Periodic Besov Spaces and Generalized Moduli of Smoothness
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Periodic Besov Spaces and Generalized Moduli of Smoothness S. Yu. Artamonov1* , K. V. Runovskii2** , and H.-J. Schmeisser3*** 1
National Research University Higher School of Economics, Moscow, 101000 Russia 2 Lomonosov Moscow State University, Moscow, 119991 Russia 3 Friedrich Schiller University of Jena, Jena, 07743 Germany Received May 3, 2020; in final form, May 3, 2020; accepted May 14, 2020
DOI: 10.1134/S0001434620090321 Keywords: Besov space, strong approximation, modulus of smoothness.
The purpose of this paper is to describe the periodic Besov spaces in terms of generalized moduli of smoothness and approximation errors. We consider the following five parameters: the dimension d ∈ N; the Lp -metric and the scale of spaces Lp := Lp (Td ) with 0 < p ≤ +∞, where Td = [0; 2π)d ; a generator ϕ(ξ), ξ ∈ Rd , of an approximation method, which belongs to the class K of complex-valued functions ϕ with compact support, which means that r(ϕ) = sup{|ξ| : ϕ(ξ) = 0} < +∞, such that ϕ(−ξ) = ϕ(ξ) for all ξ ∈ Rd , ϕ(0) = 1, and the Fourier transform of ϕ belongs to the space L1 (Rd ); next, a smoothness generator ψ(ξ), ξ ∈ Rd , which belongs to the class Hα , α > 0, of complex-valued functions ψ homogeneous of order α, continuous on Rd , infinitely differentiable on Rd \ {0}, and such that ψ(−ξ) = ψ(ξ) for all ξ ∈ Rd and ψ(ξ) = 0 for all ξ ∈ Rd \ {0}; and, finally, a generator θ(ξ), ξ ∈ Rd , of a modulus of smoothness, which belongs to the class G of complex-valued continuous functions θ(ξ) 2π-periodic in each variable on Rd and satisfying the conditions θ(−ξ) = θ(ξ),
ξ ∈ Rd ,
θ ∧ (0) = −1,
θ(0) = 0,
θ ∧ = {θ ∧ (ν)}ν∈Zd ∈ 1 ,
where θ ∧ denotes the sequence of Fourier coefficients. In [1], [2], and a number of other papers, the second-named author showed that many problems of approximation theory can be reduced to the study of three main objects constructed on the basis of the five parameters specified above. These objects are the families of linear polynomial operators Lσ;λ (f ; x) = (2N + 1)−d · (ϕ)
2N
f (tνN + λ) · Wσ (ϕ)(x − tνN − λ)
(1)
ν=0
generated by functions ϕ ∈ K, where λ ∈ ν∈Z , d
2N ν=0
≡
2N ν1 =0
···
2N
,
and
Rd , tνN
= 2πν/(2N + 1) for
W0 (ϕ)(h) = 1,
νd =0
k eikx , Wσ (ϕ)(h) = ϕ σ d
σ > 0;
k∈Z
the generalized polynomial K-functionals Kψ (f, δ)p = inf {f − T p + δα D(ψ)T p }, T ∈T1/δ
*
E-mail: [email protected] E-mail: [email protected] *** E-mail: [email protected] **
603
δ > 0,
(2)
604
ARTAMONOV et al.
where the operator D(ψ) (the so-called ψ-derivative) is defined on the harmonics by D(ψ)(eiν· )(x) = ψ(ν)eiνx ,
ν ∈ Zd ,
where T1/δ stands for the space of real-valued trigonometric polynomials of spherical degree less than or equal to 1/δ and the generalized moduli of smoothness is given by ∧ θ (ν)f (x + νh) δ ≥ 0. (3) ωθ (f, δ)p = sup , |h|≤δ
p
ν∈Zd
The families of linear polynomial operators (FLPOs) (1) were systematically studied in [3], [4], and (ϕ) many other works. Recall that an FLPO {Lσ;λ } converges in Lp ,
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