 1)^{2r}} ≤
≤ 4e^{σ|y_{1}|+τ|y_{2}|}I(f)(|x_{1}| + 1)^{2r}(|x_{2}| + 1)^{2r},

|T_{2}(z_{1}, z_{2})|≤∫_{x_{1}+1}^{∞} ∫_{x_{2}−1}^{∞} |f(u, v)|e^{σ|y_{1}|}(x_{1}−u)^{−2r}(\frac{τ}{2r} )^{2r}e^{τ|y_{2}|}du dv ≤

≤ 4(\frac{τ}{2r} )^{2r}e^{σ|y_{1}|+τ|y_{2}|}I(f)(|x_{1}| + 1)^{2r}(|x_{2}| + 1)^{2r}, etc.

Thus

S(z_{1}, z_{2}) = O((1 + x_{1}^{2r})(1 + x_{2}^{2r})e^{σ|y_{1}|+τ|y_{2}|}I(f))  (15)

uniformly in x_{1}, y_{1}, x_{2}, y_{2} ∈ R.

Evidently, the left-hand side of (15) can be replaced by

S_{μ,ν}(z_{1}, z_{2}) := ∫∫_{R^{2}} f(u, v)g_{σ}(\frac{z_{1} − u}{μ})g_{τ} (\frac{z_{2} − v}{ν} )du dv (μ, ν ∈ N).

Further,

J_{σ,τ} [f](z_{1}, z_{2}) = \frac{1}{γ_{σ}γ_{τ}} ∑_{μ=1}^{k} ∑_{ν=1}^{l}(−1)^{μ+ν} \binom{k}{m}\binom{l}{ν}\frac{1}{μν} S_{μ,ν}(z_{1}, z_{2}).

Therefore, the uniform relation (14) is established. Applying the Lebesgue dominated convergence theorem, it can easily be proved that the function J_{σ,τ} [f] is continuous at every point belonging to the space of pairs of complex numbers.

The (Lebesgue) integrals

∫_{−n}^{n} ∫_{−n}^{n} f(u, v)g_{σ}(\frac{z_{1} − u}{μ})g_{τ} (\frac{z_{2} − v}{ν} )du dv (μ, ν, n ∈ N)

define some entire functions F_{μ,ν,n} of two complex variables z_{1}, z_{1}, because they have the partial derivatives

\frac{∂F_{μ,ν,n}}{∂z_{1}} (z_{1}, z_{2}) = \frac{1}{μ} ∫_{−n}^{n} ∫_{−n}^{n} f(u, v)g′_{σ}(\frac{z_{1} − u}{μ})g_{τ} (\frac{z_{2} − v}{ν} )du dv

and

\frac{∂F_{μ,ν,n}}{∂z_{2}} (z_{1}, z_{2}) = \frac{1}{ν} ∫_{−n}^{n} ∫_{−n}^{n} f(u, v)g_{σ}(\frac{z_{1} − u}{μ})g′_{τ} (\frac{z_{2} − v}{ν} )du dv

An easy calculation shows that lim_{n→∞} F_{μ,ν,n}(z_{1}, z_{2}) = S_{μ,ν}(z_{1}, z_{2}) uniformly in z_{1}, z_{2} belonging to two arbitrary bounded sets of complex numbers. Hence the well-known Weierstrass theorem ensures that all S_{μ,ν} are entire functions of two variables. Consequently, J_{σ,τ} [f] is a bivariate entire function for every fixed pair (σ, τ) of positive numbers. Obviously, in view of (14), J_{σ,τ} [f](·, ·) ∈ E_{σ,τ}.

Proposition 5. The singular integral (12), with a positive parameter σ, has the following basic properties:

(i) J_{σ}^{1}[f](·, v) ∈ E_{σ} for almost every v ∈ R,

(ii) J_{σ}^{1}[f](z_{1}, ·) ∈ L_{loc}(R) for every complex z_{1},

(iii) J_{σ}^{1}[f](·, ·) ∈ L_{loc}(R^{2}).

Proof of (i). By Fubini’s theorem, condition (10) implies

| ∫_{−∞}^{∞} \frac{f(u, v)}{1 + u^{2r}} du| < ∞ and ∫_{−∞}^{∞} \frac{|f(u, v)|}{1 + u^{2r}}du < ∞

for almost every v ∈ R. Hence, for these v and z_{1} = x_{1} + iy_{1},

φ_{μ}(z_{1}, v) ≡ ∫_{−∞}^{∞} |f(u, v)g_{σ}(\frac{z_{1} − u}{μ})|du < ∞ (μ = 1, …, k)

and

Y (z_{1}, v) ≡ ∫_{R} f(u, v)G_{σ,k}(z_{1} − u)du =

= ∑_{μ=1}^{k}(−1)^{μ} \frac{1}{μ}\binom{k}{μ} ∫_{−∞}^{∞} f(u, v)g_{σ}(\frac{z_{1} − u}{μ})du.

Writing

φ_{μ}(z_{1}, v) = ( ∫_{x_{1}−1}^{x_{1}+1} + ∫_{x_{1}+1}^{∞} + ∫_{−∞}^{x_{1}−1} )|f(u, v)g_{σ}(\frac{z_{1} − u}{μ})|du

and proceeding as in the proof of Proposition 4, we obtain

φ_{μ}(z_{1}, v) = O((1 + x_{1}^{2r})e^{(σ/μ)|y_{1}|} ∫_{R} |f(u, v)|(1 + u^{2r})^{−1}du) (16)

uniformly in x_{1}, y_{1}, v. This immediately implies the O-relation for Y (z_{1}, v), in which the right side is as in (16) with μ = 1.

Further, Y (·, v) are entire functions, by the Weierstrass theorem. Consequently, Y (·, v) ∈ E_{σ} for almost every v. Thus the assertion (i) is obtained.

Proof of (ii). Given μ, n ∈ N and any complex number z_{1} = x_{1} + iy_{1}, we have

∫_{−∞}^{∞} ∫_{−n}^{n} |f(u, v)g_{σ}(\frac{z_{1} − u}{μ})|du dv ≤

≤ ∫_{x_{1}−1}^{x_{1}+1} ∫_{−n}^{n} |f(u, v)|(\frac{σ}{2r} )^{2r}e^{σ|y_{1}|/μ}du dv +

+( ∫_{−∞}^{x_{1}−1} ∫_{−n}^{n} + ∫_{x_{1}+1}^{∞} ∫_{−n}^{n} )|f(u, v)|(\frac{μ}{x_{1} − u})^{2r}e^{σ|y_{1}|/μ}du dv < ∞.

Applying Fubini’s theorem we conclude that Y (z_{1}, ·) ∈ L_{loc}(R), which implies (ii). Proof of (iii). Let μ, n ∈ N, x_{1} ∈ R. By Tonelli’s theorem,

∫_{−∞}^{∞} ∫_{−n}^{n} ∫_{−n}^{n} |f(u, v)g_{σ}(\frac{x_{1} − u}{μ})|du dv dx_{1} =

= ∫_{−n}^{n} \{ ∫_{−∞}^{∞} ∫_{−n}^{n} |f(u, v)|g_{σ}(\frac{x_{1} − u}{μ})du dv\}dx_{1}.  (17)

The inner double integral of the right-hand side of (17) does not exceed

∫_{x_{1}−1}^{x_{1}+1} ∫_{−n}^{n} |f(u, v)|(\frac{σ}{2r} )^{2r}du dv +

+( ∫_{−∞}^{x_{1}−1} ∫_{−n}^{n} + ∫_{x_{1}+1}^{∞} ∫_{−n}^{n} )|f(u, v)|(\frac{μ}{x_{1} − u})^{2r}du dv ≤

≤ (\frac{σ}{2r} )^{2r} ∫_{−n−1}^{n+1} ∫_{−n}^{n} |f(u, v)|du dv + 2μ^{2r}(1 + n)^{2r})(1 + n^{2r})I(f).

Thus the left-hand side of (17) is finite.

Now, the Fubini theorem ensures that Y (·, ·) ∈ L_{loc}(R^{2}), and (iii) follows.

Analogous properties of the singular integral (13) can easily be formulated. They will also be used in the sequel.

4. Approximations by the sums of pseudoentire functions. Let f and the operators J_{σ,τ}, J_{σ}^{1}, J_{τ}^{2} be as in Section 3. Putting

Φ(z_{1}, v) := −J_{σ,τ} [f](z_{1}, v) − J_{σ}^{1}[f](z_{1}, v),
Ψ(u, z_{2}) := J_{τ}^{2}[f](u, z_{2}),

where z_{1} = x_{1} + iy_{1}, z_{2} = x_{2} + iy_{2} (x_{j}, y_{j} ∈ R), and u, v ∈ R, it can easily be observed (see Propositions 4,5) that Φ(·, ·) ∈ W_{σ}^{1}, Ψ(·, ·) ∈ W_{τ}^{2}.

Introduce the approximant Q_{σ,τ} [f](x_{1}, x_{2}) := Φ(x_{1}, x_{2})+Ψ(x_{1}, x_{2}), which is defined almost everywhere on R^{2} and Q_{σ,τ} [f] ∈ L_{loc}(R^{2}). Assuming that k, l ≤ 2r − 2 (k, l, r ∈ N) and 1 ≤ p ≤ ∞, we will present some Jackson type estimates.

In discussions below, the symbols C_{j}(q, …) will mean positive constants depending on the indicated parameters q, …, only.

Theorem 1. Under the restriction 0 < σ, τ < ∞, we have

∥f − Q_{σ,τ} [f]∥_{p} ≤ 2^{k+l}C_{1}(r)ω_{k,l}(1/σ, 1/τ; f)_{p}.

Proof. Take into account the real numbers x_{1}, x_{2} for which J_{σ}^{1}[f](·, x_{2})∈ E_{σ}, J_{τ}^{2}[f](x_{1}, ·) ∈ E_{τ}. In this case,

f(x_{1}, x_{2}) − Q_{σ,τ} [f](x_{1}, x_{2}) =
= \frac{(−1)^{k+l}}{γ_{σ}γ_{τ}} ∫∫_{R^{2}} Δ_{s,t}^{k,l}f(x_{1}, x_{2})g_{σ}(s)g_{τ} (t) ds dt.  (18)

Hence, by Minkowski’s inequality and (5),

γ_{σ}γ_{τ} ∥f − Q_{σ,τ} [f]∥_{p} ≤
≤ ω_{k,l}(1/σ, 1/τ; f)_{p} ∫∫_{R^{2}}(σ|s| + 1)^{k}(τ|t| + 1)^{l}g_{σ}(s)g_{τ} (t) ds dt.

Further,

\frac{1}{γ_{σ}} ∫_{R}(σ|s| + 1)^{k}g_{σ}(s) ds ≤ \frac{2}{γ_{σ}} \{2^{k} ∫_{0}^{1/σ} g_{σ}(s) ds +

+2^{k}σ^{k} ∫_{1/σ}^{∞} s^{k}g_{σ}(s) ds\} ≤ 2^{k}\{1 + \frac{2}{γ_{σ}} σ^{k} ∫_{1/σ}^{∞} s^{k−2r}ds\} ≤

≤ 2^{k}\{1 + \frac{2}{γ_{σ}} σ^{2r−1}\}.

Observing that

γ_{σ} ≥ 2(\frac{σ}{2r} )^{2r−1} ∫_{0}^{π/2} (\frac{2}{π} )^{2r}dv = 2(\frac{σ}{πr} )^{2r−1},

we obtain

\frac{1}{γ_{σ}} ∫_{R}(σ|s| + 1)^{k}g_{σ}(s) ds ≤ 2^{k}\{1 + (πr)^{2r−1}\}.

Thus

∥f − Q_{σ,τ} [f]∥_{p} ≤ ω_{k,l}(1/σ, 1/τ; f)_{p} · 2^{k+l}{1 + (πr)^{2r−1}}^{2},

and the proof is complete.

Corollary 1. Let f satisfy all conditions of Proposition 2, with some positive integers m ≤ k, n ≤ l. Then

∥f − Q_{σ,τ} [f]∥_{p} ≤ 2^{k+l}C_{1}(r)σ^{−m}τ^{−n}ω_{k−m,l−n}(1/σ, 1/τ; f^{(m,n)})_{p}

This estimate is an immediate consequence of Theorem 1 and inequality (7).

Corollary 2. Consider the bivariate functions f_{0}, f_{1}, …, f_{ρ} defined in Proposition 3, with f_{0}(·, 0), f_{0}(0, ·)∈L_{loc}(R) and f_{ρ−j}(·, 0), f_{ρ−j}(0, ·)∈ C(R) when 1 ≤ j ≤ ρ − 1. Suppose that for some non-negative numbers a, b < 2r − ρ − 1 and for j = 1, …, ρ the relations

f_{ρ}(x, y) = O((1 + |x|^{a})(1 + |y|^{b})),
f_{ρ−j}(x, 0) = O(1 + |x|^{a+j}), f_{ρ−j}(0, y) = O(1 + |y|^{b+j}))

hold uniformly in x, y ∈ R. Then if k = l ≥ ρ, the function f = f_{0} and its approximant Q_{σ,τ} [f] have weak derivatives of order ρ − 1 everywhere on R^{2} and

∥f^{(μ)} − Q_{σ,τ}^{(μ)}[f]∥_{p} ≤ C_{2}(k, r)(στ)^{μ−ρ}ω_{k−ρ+μ,k−ρ+μ}(\frac{1}{σ} , \frac{1}{τ}; f_{ρ})_{p}

for μ = 0, …, ρ − 1. Under the additional assumption f_{ρ} ∈ C(R^{2}), also the derivatives f^{(ρ)} and Q_{σ,τ}^{(ρ)} [f] exist on R^{2} and the last inequality remains valid for μ = ρ.

Indeed, an easy calculation shows that

f^{(μ)}(x, y) = f_{μ}(x, y) and Q_{σ,τ}^{(μ)}[f](x, y) = Q_{σ,τ} [f_{μ}](x, y)

for all (x, y) ∈ R^{2}, whenever 0 ≤ μ ≤ ρ − 1 or μ = ρ and f_{ρ} ∈ C(R^{2}). By Theorem 1 and a suitable estimate analogous to (9), we obtain

∥f_{μ} − Q_{σ,τ} [f]∥_{p} ≤ C_{2}(k, r)ω_{k,k}(1/σ, 1/τ; f_{μ})_{p} ≤
≤ C_{2}(k, r)(στ)^{μ−ρ}ω_{k−ρ+μ,k−ρ+μ}(1/σ, 1/τ; f_{ρ})_{p} if 0 ≤ μ ≤ ρ.

The conclusion is now evident.

Theorem 2. Let f ∈ H_{φ,ψ,p}^{(k,l)}, where φ, ψ are as in Section 1. Denote by α and β two non-negative numbers such that t^{−α}φ(t), t^{−β}ψ(t) are nondecreasing on (0, 1]. Then, for σ, τ ≥ 1,

∥f − Q_{σ,τ} [f]∥_{α,β,p}^{(k,l)} ≤ C_{3}(k, l, r)∥f∥_{φ,ψ,p}^{(k,l)}{σ^{α}φ(1/σ) + τ^{β}ψ(1/τ)}.

226 ROMAN TABERSKI

Proof. Write D_{σ,τ} (x, y) := f(x, y)−Q_{σ,τ} [f](x, y) (x, y ∈ R) In view of (18),

Δ_{λ,η}^{k,l}D_{σ,τ} (x, y) = \frac{1}{γ_{σ}γ_{τ}} ×

× ∫∫_{R^{2}} g_{σ}(s)g_{τ} (t) ∑_{μ=0}^{k} ∑_{ν=0}^{l}(−1)^{μ+ν} \binom{k}{μ}\binom{l}{ν}Δ_{s,t}^{k,l}f(x + μλ, y + νη)ds dt

for all λ, η ∈ R and almost all (x, y) ∈ R^{2}; whence, by Minkowski’s inequality,

∥Δ_{λ,η}^{k,l}D_{σ,τ} ∥ ≤ \frac{2^{k+l}}{γ_{σ}γ_{τ}} ∫∫_{R^{2}} g_{σ}(s)g_{τ} (t)∥Δ_{s,t}^{k,l}f∥_{p}ds dt.

Proceeding now as in the proof of Theorem 1, we obtain

∥Δ_{λ,η}^{k,l}D_{σ,τ} ∥_{p} ≤ 4^{k+l}C_{1}(r)∥f∥_{φ,ψ,p}^{(k,l)}φ(1/σ)ψ(1/τ).  (19)

On the other hand, equality (18) leads to

Δ_{λ,η}^{k,l}D_{σ,τ} (x, y) = \frac{1}{γ_{σ}γ_{τ}} ×

× ∫∫_{R^{2}} g_{σ}(s)g_{τ} (t) ∑_{μ=0}^{k} ∑_{ν=0}^{l}(−1)^{μ+ν} \binom{k}{μ}\binom{l}{ν}Δ_{λ,η}^{k,l}f(x + μs, y + νt)ds dt

for almost all (x, y) ∈ R^{2}, which implies

∥Δ_{λ,η}^{k,l}D_{σ,τ} ∥_{p} ≤ 2^{k+l}∥f∥_{φ,ψ,p}^{(k,l)}φ(λ)ψ(η) when λ, η ∈ (0, 1].  (20)

Next, ∥D_{σ,τ} ∥_{α,β,p}^{(k,l)} = sup {Ω(λ, η) : 0 < λ, η ≤ 1}, where Ω(λ, η) = λ^{−α}η^{−β}∥Δ_{λ,η}^{k,l}D_{σ,τ} ∥_{p}. If 1/σ ≤ λ ≤ 1 and 1/τ ≤ η ≤ 1 then, by (19), Ω(λ, η) ≤ 4^{k+l}C_{1}(r)∥f∥_{φ,ψ,p}^{(k,l)}σ^{α}φ(1/σ)τ^{β}ψ(1/τ). From (20) it follows that

Ω(λ, η) ≤ 2^{k+l}∥f∥_{φ,ψ,p}^{(k,l)} ×

× \begin{cases}σ^{α}φ(1/σ)τ^{β}ψ(1/τ) & if 0 < λ ≤ 1/σ, 0 < η ≤ 1/τ, \\ σ^{α}φ(1/σ) & if 0 < λ ≤ 1/σ, 1/τ ≤ η ≤ 1, \\ τ^{β}ψ(1/τ) & if 1/σ ≤ λ ≤ 1, 0 < η ≤ 1/τ. \end{cases}

Hence

∥D_{σ,τ} ∥_{α,β,p}^{(k,l)} ≤ 2^{k+l}∥f∥_{φ,ψ,p}^{(k,l)}{2^{k+l}C_{1}(r)σ^{α}φ(1/σ)τ^{β}ψ(1/τ) +
+σ^{α}φ(1/σ)τ^{β}ψ(1/τ) + σ^{α}φ(1/σ) + τ^{β}ψ(1/τ)},

and the proof is completed (cf. [4], Theorem 2).

Corollary 3. If φ(δ) = δ^{α′}, ψ(δ) = δ^{β′} for all δ ∈ (0, 1] then, in the case α′ ≥ α ≥ 0 and β′ ≥ β ≥ 0, ∥f − Q_{σ,τ} [f]∥_{α,β,p}^{(k,l)} = O(σ^{α−α′} + τ^{β−β′}) uniformly in σ, τ ≥ 1.

References

1. S.N.Bernstein, Collected Papers, vol. II. (Russian) Izdat. Akad. Nauk SSSR, Moscow, 1954.

2. Yu.A.Brudnyǐ, Approximation of functions of n variables by quasipolynomials. (Russian) Izv. Akad. Nauk SSSR, Ser. Mat. 34(1970), 564-583.

3. H.Gonska and K.Jetter, Jackson-type theorems on approximation by trigonometric and algebraic pseudopolynomials. J. Approx. Theory 48(1986), 396-406.

4. R.Taberski, Approximation properties of the integral Bernstein operators and their derivatives in some classes of locally integrable functions. Funct. Approx. Comment. Math. 21 (in press).

5. A.F.Timan, Theory of Approximation of Functions of a Real Variable. (Russian) Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1960.

(Received 20. XI. 1992)

Author’s address: Institute of Mathematics Adam Mickiewicz University Matejki 48/49 60-769 Poznań, Poland