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Original Research ARTICLE

Front. Appl. Math. Stat., 28 March 2017 | https://doi.org/10.3389/fams.2017.00004

Characterization of Local Besov Spaces via Wavelet Basis Expansions

  • 1Department of Theory of Functions, Institute of Mathematics of NAS of Ukraine, Kyiv, Ukraine
  • 2Institut für Mathematik, Universität zu Lübeck, Lübeck, Germany

In this paper we deal with local Besov spaces of periodic functions of one variable. We characterize these spaces in terms of summability conditions on the coefficients in series expansions of their elements with respect to an orthogonal Schauder basis of trigonometric polynomials. We consider a Schauder basis that was constructed by using ideas of a periodic multiresolution analysis and corresponding wavelet spaces. As an interim result we obtain a characterization of local Besov spaces via operators of the orthogonal projection on the corresponding scaling and wavelet spaces. In order to achieve our new results, we substantially use a theorem on the discretization of scaling and wavelet spaces as well as a connection between local and usual classical Besov spaces. The corresponding characterizations are also given for the classical Besov spaces.

1. Introduction

One of the crucial problems in the theory of approximation is to describe the smoothness properties of functions by the behavior of the coefficients in their series expansions in terms of given bases or frames. Besov spaces and their generalizations are particularly suitable for such studies. Recent papers describing the smoothness of functions from these spaces by the decay of the coefficient sequences are e.g., Bazarkhanov [1] for Meyer wavelets, Dinh [2] for mixed B-splines, and Hinrichs et al. [3] for Faber-Schauder bases.

In the present paper we consider this problem for local Besov spaces of periodic functions of one variable with respect to some orthogonal trigonometric Schauder basis. Let us first give some motivation of our work. Let

n = -an(f)einx

be the Fourier series of some function fL2π2, an(f) the Fourier coefficients of f.

In view of the Parseval's equality, it is easy to obtain the following result about the description of the usual classical Besov spaces B2,θα of periodic functions: fB2,θα, α > 0, if and only if fL2π2 and the norm

{(|a0(f)|θ+n=1[2nα(2n1|k|<2n|ak(f)|2)1/2]θ)1/θ,1θ<,supn(|a0(f)|+2nα(2n1|k|<2n|ak(f)|2)1/2),θ=,    (1.1)

is finite.

On the other hand, local properties of functions from the Besov spaces can be investigated by expanding them in a series with respect to the Haar basis. To give a short description of these results let for n ∈ ℕ, ν(n) = {k: k = 0, …, 2n−1−1}, ν(0) = {0} be the sets of indices and {hn, k}, n ∈ ℕ0, k ∈ ν(n) be the Haar system. This system is an orthogonal Schauder basis in the space Lp([0, 1]), 1 ≤ p < ∞, and for every fLp([0, 1])

f(x)=n = 0kν(n)bn,k(f)hn,k(x),

in the sense of the norm of Lp([0, 1]) [4, Chap. 3].

Romanyuk [5] obtained necessary and sufficient conditions on the Fourier-Haar coefficients bn, k(f) at which functions from Lp([0, 1]) belong to the Besov spaces. Namely, let 1 ≤ p, θ < ∞, 0 < α < 1/p, then fBp,θα if and only if fLp([0, 1]) and the norm

(n = 0[2n(α-1/p+1/2)(kν(n)|bn,k(f)|p)1/p]θ)1/θ    (1.2)

is finite. In this case it is evident that the Fourier-Haar coefficients bn, k(f) describe the local behavior of the function f. Note that V. Romanyuk considered the multivariate case, but since in the present paper we investigate functions of one variable, we formulate his result only in the univariate case.

Our aim in this paper is to combine these two approaches and to describe local smoothness of periodic functions in terms of summability conditions on the Fourier coefficients with respect to an orthogonal Schauder basis of trigonometric polynomials in the space L2πp for all 1 ≤ p ≤ ∞. The local smoothness is understood in the sense of Besov spaces. We call these Besov-type spaces as local Besov spaces (see Subsection 2.1 for a definition).

Note that some results in this direction were obtained by Mhaskar and Prestin [6]. There expansions of functions from the local Besov spaces in series with respect to a system of trigonometric frames were considered and these spaces were described via coefficients of these expansions. However, this system is not a Schauder basis.

Let us sketch the main results of the present paper. Let ψ0 be a scaling function of a periodic multiresolution analysis (PMRA) generated by de la Vallée Poussin means and ψn, n ∈ ℕ, be corresponding wavelets [7]. Let Nn:=3·2n for n ∈ ℕ and N0: = N1. By ψn, s we denote shifts of ψn:

ψn,s(·):=ψn(·-sπNn),  n0, s=0,,2Nn-1.    (1.3)

We show that for a particular choice of ψn the system {ψn, s} constitutes an orthogonal trigonometric Schauder basis in the space L2πp, 1 ≤ p ≤ ∞, and a function fL2πp can be represented by a series (for more detailed information see Subsection 2.2)

f(x)=n = 0s = 02Nn-1an,s(f)ψn,s(x),  x,    (1.4)

converging in the norm of the space L2πp, where the coefficient functionals an, s(f) are Fourier coefficients of f with respect to the basis {ψn, s}:

an,s(f)=f,ψn,s=12π02πf(x)ψn,s(x)dx.

Because ψn is even (see Subsection 2.2 for definition) and from (1.3) we conclude that an, s(f) can be represented in the following way:

an,s(f)=12π02πf(x)ψn(sπNn-x)dx=(f*ψn)(sπNn),

where f * g means the convolution

(f*g)(x):=12π02πf(y)g(x-y)dy,  x.

For more information regarding trigonometric Schauder bases we refer to Lorentz and Saakyan [8], Prestin and Selig [7], Selig [9] and the references cited there.

Let I ⊂ ℝ, |I| < 2π, be some segment and n ∈ ℕ. By κ(I, n) we denote the set of indices s which satisfy the properties s = 0, …, 2Nn − 1 and there exists k ∈ ℤ such that the point (s+2kNn)πNn belongs to the segment I. For 1 ≤ p ≤ ∞, n ∈ ℕ0, and the segment I we define the following sequence

cn(I,p):={Nn1/2-1/p(sκ(I,n)|an,s(f)|p)1/p,1p<,Nn1/2maxsκ(I,n)|an,s(f)|,p=.    (1.5)

Let 1 ≤ p ≤ ∞, fL2πp, x0 ∈ [0, 2π), 0 < α < 1, 0 < θ ≤ ∞. Then the main result of this paper is written as follows: A function f belongs to the local Besov spaces Bp,θα(x0) if and only if there exists an interval I ⊂ ℝ, |I| < 2π, centered at x0, such that the norm

{(n = 02nθα|cn(I,p)|θ)1/θ,0<θ<, supn02nα|cn(I,p)|,θ=,

is finite.

We adopt the following convention regarding constants. The letters C, Ci, i = 1, 2, …, mean positive constants which may depend on parameters fixed for the spaces. Their values are not necessarily the same in different parts of the text. When constants depend on functions f, ζ or some intervals I, we indicate this in brackets.

The present paper is organized as follows: In Subsection 2.1 we define the local Besov spaces and formulate their connection with classical Besov spaces. In Subsection 2.2 we give definitions of the orthogonal trigonometric Schauder basis that we work with and describe expansions of functions from L2πp in a series with respect to this basis. In Section 3 we formulate the main results of this paper. In Section 4 we prove the main auxiliary statements. In Section 5 we give proofs of the main results of the present paper.

2. Preliminaries

As stated in Section 1, our aim in this paper is to describe the local Besov spaces of functions f in terms of summability conditions on the coefficients in a series expansion of f as in (1.4).

Let us first agree about the notation. As usual ℕ is reserved for the natural numbers, by ℕ0 we denote the natural numbers including 0, by ℤ the set of all integers and by ℝ the set of all real numbers.

Let 1 ≤ p ≤ ∞ and A ⊂ ℝ be a Lebesgue measurable set. By Lp(A) we denote the space of functions f : A → ℝ Lebesgue measurable on A with the finite norm

fA,p:={(A|f(t)|pdt)1/p,1p<,ess suptA|f(t)|,p=.

When A = [0, 2π), we understand by Lp([0, 2π)) the space of functions f defined on the segment [0, 2π) and extended 2π-periodically to the real line with natural modification for the norm f[0,2π),p:=(12π02π|f(t)|pdt)1/p. For simplicity, by L([0, 2π)) we denote the space of 2π-periodic continuous functions (equipped with ||f||[0, 2π), ∞ as its norm). Further, we will write L2πp instead of Lp([0, 2π)) and ||f||p instead of ||f||[0, 2π), p.

2.1. The Local Besov Spaces of Functions and Their Connection with Classical Besov Spaces

Let us introduce the local Besov spaces Bp,θα(x0) of periodic functions. For a function fLp(I) where I = [a, b] ⊆ ℝ we define the rth difference operator Δtr by

Δtrf(·)=k=0r(1)rk(rk)f(·+kt),  r,t,

and for 0 < δ < (ba)/r we define the modulus of smoothness by

ωI,r,p(f,δ):=sup0<tδΔtrf(·)[a,b-rδ],p.

If δ ≥ (ba)/r, we put

ωI,r,p(f,δ):=inff-PI,p,

where the infimum is taken over all algebraic polynomials of degree at most r−1.

It will be convenient for us to use a sequential version of the Besov spaces which we now define. For a sequence a={an}n=0 and numbers α, θ > 0 by ||a||θ, α we denote the following norm

aθ,α:={(n=0(2nα|an|)θ)1/θ,0<θ<,supn02nα|an|,θ=.

The notation abθ,α means that the norm ||a||θ,α is finite.

Let 1 ≤ p ≤ ∞, 0 < θ ≤ ∞, α > 0 and r = [α]+1. For x0 ∈ ℝ the local Besov space Bp,θα(x0) is the collection of functions f which satisfy the following properties:

(1) fL2πp;

(2) there is a non-degenerate interval I ⊂ ℝ, |I| < 2π, centered at x0, such that

{ωI,r,p(f,2-n)}n=0bθ,α.

The spaces Bp,θα(x0) were considered in Mhaskar and Prestin [6]. By periodicity we can restrict ourselves to points x0 ∈ [0, 2π).

In order to prove our main results, we use a connection between the local and the classical Besov spaces of periodic functions. Let us define these Besov spaces.

Let fL2πp, 1 ≤ p ≤ ∞. For δ > 0 we define the modulus of smoothness by

ωr,p*(f,δ):=sup0<tδΔtrf(·)p,

where, in contrast to the modulus of smoothness ωI, r, p(f, δ), the norm is taken over the entire period of f, using the periodicity of f in the case when the translates go outside of [0, 2π). Let 1 ≤ p ≤ ∞, α > 0, 0 < θ ≤ ∞ and r be some integer number greater than α. The classical Besov space Bp,θα* consists of functions f such that fL2πp and the sequence {ωr,p*(f,2-n)}n=0bθ,α for some integer r > α. The space Bp,θα* is independent of the choice of r as long as r > α (see [10, Theorem 10.1, Chapter 2]). One can find more information about Besov spaces in the monographs [10, 11]. The following statement about the connection between the two spaces mentioned above is proved in the paper [6].

Proposition A. Let 1 ≤ p ≤ ∞, fL2πp, α > 0, 0 < θ ≤ ∞, x0 ∈ [0, 2π). Then fBp,θα(x0) if and only if there exists an interval I ⊂ ℝ, |I| < 2π, centered at x0 such that for every infinitely differentiable onfunction ζ supported on I and extended as a-periodic function, the function fζ is in Bp,θα*.

2.2. Expansions in a Series

Let us first give some necessary definitions. Let

g1(x):={2,-π2+π6xπ2-π6,1-3(2|x|π-1)1+(3(2|x|π-1))2,π2-π6|x|π2+π6,0,π2+π6|x|π,    (2.1)

be a coefficient function given on the segment [−π, π] and let as above Nn=3·2n and Mn:=2n, n ∈ ℕ. By φn we denote the following function

φn(·):=12Nnk=-Nn-MnNn+Mng1(kπ2Nn)cosk·,

which is a modification of the de la Vallée Poussin kernel.

Let as above φn, s denote shifts of the function φn:

φn,s(·):=φn(·-sπNn), n, s = 0,,2Nn-1.

In Prestin and Selig [7] it is proved that the sequence of spaces {Vn}n=1, defined as

Vn:=span{φn,s,s = 0,,2Nn-1},

provides a PMRA in L2π2, i.e., the system {φn, s, s = 0, …, 2Nn−1} is a basis in the space Vn; VnVn + 1 for all n ∈ ℕ; it holds that closL2π2(nVn) = L2π2. The functions φn are called scaling functions of this PMRA and the spaces Vn are called scaling spaces.

The wavelet space Wn which is defined to be the orthogonal complement of Vn with respect to Vn + 1, i.e., Wn = Vn + 1Vn, is spanned by the translates of the function ψn [7]:

Wn=span{ψn,s,s = 0,,2Nn-1},

where

ψn(·)=1Nnk=-Mn2Nn-Mng2(kπNn)cos(Nn+k)·,

and

g2(x)={1+3xπ1+(3xπ)2,-π3xπ3,2-3(xπ-1)4+(3(xπ-1))2,π3<x2π-π3,    (2.2)

is a coefficient function given on the segment [-π3,2π-π3]. The functions ψn are called wavelets. Note that the functions g1 and g2 defined on the segments with lengths equal 2π can be extended to ℝ as 2π-periodic functions (and continuous since g1(−π) = g1(π) = 0 and g2(-π3)=g1(2π-π3)=0).

For simplicity of notations we denote ψ0: = φ1, N0: = N1 and for n ∈ ℕ0, s = 0, …, 2Nn−1,

t2Nn+s(·):=ψn,s(·).

For fL2πp, 1 ≤ p ≤ ∞, we define some trigonometric polynomial operators σn. Let n ∈ ℕ0, x ∈ ℝ and

σn(f,x):=k = 02Nn-2-1f,tktk(x),  n3,

where f,tk=12π02πf(x)tk(x)dx are Fourier coefficients of the function f with respect to the basis {tk}k = 0. It is convenient for us to put σn(f, x) ≡ 0 if n = 0, 1, 2.

We also use the following representation of the operators σn:

σn(f,x)=12π02πf(y)Kn(x,y)dy,    (2.3)

where Kn(x,y)=k = 02Nn-2-1tk(x)tk(y).

Using a similar technique as in Prestin and Selig [7], one can prove that the system of polynomials {tk}k = 0 is an orthogonal trigonometric Schauder basis in the space L2π. In view of Theorem 9 [4, p. 12] and Theorem 6 [4, p. 10] we get that {tk}k = 0 is a Schauder basis in the space L2πp, 1 ≤ p < ∞, and

σn(f,·)pCf(·)p,    (2.4)

with some constant C > 0.

By 𝕋n, n ∈ ℕ, we denote the set of all trigonometric polynomials of the form

T(x)=k = 0n(akcoskx+bksinkx), x.

From (2.3) and the representation of the kernel Kn [7, p. 421], it can be derived that

σn(T,·)=T(·),  TT2n-1.    (2.5)

Let

En(f)p:=infTTnf-Tp

be the best approximation of a function fL2πp by trigonometric polynomials from 𝕋n.

A sequence of linear operators Un: L2πpT2n, n ∈ ℕ0, is called a sequence of near best approximation (with the constant λ > 0) for L2πp if it satisfies the following condition:

f-Un(f)pλE2n-1(f)p.

For example, in the case p = ∞ the operators of de la Vallée Poussin Vn2n determine a sequence of near best approximation with the constant λ=43+23π [12, Chap. 5, §2].

The following lemma results from the properties (2.4) and (2.5).

Lemma 2.1. {σn(f)}n = 0 is a sequence of near best approximation (with some constant) for L2πp, 1 ≤ p ≤ ∞.

Further, for fL2πp, 1 ≤ p ≤ ∞, we define operators τn, n ∈ ℕ0, as follows:

τn(f,x)=σn+3(f,x)-σn+2(f,x), x.

From Lemma 2.1 it is easy to see that a function fL2πp, 1 ≤ p ≤ ∞, can be represented by the series

f(x)=n = 0τn(f,x), x,    (2.6)

where convergence is understood in the metric of the space L2πp.

Using the definition of the operators τn, we can represent them in the following form:

τ0(f,x)=k = 02N1-1f,tktk(x)=s = 02N0-1f,ψ0,sψ0,s(x),    (2.7)

and for n ∈ ℕ

τn(f,x)=2Nnk<2Nn+1-1f,tktk(x)=s = 02Nn-1f,ψn,sψn,s(x).    (2.8)

From (2.6)–(2.8) we get representation (1.4).

3. Formulations of the Main Results

In this section we formulate the main results of this paper. Let us first explain the relationship between the local Besov spaces Bp,θα(x0) and the behavior of the operators τn near the point x0. This behavior will be described by the condition that certain norms of the operators belong to a sequential version of the Besov spaces.

Theorem 3.1. Let 1 ≤ p ≤ ∞, fL2πp, x0 ∈ [0, 2π), 0 < α < 1, 0 < θ ≤ ∞. The following statements are equivalent:

(a) fBp,θα(x0);

(b) There exists an interval I ⊂ ℝ, |I| < 2π, centered at x0 such that the sequence {τn(f,·)I,p}n=0bθ,α;

(c) There exists an interval I ⊂ ℝ, |I| < 2π, centered at x0 such that for every infinitely differentiable onfunction ζ supported on I and extended as a 2π-periodic function, the sequence {τn(fζ,·)p}n=0bθ,α.

Let further for 1 ≤ p ≤ ∞ and a segment I, {cn(I,p)}n=0 be a sequence defined by the formula (1.5). And let ζ be some infinitely differentiable on ℝ function supported on I and extended as a 2π-periodic function. By {dn(I,ζ,p)}n=0 we denote the sequence:

dn(I,ζ,p):={Nn1/2-1/p(s = 02Nn-1|an,s(fζ)|p)1/p,1p<,Nn1/2maxs = 0,,2Nn-1|an,s(fζ)|,p=.

Theorem 3.2. Let 1 ≤ p ≤ ∞, fL2πp, x0 ∈ [0, 2π), 0 < α < 1, 0 < θ ≤ ∞. The following statements are equivalent:

(a) fBp,θα(x0);

(b) There exists an interval I ⊂ ℝ, |I| < 2π, centered at x0 such that the sequence {cn(I,p)}n = 0bθ,α;

(c) There exists an interval I ⊂ ℝ, |I| < 2π, centered at x0 such that for every infinitely differentiable onfunction ζ supported on I and extended as a 2π-periodic function, the sequence {dn(I,ζ,p)}n = 0bθ,α.

This theorem is the discrete version of Theorem 3.1 in the sense that the L2πp norm of the operators τn is replaced by a corresponding discrete norm (see Theorem 4.1).

For classical Besov spaces Bp,θα*, α > 0, we can obtain a result similar to Theorems 3.1 and 3.2 which is essentially of the same kind as (1.1) and (1.2). To formulate this equivalence we introduce the following sequence

an(p):={Nn1/2-1/p(s = 02Nn-1|an,s(f)|p)1/p,1p<,Nn1/2maxs = 0,,2Nn-1|an,s(f)|,p=.

Theorem 3.3. Let 1 ≤ p ≤ ∞, fL2πp, α > 0, 0 < θ ≤ ∞. The following statements are equivalent:

(a) fBp,θα*;

(b) The sequence {τn(f,·)p}n = 0bθ,α;

(c) The sequence {an(p)}n = 0bθ,α.

4. The Main Auxiliary Statements

4.1. A Property of the Kernel Kn(x, y)

In this subsection we present some estimates for the kernel Kn(x, y). Further, by V[f] we mean the total variation of a function f:ℝ → ℝ and by Vab[f] the total variation of f defined on the segment [a, b].

Lemma 4.1. There exists a constant C > 0 such that for every n ∈ ℕ, n ≥ 3:

|Kn(x,y)|C2n|x-y|2,  0<|x-y|2π.

Proof. Let N = 3·2n−2, M = 2n−2, n ≥ 3. From Selig [9, pp. 91–93] it is known that

Kn(x,y)=QNM(y-x)+cosN(y+x) RM(y-x),    (4.1)

where

QNM(x)=1+2k=1N-Mcos kx + k = -M+1M-1(M-k)2M2+k2 cos (N+k)x            =k = -N+MN-Mcos kx                +12|k|=N-M+1N+M-1(M-(|k|-N))2M2+(|k|-N)2 cos kx,RM(x)=1+2k = 1M-1M2-k2M2+k2 cos kx            =k = -M+1M-1M2-k2M2+k2 cos kx.

Defining continuous coefficient functions by

g(x)={1-x21+x2,-1x1,0,          else;
fβ(x)={1,|x|1-β,(β-(|x|-1))22(β2+(|x|-1)2),1-β|x|1+β,0,else,

where β ∈ (0, 1), we can rewrite the polynomials RM and QNM in the following forms:

RM(x)=kg(kM) coskx,  QNM(x)=kfMN(kN) coskx.

Let us first estimate |RM(yx)|. Using Proposition 2.2 [13], for 0 < |xy| ≤ 2π we get

|RM(yx)|=|kg(kM) cosk(yx)|                           C|xy|2k|Δ2g(kM)|,    (4.2)

where Δ2g(kM)=g(kM)-2g(k+1M)+g(k+2M).

Then, using the mean value theorem, we obtain that there exist points lk ∈ (k, k + 1), k ∈ ℤ, such that

Δ2g(kM)=g(kM)-2g(k + 1M)+g(k+2M)               =1M(g(lk + 1M)-g(lkM)).

Since the system of the points {lkM,k} is some partition of the real line and g is a function having a first derivative of finite total variation V[g′], it holds that

k|Δ2g(kM)|=k1M|(g(lk+1M)g(lkM))|                             1MV[g]=C2nV[g].    (4.3)

Since g′ = 0 outside of the segment [−1, 1], it holds that V[g]=V-11[g].

From the first and second derivatives

g(x)= -4x(1 + x2)2,     g(x)=4(3x2-1)(1+x2)3,

we see that g″ = 0 at the points -33 and 33 and g′ is monotonously increasing on [-1,-33] and on [33,1] and monotonously decreasing on [-33,33]. Therefore, the extrema are at -33 and 33:

supx[1,1]g(x)=g(33)=g(33)=334.

Since g′ (−1) > 0 and g′(1) < 0, the total variation of g′ is

V[g]=V-11[g]=4supx[-1,1]g(x)=33.    (4.4)

From (4.2), (4.3) and (4.4), we get

|RM(y-x)|C2n|x-y|2,  0<|x-y|2π.    (4.5)

Analogously, we can estimate |QNM(y-x)|. Let us calculate V[fβ] where

fβ(x)={-β(β2-(x-1)2)β2+(x - 1)2,1-βx1+β,β(β2-(-x-1)2)β2+(-x -1)2,-1-βx-1+β,0,else.

Since fβ=0 outside of the segments [−1 − β, −1 + β] and [1 − β, 1 + β] and fβ is an odd function, it holds that V[fβ]=2V-1-β-1+β[fβ].

From the second derivative

fβ(x)=2β(1+x)(-3β2+(1+x)2)(β2+(1 + x)2)3,   x[-1-β,-1+β],

we can see that fβ=0 at the point −1 and fβ is monotonously increasing on [−1−β, −1] and monotonously decreasing on [−1, −1+β]. This means that

supx[-1-β,-1+β]fβ(x)=fβ(-1)=1β.

Since fβ(-1-β)=fβ(-1+β)=0, it holds that

V-1-β-1+β[fβ]=2supx[-1-β,-1+β]fβ(x)=2β,

and the total variation of fβ is

V[fβ]=4β.    (4.6)

Then, using Proposition 2.2 [13], for 0 < |xy| ≤ 2π we get

|QNM(yx)|=|kfMN(kN)cosk(yx)|                            C|xy|2k|Δ2fMN(kN)|,    (4.7)

where Δ2fMN(kN)=fMN(kN)-2fMN(k+1N)+fMN(k+2N).

From the mean value theorem it follows that there exist points sk ∈ (k, k + 1), k ∈ ℤ, such that

Δ2fMN(kN)=1N(fMN(sk+1N)fMN(skN)).    (4.8)

Since the system of the points {skN,k} is some partition of the real line and fMN is a function having a first derivative of finite total variation, from (4.6) and (4.8) it can be derived that

k|Δ2fMN(kN)|k1N|(fMN(sk+1N)fMN(skN))|                                     1NV[fMN]=C2n.    (4.9)

In view of (4.7) and (4.9), we have

|QNM(y-x)|C2n|x-y|2,  0<|x-y|2π.    (4.10)

Finally, from (4.1), (4.5) and (4.10) we conclude that

|Kn(x,y)|C2n|x-y|2,  0<|x-y|2π.

 

4.2. A Property of the Spaces Vn and Wn

In this subsection we formulate and prove the main auxiliary statement. Let us make some preparations for this. We use the Minkowski inequality in the following form:

(l = 1m|abfl(x)dx|p)1/pab(l = 1m|fl(x)|p)1/pdx, 1p<    (4.11)

(under appropriate conditions on the functions fl which appear above). One can prove this inequality by using a similar technique as in the proof of the generalized Minkowski inequality [14, pp. 18–19].

Since the coefficient functions g1 and g2 defined by the formulas (2.1) and (2.2) on the segments of length 2π have the first derivatives (on corresponding segments) of finite total variation, by similar techniques as in Lemma 4.3 [7] one can prove the following estimates (with some constants C1 and C2):

s=02Nn1|φn,s(·)|C1Nn,    (4.12)
s=02Nn1|ψn,s(·)|C2Nn.    (4.13)

Theorem 4.1. Let n ∈ ℕ and (an,s)s = 02Nn-1 be an arbitrary sequence of real numbers. Then, there exist constants Ci > 0, i = 1, 2, 3, 4, such that for 1 ≤ p < ∞ the following inequalities hold:

C12n(121p)(s=02Nn1|an,s|p)1ps=02Nn1an,sψn,s(·)pC22n(121p)(s=02Nn1|an,s|p)1p.    (4.14)

For p = ∞ we have

C32n2maxs=0,,2Nn1|an,s|s=02Nn1an,sψn,s(·)                                 C42n2maxs=0,,2Nn1|an,s|.    (4.15)

Proof. For the sake of simplicity we denote

τ(x)=s = 02Nn-1an,sψn,s(x).    (4.16)

First, we prove the right-hand side of (4.14). For p = 1 we have

τ(·)1=12π02π|s=02Nn1an,sψn,s(x)|dx    s=02Nn1|an,s|12π02π|ψn,s(x)|dx=s=02Nn1|an,s|ψn(·sπNn)1=ψn(·)1s=02Nn1|an,s|.    (4.17)

Let us estimate ||ψn(·)||1. For the L2π1-norm of a polynomial T ∈ 𝕋n it holds that [15, p. 228]

T(·)1supx2πms=0m1|T(x2πsm)|,  m.

Applying this estimate to ψn with m = 2Nn and using the inequality (4.13), we obtain

ψn(·)112Nns=02Nn1|ψn(·sπNn)|C2NnC2n/2.    (4.18)

From (4.17) and (4.18) we derive

τ(·)1C2-n/2s = 02Nn-1|an,s|.

Let 1 < p < ∞. Using the inequality of Hölder for sums

k|akbk|(k|ak|p)1/p(k|bk|p)1/p,  1p+1p=1,

and the inequality (4.13), we obtain

|s=02Nn1an,sψn,s(x)|s=02Nn1|an,s||ψn,s(x)|=s=02Nn1|an,s||ψn,s(x)|1p+1p(s=02Nn1|an,s|p|ψn,s(x)|)1p×(s=02Nn1|ψn,s(x)|)1p(s=02Nn1|an,s|p|ψn,s(x)|)1p×(supx[0,2π)s=02Nn1|ψn,s(x)|)1pC2n2p(s=02Nn1|an,s|p|ψn,s(x)|)1p.

Therefore,

|s=02Nn1an,sψn,s(x)|pC2np2ps=02Nn1|an,s|p|ψn,s(x)|.

In view of (4.18), we have

τ(·)pp=12π02π|s = 02Nn1an,sψn,s(x)|pdxC2np2ps = 02Nn1|an,s|p12π02π|ψn,s(x)|dx= C2np2pψn(·)1s = 02Nn1|an,s|pC2np2p2n2s = 02Nn1|an,s|p=C2np(121p)s = 02Nn1|an,s|p,

and this implies that

τ(·)pC2n(121p)(s=02Nn1|an,s|p)1/p.

Now, we prove the left-hand side of (4.14). Taking the inner product in (4.16) with Tnsψn, we get that for all s = 0, …, 2Nn − 1

an,s=τ,ψn,s=12π02πτ(x)ψn(x-sπNn)dx.    (4.19)

Let p = 1. Using (4.19) and the estimation (4.13), we obtain

s=02Nn1|an,s|=s=02Nn112π|02πτ(x)ψn(xsπNn)dx|12π02π|τ(x)|s=02Nn1|ψn(xsπNn)|dxsupx[0,2π)s=02Nn1|ψn(xsπNn)|×12π02π|τ(x)|dxC2n/2τ(·)1.

Finally, C2-n/2s = 02Nn-1|an,s|τ(·)1.

Let now 1 < p < ∞. From (4.19) and the inequality (4.11) we derive that

(s = 02Nn-1|an,s|p)1/p=(s = 02Nn-1|12π02πτ(x)ψn(x-sπNn)dx|p)1/p=(s = 02Nn-1|12π02πτ(x-sπNn)ψn(x)dx|p)1/p12π02π(s = 02Nn-1|τ(x-sπNn)|p|ψn(x)|p)1/pdx=12π02π|ψn(x)|(s = 02Nn-1|τ(x-sπNn)|p)1/pdxsupx[0,2π)(s = 02Nn-1|τ(x-sπNn)|p)1/pψn(·)1C2npsupx[0,2π)(12Nns = 02Nn-1|τ(x-sπNn)|p)1/pψn(·)1.

For the L2πp-norm of a polynomial TTn we use the following inequality [15, p. 228]:

supx(2πms = 0m-1|T(x-2πsm)|p)1p(1+2nπm)T(·)p,  m.

Applying this estimation to τ with m = 2Nn and using the inequality (4.18), we obtain that

(s = 02Nn-1|an,s|p)1/pC2n(1p-12)τ(·)p.

Let p = ∞. The right-hand side of the inequality (4.15) follows from obvious inequalities and the inequality (4.13):

τ(·)=supx[0,2π)|s=02Nn1an,sψn,s(x)|      supx[0,2π)s=02Nn1|an,s||ψn,s(x)|maxs=0,,2Nn1|an,s|supx[0,2π]s=02Nn1|ψn(xsπNn)|=maxs=0,,2Nn1|an,s|s=02Nn1|ψn(·sπNn)|C2Nnmaxs=0,,2Nn1|an,s|C2n/2maxs=0,,2Nn1|an,s|.

Let us prove the left-hand side of (4.15). From the inequality (4.19) for s = 0, …, 2Nn − 1 we get

|an,s|12π 02π|τ(x)ψn(x-sπ/Nn)|dx           τψn1C2-n/2τ.

Finally, we conclude:

C2n/2maxs = 0,,2Nn-1|an,s|τ.

 

We formulated and proved Theorem 4.1 for a polynomial τ ∈ Wn, but using the same techniques and the inequality (4.12) instead of (4.13) in the corresponding places of the proof, one can prove a similar theorem for a polynomial τ ∈ Vn.

5. Proof of the Main Results

5.1. Proof of Theorem 3.1

In order to prove Theorem 3.1, we need some known statements from the paper Mhaskar and Prestin [6], Theorem A and Lemma A, and the following Lemma 5.1. Note that in the proofs of this Lemma and Theorem 3.1 we use similar considerations as in the proofs of Lemma 4.2 and Theorem 2.1 in Mhaskar and Prestin [6].

Theorem A. Let 1 ≤ p ≤ ∞, fL2πp, α > 0, 0 < θ ≤ ∞, and {Un} be a sequence of near best approximation (with some constant) for L2πp. The following statements are equivalent:

(a) fBp,θα*;

(b) {E2n(f)p}bθ,α;

(c) {||Un(f) − Un − 1(f)||p} ∈ bθ, α.

Lemma A. Let 1 ≤ p ≤ ∞, fL2πp, α > 0, 0 < θ ≤ ∞, and {Un} be a sequence of near best approximation (with some constant) for L2πp. If, for some interval I centered at x0, the sequence {||Un(f) − Un−1(f)||I,p} ∈ bθ,α, then for every infinitely differentiable on ℝ function ζ supported on I and extended 2π-periodically, the function is in Bp,θα*.

Lemma 5.1. Let I ⊂ ℝ, |I| < 2π, be an interval centered at x0, J1 and J be intervals centered at x0 such that JJ1I, ζ be an infinitely differentiable on ℝ function supported on I and extended as a 2π-periodic function such that ζ(x) = 1 for all xJ1 and let fL2π1. Then, for the operator

σn(f,x)=12π02πf(y)Kn(x,y)dy,  n3,

we have

σn((1-ζ)f,·)J,C(I,ζ,f)2n.

Proof. Without loss of generality we can assume that J1 is an interval with length equal to |I|/2 and J is an interval with length equal to |I|/4. For xJ, we have

|σn((1-ζ)f,x)|12π02π|f(y)(1-ζ(y))Kn(x,y)|dy                            =12π[0,2π)\J1|f(y)(1-ζ(y))Kn(x,y)|dy.

Since xJ and y ∈ [0, 2π)\J1, then |I|/8 < |xy|. Using Lemma 4.1, we get

|σn((1-ζ)f,x)|12π[0,2π)\J1|f(y)(1-ζ(y))|C2n|x-y|2dy                            C2π(|I|/8)212n[0,2π)\J1|f(y)(1-ζ(y))|dy                            C(I,ζ,f)2n.

Therefore,

σn((1-ζ)f,·)J,C(I,ζ,f)2n.

 

Proof of Theorem 3.1. Let part (a) hold. In view of Proposition A, it is equivalent to the fact that there exists an interval I centered at x0 such that for every infinitely differentiable on ℝ function ζ supported on I and extended as a 2π-periodic function, the function is in Bp,θα*. According to Theorem A (with σn instead of Un) it is equivalent to part (c). Thus, parts (a) and (c) are equivalent.

Let part (c) hold and let I be the interval chosen as in that part, let J1 and J be intervals centered at x0 such that JJ1I and ζ be an infinitely differentiable on ℝ function supported on I and extended as 2π-periodic such that ζ(x) = 1 for all xJ1.

From the obvious inequalities, we get

τn(f,·)J,pτn(fζ,·)J,p+τn(f(1-ζ),·)J,p                          τn(fζ,·)p+τn(f(1-ζ),·)J,                          τn(fζ,·)p                          + σn(f(1-ζ),·)J,+ σn-1(f(1-ζ),·)J,,

and using Lemma 5.1, we obtain

τn(f,·)J,pτn(fζ,·)p+C(I,ζ,f)2n.

Since α ∈ (0, 1), {2-n}n=1bθ,α and from the condition of part (c) we know that {τn(fζ,·)p}n=0bθ,α. Therefore, {τn(f,·)J,p}n=0bθ,α and part (b) is proved.

Let part (b) hold, and I be the interval chosen as in that part and let ζ be an infinitely differentiable on ℝ function supported on I and extended as a 2π-periodic function. In view of Lemma A (applied with σn in place of Un), we get that fϕBp,θα*. According to Theorem A this means that {τn(fζ,·)p}n=0bθ,α. This proves part (c).

 

5.2. Proof of Theorem 3.2

Let us first formulate and prove some auxiliary statements.

Lemma 5.2. There exist constants C1, C2 > 0 such that for every n ∈ ℕ and 0 < x ≤ 2π:

|φn(x)|C123n/2|x|2,    (5.1)
|ψn(x)|C223n/2|x|2.    (5.2)

Proof. First, we prove estimation (5.1). Note that we can define the scaling functions φn using an aperiodic coefficient function

φn(·)=12Nnkg3(kNn)cos k·,

where

g3(x)={2,|x|1-13,1-3(|x|-1)1+9(|x|-1)2,1-13|x|1+13,0,else.

Let us calculate V[g3]. Since g3=0 outside of the segment [−4/3, 4/3], it holds that V[g3]=V-4343[g3]. From the first and second derivatives of g3

g3(x)={6-9x(9x2-18x+10)3/2,23x43,-6+9x(9x2+18x+10)3/2,-43x-23,0,else,
g3(x)={9(18x2-27x+8)(9x2-18x+10)5/2,23x43,-9(18x2+27x+8)(9x2+18x+10)5/2,-43x-23,0,else,

we get that g3=0 at the points -112(9+17) and -112(9+17) and g3 is monotonously increasing on the segments [-43,-112(9+17)] and [112(9+17),43] and monotonously decreasing on the segments [-112(9+17),-23] and [23,-112(9+17)]. This means that the extrema are at the points -112(9+17) and 112(9+17):

supx[-4/3,4/3]g3(x)=g3(-112(9+17))                                  =-g3(112(9+17))                                  =(3+17)7+1743.

Since g3(-43)>0 and g3(43)<0, the total variation of g3 is

V[g3]=V-4343[g3]=4supx[-4/3,4/3]g3(x)=(3+17)7+173.    (5.3)

Then, using Proposition 2.2 [13], for x ∈ (0, 2π] we get

|φn(x)|=12Nn|kg3(kNn)coskx|     CNn|x|2k|Δ2g3(kNn)|,    (5.4)

where Δ2g3(kNn)=g3(kNn)-2g3(k+1Nn)+g3(k+2Nn).

From the mean value theorem we can deduce that there exist points mk ∈ (k, k + 1) such that

Δ2g3(kNn)=1Nn(g3(mk+1Nn)-g3(mkNn)).

Since the system of the points {mkNn,k} is some partition of the real line and g3 is a function having a first derivative of finite total variation V[g3], it holds that

k|Δ2g3(kNn)|=k1Nn|(g3(mk+1Nn)g3(mkNn))|        1NnV[g3]=C2n.    (5.5)

Thus, from (5.3)–(5.5) we get

|φn(x)|C23n/2|x|2.

The function ψn can be estimated in the same way. We have that

ψn(·)=1Nnkg4(kNn)cosk·,

where

g4(x)={1+3(x-1)1+9(x-1)2,1-13x1+13,2-3(x-2)4+9(x-2)2,1+13<x2+23,0,else,

is an aperiodic coefficient function.

From the first and second derivatives g4 and g4, we get that the points 16(9+17) and 112(15-17) are extrema of the function g4. Since g4(1-13)>0 and g4(2+23)<0, the total variation of g4 is

V[g4]=2g4(112(15-17))-2g4(16(9+17))           =3(17+3)7+174.

Using the same consideration as above, we have

|ψn(x)|C23n/2|x|2V[g4]C23n/2|x|2.

 

The main ingredient to prove the following lemmata is using estimations (5.1) and (5.2). Therefore, we formulate and prove these results for functions ψn, n ∈ ℕ, but they are true also for functions φn, n ∈ ℕ.

Further, by I′ we denote the complement of the interval I ⊂ [0, 2π) to the segment [0, 2π), i.e., I′: = [0, 2π) \ I. In the case when x0 = 0 we use corresponding modification: I′: = [−π, π) \ I, where I ⊂ [−π, π).

Lemma 5.3. Let I ⊂ ℝ, |I| < 2π, be an interval centered at x0, J be an interval centered at x0 such that JI, and ζ be an infinitely differentiable onfunction supported on J and extended 2π-periodically, and let fL2π1. Then, for 1 ≤ p < ∞

(s=02Nn-1|((fζ)*ψn)(sπNn)|p)1p(sκ(I,n)|((fζ)*ψn)(sπNn)|p)1p+C1(I,ζ,f)2-n(32-1p),

and for p = ∞

maxs=0,,2Nn-1|((fζ)*ψn)(sπNn)|maxsκ(I,n)|((fζ)*ψn)(sπNn)|                                                                 + C2(I,ζ,f)2-3n2.

Proof. Without loss of generality, we can assume that the interval J has a length equal to |I|/2. First, we consider the case 1 ≤ p < ∞. Using the inequality (a + b)qaq + bq, a, b > 0, 0 < q ≤ 1, with q = 1/p, we have

      (s = 02Nn-1|((fζ)*ψn)(sπNn)|p)1p(sκ(I,n)|((fζ)*ψn)(sπNn)|p)1p +(sκ(I,n)|((fζ)*ψn)(sπNn)|p)1p.

Let us estimate the second term in this inequality. From the inequality (4.11), for 1 ≤ p < ∞ we have

(sκ(I,n)|((fζ)*ψn)(sπNn)|p)1/p=(sκ(I,n)|12π02πf(x)ζ(x)ψn(x-sπNn)dx|p)1/p=(sκ(I,n)|12πJf(x)ζ(x)ψn(x-sπNn)dx|p)1/p12πJ(sκ(I,n)|f(x)|p|ζ(x)|p|ψn(x-sπNn)|p)1/pdx.

Since xJ and (s+2kNn)πNnI for some k ∈ ℤ, we have that |I|/4<|x-(s+2kNn)πNn|. Using Lemma 5.2, we obtain

(sκ(I,n)|((fζ)*ψn)(sπNn)|p)1/pC23n/2J(sκ(I,n)|f(x)|p|ζ(x)|p1|x-(s+2kNn)πNn|2p)1/pdxC23n/2(|I|/4)2J(sκ(I,n)|f(x)|p|ζ(x)|p)1/pdxC(I)2n(32-1p)J|f(x)ζ(x)|dxC(I,ζ,f)2n(32-1p).

Let p = ∞. From the obvious inequality maxsK|as|maxsK1|as|+maxsK2|as|, K = K1K2, K1K2 = ∅, where K is a finite set of indices, we get

maxs=0,,2Nn-1|((fζ)*ψn)(sπNn)|maxsκ(I,n)|((fζ)*ψn)(sπNn)|                                                                 +maxsκ(I,n)|((fζ)*ψn)(sπNn)|.

Let us estimate the second term in the last inequality. Using a similar consideration (with corresponding modification) as in the case 1 ≤ p < ∞, we have

maxsκ(I,n)|((fζ)*ψn)(sπNn)|maxsκ(I,n)12πJ|f(x)ζ(x)||ψn(x-sπNn)| dxC23n/2maxsκ(I,n)J|f(x)ζ(x)|dx|x-(s+2kNn)πNn|2C(I,ζ,f)23n/2.

 

Lemma 5.4. Let I ⊂ ℝ, |I| < 2π, be an interval centered at x0, J1 and J be intervals centered at x0 such that JJ1I, and ζ be an infinitely differentiable onfunction supported on I and extended 2π-periodically such that ζ(x) = 1 for all xJ1, and let fL2π1. Then, for 1 ≤ p < ∞, we have

(sκ(J,n)|(f*ψn)(sπNn)|p)1p(sκ(J,n)|((fζ)*ψn)(sπNn)|p)1p+C1(I,ζ,f)2-n(3/2-1/p),

and for p = ∞

maxsκ(J,n)|(f*ψn)(sπNn)|maxsκ(J,n)|((fζ)*ψn)(sπNn)|                                                + C2(I,ζ,f)2-3n2.

Proof. Without loss of generality we can assume that |J1| = |I|/2 and |J| = |I|/4. Applying Minkowski's inequality for sums (with corresponding modification for p = ∞)

(k|ak+bk|p)1p(k|ak|p)1p+(k|bk|p)1p,

we have that

     (sκ(J,n)|(f*ψn)(sπNn)|p)1/p(sκ(J,n)|((fζ)*ψn)(sπNn)|p)1/p   +(sκ(J,n)|((f(1-ζ))*ψn)(sπNn)|p)1/p.

Let us estimate the second term in this inequality. From inequality (4.11), for 1 ≤ p < ∞ we have

(sκ(J,n)|((f(1-ζ))*ψn)(sπNn)|p)1/p=(sJn|12π02πf(x)(1-ζ(x))ψn(x-sπNn) dx|p)1/p=(sκ(J,n)|12π[0,2π)\J1f(x)(1-ζ(x))ψn(x-sπNn) dx|p)1/p12π[0,2π)\J1(sκ(J,n)|f(x)|p|1-ζ(x)|p|ψn(x-sπNn)|p)1/pdx.

Since for x ∈ [0, 2π) \ J1 and (s+2kNn)πNnJ for some k ∈ ℤ, it holds that |I|/8<|x-(s+2kNn)πNn|, from Lemma 5.2 we get

(sκ(J,n)|((f(1-ζ))*ψn)(sπNn)|p)1/pC23n/2[0,2π)\J1(sκ(J,n)|f(x)|p|1-ζ(x)|p|x-(s+2kNn)πNn|2p)1/pdxC23n/2(|I|/8)2[0,2π)\J1(sκ(J,n)|f(x)|p|1-ζ(x)|p)1/pdxC(I)2n(32-1p)[0,2π)\J1|f(x)1-ζ(x)|dxC(I,ζ,f)2n(32-1p).

Using a similar consideration (with corresponding modification), for p = ∞ we have

maxsκ(J,n)|((f(1-ζ))*ψn)(sπNn)|maxsκ(J,n)12π[0,2π)\J1|f(x)1-ζ(x)||ψn(x-sπNn)| dxC23n/2maxsκ(J,n)[0,2π)\J1|f(x)1-ζ(x)|dx|x-(s+2kNn)πNn|2C(I,ζ,f)23n/2.

 

Proof of Theorem 3.2. The equivalence between (a) and (c) follows from Theorem 4.1 and the equivalence between (a) and (c) of Theorem 3.1.

Let now part (c) hold and I be the interval chosen as in that part. Let J1 and J be intervals centered at x0 such that JJ1I, and ζ be an infinitely differentiable on ℝ function supported on I and extended as a 2π-periodic function such that ζ(x) = 1 for all xJ1. Using Lemma 5.4 for n ∈ ℕ and 1 ≤ p ≤ ∞ (with corresponding modification for p = ∞), we get

Nn1/2-1/p(sκ(J,n)|(f*ψn)(sπNn)|p)1pNn1/2-1/p(sκ(J,n)|((fζ)*ψn)(sπNn)|p)1p + C(I,ζ,f)2-nNn1/2-1/p(s = 02Nn-1|((fζ)*ψn)(sπNn)|p)1p + C(I,ζ,f)2-n.

Since {2-n}n = 0bθ,α for 0 < α < 1 and from the conditions of part (c), we have that part (b) also holds.

Let part (b) hold. Let I be the interval as in that part and I1 be an interval centered at x0 such that I1I. Let ζ be an infinitely differentiable on ℝ function supported on I1 and extended 2π-periodically. Then, from Lemma 5.3 for n ∈ ℕ and 1 ≤ p ≤ ∞, we have

Nn1/2-1/p(s = 02Nn-1|((fζ)*ψn)(sπNn)|p)1pNn1/2-1/p(sκ(I,n)|((fζ)*ψn)(sπNn)|p)1p+ C(I,ζ,f)2-nNn1/2-1/p(sκ(I,n)|(f*ψn)(sπNn)|p)1p+ C(I,ζ,f)2-n.

Since {2-n}n=0bθ,α for 0 < α < 1 and from the conditions of part (b) we have that part (c) also holds. 

Note that Theorem 3.1 and Theorem 3.2 were proved for the smoothness parameter α which takes values from the interval (0, 1). The proofs depends on the smoothness properties of the coefficient functions g1 and g2 (see (2.1) and (2.2)). One can prove these theorems for other values of the parameter α by taking “smoother” coefficient functions (2.7) and (2.8).

5.3. Proof of Theorem 3.3

The equivalence between parts (a) and (b) follows from Lemma 2.1 and the equivalence between parts (a) and (c) of Theorem A. To get equivalence between parts (b) and (c) we use Theorem 4.1 and formulas (2.7) and (2.8).  

Author Contributions

All authors listed, have made substantial, direct and intellectual contribution to the work, and approved it for publication.

Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Acknowledgments

The authors were supported by FP7-People-2011-IRSES Project number 295164 (EUMLS: EU-Ukrainian Mathematicians for Life Sciences) and H2020-MSCA-RISE-2014 Project number 645672 (AMMODIT: Approximation Methods for Molecular Modeling and Diagnosis Tools). The authors would like to thank Viktor Romanyuk (Institute of Mathematics of NAS of Ukraine) for many valuable scientific discussions. Moreover, the authors would like to thank the referees for their helpful remarks and corrections.

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Keywords: local Besov spaces, Schauder basis, Fourier coefficients, periodic multiresolution analysis, wavelets, scaling functions, trigonometric polynomials

2010 Mathematics Subject Classification: 42A10, 42C40.

Citation: Derevianko N, Myroniuk V and Prestin J (2017) Characterization of Local Besov Spaces via Wavelet Basis Expansions. Front. Appl. Math. Stat. 3:4. doi: 10.3389/fams.2017.00004

Received: 21 December 2016; Accepted: 13 March 2017;
Published: 28 March 2017.

Edited by:

Lixin Shen, Syracuse University, USA

Reviewed by:

Jian Lu, Shenzhen University, China
Xiaosheng Zhuang, City University of Hong Kong, Hong Kong

Copyright © 2017 Derevianko, Myroniuk and Prestin. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Jürgen Prestin, prestin@math.uni-luebeck.de