Characterization of Local Besov Spaces via Wavelet Basis Expansions

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.


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. [1] for Meyer wavelets, [3] for mixed B-splines and [5] 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=−∞ a n (f )e inx be the Fourier series of some function f ∈ L 2 2π , a n (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 B α 2,θ of periodic functions: f ∈ B α 2,θ , α > 0, if and only if f ∈ L 2 2π and the norm (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, . . . , 2 n−1 − 1}, ν(0) = {0} be the sets of indices and {h n,k }, n ∈ N 0 , k ∈ ν(n) be the Haar system. This system is an orthogonal Schauder basis in the space L p ([0, 1]), 1 ≤ p < ∞, and for every f ∈ L p ([0, 1]) in the sense of the norm of L p ([0, 1]) [9,Chap. 3].
V. Romanyuk [12] obtained necessary and sufficient conditions on the Fourier-Haar coefficients b n,k (f ) at which functions from L p ([0, 1]) belong to the Besov spaces. Namely, let 1 ≤ p, θ < ∞, 0 < α < 1/p, then f ∈ B α p,θ if and only if f ∈ L p ([0, 1]) and the norm is finite. In this case it is evident that the Fourier-Haar coefficients b n,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 L p 2π 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 H.N. Mhaskar et al. in [8]. 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 ψ 0 be a scaling function of a periodic multiresolution analysis (PMRA) generated by de la Vallée Poussin means and ψ n , n ∈ N, be corresponding wavelets [11]. Let N n := 3 · 2 n for n ∈ N and N 0 := N 1 . By ψ n,s we denote shifts of ψ n : We show that for a particular choice of ψ n the system {ψ n,s } constitutes an orthogonal trigonometric Schauder basis in the space L p 2π , 1 ≤ p ≤ ∞, and a function f ∈ L p 2π can be represented by a series (for more detailed information see Subsection 2.2) converging in the norm of the space L p 2π , where the coefficient functionals a n,s (f ) are Fourier coefficients of f with respect to the basis {ψ n,s }: Because ψ n is even (see Subsection 2.2 for definition) and from (1.3) we conclude that a n,s (f ) can be represented in the following way: where f * g means the convolution For more information regarding trigonometric Schauder bases we refer to [6], [11], [13] 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, . . . , 2N n − 1 and there exists k ∈ Z such that the point (s+2kNn)π Nn belongs to the segment I. For 1 ≤ p ≤ ∞, n ∈ N 0 , and the segment I we define the following sequence (1.5) c n (I, p) := Then the main result of this paper is written as follows: A function f belongs to the local Besov spaces B α p,θ (x 0 ) if and only if there exists an interval I ⊂ R, |I| < 2π, centered at x 0 , such that the norm is finite. We adopt the following convention regarding constants. The letters C, C i , 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 L p 2π 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.

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 N 0 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 L p (A) we denote the space of functions f : A → R Lebesgue measurable on A with the finite norm If δ ≥ (b − a)/r, we put ω I,r,p (f, δ) := inf f − P I,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 = {a n } ∞ n=0 and numbers α, θ > 0 by a θ,α we denote the following norm a θ,α := The notation a ∈ b θ,α means that the norm a θ,α is finite. Let 1 ≤ p ≤ ∞, 0 < θ ≤ ∞, α > 0 and r = [α] + 1. For x 0 ∈ R the local Besov space B α p,θ (x 0 ) is the collection of functions f which satisfy the following properties: 1) f ∈ L p 2π ; 2) there is a nondegenerate interval I ⊂ R, |I| < 2π, centered at x 0 , such that The spaces B α p,θ (x 0 ) were considered in [8]. By periodicity we can restrict ourselves to points x 0 ∈ [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 f ∈ L p 2π , 1 ≤ p ≤ ∞. For δ > 0 we define the modulus of smoothness by where, in contrast to the modulus of smoothness ω I,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 B α * p,θ consists of functions f such that f ∈ L p 2π and the sequence ω * r,p (f, 2 −n ) ∞ n=0 ∈ b θ,α for some integer r > α. The space B α * p,θ is independent of the choice of r as long as r > α (see [2, Theorem 10.1, Chapter 2]). One can find more information about Besov spaces in the monographs [2,10]. The following statement about the connection between the two spaces mentioned above is proved in the paper [8].
if and only if there exists an interval I ⊂ R, |I| < 2π, centered at x 0 such that for every infinitely differentiable on R function ζ supported on I and extended as a 2π-periodic function, the function f ζ is in B α * p,θ .

2.2.
Expansions in a series. Let us first give some necessary definitions. Let 0, π 2 + π 6 ≤ |x| ≤ π, be a coefficient function given on the segment [−π, π] and let as above N n = 3 · 2 n and M n := 2 n , n ∈ N. By ϕ n we denote the following function which is a modification of the de la Vallée Poussin kernel. Let as above ϕ n,s denote shifts of the function ϕ n : ϕ n,s (·) := ϕ n · − sπ N n , n ∈ N, s = 0, . . . , 2N n − 1.
In [11] it is proved that the sequence of spaces {V n } ∞ n=1 , defined as The functions ϕ n are called scaling functions of this PMRA and the spaces V n are called scaling spaces.
The wavelet space W n which is defined to be the orthogonal complement of V n with respect to V n+1 , i.e., W n = V n+1 V n , is spanned by the translates of the function ψ n [11]: is a coefficient function given on the segment [− π 3 , 2π − π 3 ]. The functions ψ n are called wavelets. Note that the functions g 1 and g 2 defined on the segments with lengths equal 2π can be extended to R as 2π-periodic functions (and continuous since g 1 (−π) = g 1 (π) = 0 and . For simplicity of notations we denote ψ 0 := ϕ 1 , N 0 := N 1 and for n ∈ N 0 , It is convenient for us to put σ n (f, x) ≡ 0 if n = 0, 1, 2. We also use the following representation of the operators σ n : Using a similar technique as in [11], one can prove that the system of polynomials is an orthogonal trigonometric Schauder basis in the space L ∞ 2π . In view of Theorem 9 [9, p. 12] and Theorem 6 [9, p. 10] we get that {t k } ∞ k=0 is a basis in the space L p 2π , 1 ≤ p < ∞, and with some constant C > 0. By T n , n ∈ N, we denote the set of all trigonometric polynomials of the form From (2.3) and the representation of the kernel K n [11, p. 421], it can be derived that be the best approximation of a function f ∈ L p 2π by trigonometric polynomials from T n . A sequence of linear operators U n : L p 2π → T 2 n , n ∈ N 0 , is called a sequence of near best approximation (with the constant λ > 0) for L p 2π if it satisfies the following condition: For example, in the case p = ∞ the operators of de la Vallée Poussin V 2n n determine a sequence of near best approximation with the constant λ = 4 The following lemma results from the properties (2.4) and (2.5).
is a sequence of near best approximation (with some constant) for Further, for f ∈ L p 2π , 1 ≤ p ≤ ∞, we define operators τ n , n ∈ N 0 , as follows: From Lemma 2.1 it is easy to see that a function f ∈ L p 2π , 1 ≤ p ≤ ∞, can be represented by the series where convergence is understood in the metric of the space L p 2π . Using the definition of the operators τ n , we can represent them in the following form:

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 B α p,θ (x 0 ) and the behavior of the operators τ n near the point x 0 . This behavior will be described by the condition that certain norms of the operators belong to a sequential version of the Besov spaces.
The following statements are equivalent: (a) f ∈ B α p,θ (x 0 ); (b) There exists an interval I ⊂ R, |I| < 2π, centered at x 0 such that the sequence { τ n (f, ·) I,p } ∞ n=0 ∈ b θ,α ; (c) There exists an interval I ⊂ R, |I| < 2π, centered at x 0 such that for every infinitely differentiable on R function ζ supported on I and extended as a 2π-periodic function, the sequence { τ n (f ζ, ·) p } ∞ n=0 ∈ b θ,α . Let further for 1 ≤ p ≤ ∞ and a segment I, {c n (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 {d n (I, ζ, p)} ∞ n=0 we denote the sequence: The following statements are equivalent: (a) f ∈ B α p,θ (x 0 ); (b) There exists an interval I ⊂ R, |I| < 2π, centered at x 0 such that the sequence {c n (I, p)} ∞ n=0 ∈ b θ,α ; (c) There exists an interval I ⊂ R, |I| < 2π, centered at x 0 such that for every infinitely differentiable on R function ζ supported on I and extended as a 2π-periodic function, This theorem is the discrete version of Theorem 3.1 in the sense that the L p 2π norm of the operators τ n is replaced by a corresponding discrete norm (see Theorem 4.1).
For classical Besov spaces B α * p,θ , 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 a n (p) := The following statements are equivalent: Lemma 4.1. There exists a constant C > 0 such that for every n ∈ N, n ≥ 3: Proof. Let N = 3 · 2 n−2 , M = 2 n−2 , n ≥ 3. From [13, pp. 91-93] it is known that Defining continuous coefficient functions by where β ∈ (0, 1), we can rewrite the polynomials R M and Q M N in the following forms: Let us first estimate |R M (y − x)|. Using Proposition 2.2 [7], for 0 < |x − y| ≤ 2π we get Then, using the mean value theorem, we obtain that there exist points l k ∈ (k, k + 1), k ∈ Z, such that Since the system of the points l k M , k ∈ Z 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 . From the first and second derivatives we see that g = 0 at the points − .
Since f β = 0 outside of the segments [−1 − β, −1 + β] and [1 − β, 1 + β] and f β is an odd From the second derivative we can see that f β = 0 at the point −1 and f β is monotonously increasing on [−1 − β, −1] and monotonously decreasing on [−1, −1 + β]. This means that and the total variation of f β is Then, using Proposition 2.2 [7], for 0 < |x − y| ≤ 2π we get . From the mean value theorem it follows that there exist points s k ∈ (k, k + 1), k ∈ Z, such that Since the system of the points s k N , k ∈ Z is some partition of the real line and f M N is a function having a first derivative of finite total variation, from (4.6) and (4.8) it can be derived that In view of (4.7) and (4.9), we have Finally, from (4.1), (4.5) and (4.10) we conclude that

4.2.
A property of the spaces V n and W n . 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: (under appropriate conditions on the functions f l which appear above). One can prove this inequality by using a similar technique as in the proof of the generalized Minkowski inequality [15, pp. 18-19].
Since the coefficient functions g 1 and g 2 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 [11] one can prove the following estimates (with some constants C 1 and C 2 ):  be an arbitrary sequence of real numbers. Then, there exist constants C i > 0, i = 1, 2, 3, 4, such that for 1 ≤ p < ∞ the following inequalities hold: For p = ∞ we have First, we prove the right-hand side of (4.14). For p = 1 we have τ (·) 1 = 1 2π Let us estimate ψ n (·) 1 . For the L 1 2π -norm of a polynomial T ∈ T n it holds that [14, p. 228] Applying this estimate to ψ n with m = 2N n and using the inequality (4.13), we obtain and this implies that |a n,s | p 1/p . Now, we prove the left-hand side of (4.14). Taking the inner product in (4.16) with T s n ψ n , we get that for all s = 0, ..., 2N n − 1 (4.19) a n,s = τ, ψ n,s = 1 2π Let p = 1. Using (4.19) and the estimation (4.13), we obtain Finally, C 2 −n/2 2Nn−1 s=0 |a n,s | ≤ τ (·) 1 .
Let now 1 < p < ∞. From (4.19) and the inequality (4.11) we derive that For the L p 2π -norm of a polynomial T ∈ T n we use the following inequality [14, p. 228]: Applying this estimation to τ with m = 2N n and using the inequality (4.18), we obtain that Let p = ∞. The right-hand side of the inequality (4.15) follows from obvious inequalities and the inequality (4.13): Let us prove the left-hand side of (4.15). From the inequality (4.19) for s = 0, ..., 2N n − 1 we get Finally, we conclude: C 2 n/2 max s=0,...,2Nn−1 |a n,s | ≤ τ ∞ .
We formulated and proved Theorem 4.1 for a polynomial τ ∈ W n , 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 ∈ V n .

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 [8], 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 [8].
2π , α > 0, 0 < θ ≤ ∞, and {U n } be a sequence of near best approximation (with some constant) for L p 2π . The following statements are equivalent: 2π , α > 0, 0 < θ ≤ ∞, and {U n } be a sequence of near best approximation (with some constant) for L p 2π . If, for some interval I centered at x 0 , the sequence { U n (f ) − U n−1 (f ) I,p } ∈ b θ,α , then for every infinitely differentiable on R function ζ supported on I and extended 2π-periodically, the function f ζ is in B α * p,θ .
Lemma 5.1. Let I ⊂ R, |I| < 2π, be an interval centered at x 0 , J 1 and J be intervals centered at x 0 such that J ⊂ J 1 ⊂ I, 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 x ∈ J 1 and let f ∈ L 1 2π . Then, for the operator we have Proof. Without loss of generality we can assume that J 1 is an interval with length equal to |I|/2 and J is an interval with length equal to |I|/4. For x ∈ J, we have 2π 0 |f (y)(1−ζ(y))K n (x, y)|dy = 1 2π [0,2π)\J 1 |f (y)(1−ζ(y))||K n (x, y)|dy.
Since x ∈ J and y ∈ [0, 2π)\J 1 , then |I|/8 < |x − y|. Using Lemma 4.1, we get Therefore, 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 x 0 such that for every infinitely differentiable on R function ζ supported on I and extended as a 2π-periodic function, the function f ζ is in B α * p,θ . According to Theorem A (with σ n instead of U n ) 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 J 1 and J be intervals centered at x 0 such that J ⊂ J 1 ⊂ I and ζ be an infinitely differentiable on R function supported on I and extended as 2π-periodic such that ζ(x) = 1 for all x ∈ J 1 .
From the obvious inequalities, we get and using Lemma 5.1, we obtain Since α ∈ (0, 1), {2 −n } ∞ n=1 ∈ b θ,α and from the condition of part (c) we know that { τ n (f ζ, ·) p } ∞ n=0 ∈ b θ,α . Therefore, { τ n (f, ·) J,p } ∞ n=0 ∈ b θ,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 σ n in place of U n ), we get that f φ ∈ B α * p,θ . According to Theorem A this means that { τ n (f ζ, ·) p } ∞ n=0 ∈ b θ,α . This proves part (c).

Proof of Theorem 3.2.
Let us first formulate and prove some auxiliary statements.
Lemma 5.2. There exist constants C 1 , C 2 > 0 such that for every n ∈ N and 0 < x ≤ 2π: Proof. First, we prove estimation (5.1). Note that we can define the scaling functions ϕ n using an aperiodic coefficient function else.
Let us calculate V[g 3 ]. Since g 3 = 0 outside of the segment [−4/3, 4/3], it holds that From the first and second derivatives of g 3 we get that g 3 = 0 at the points − Since g 3 − 4 3 > 0 and g 3 4 3 < 0, the total variation of g 3 is Then, using Proposition 2.2 [7], for x ∈ (0, 2π] we get . From the mean value theorem we can deduce that there exist points m k ∈ (k, k + 1) such that Since the system of the points { m k Nn , k ∈ Z} is some partition of the real line and g 3 is a function having a first derivative of finite total variation V[g 3 ], it holds that Thus, from (5.3)-(5.5) we get The function ψ n can be estimated in the same way. We have that is an aperiodic coefficient function. From the first and second derivatives g 4 and g 4 , we get that the points 1 6 (9 + √ 17) and Using the same consideration as above, we have 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 ψ n , n ∈ N, but they are true also for functions ϕ n , 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 x 0 = 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 x 0 , J be an interval centered at x 0 such that J ⊂ I, and ζ be an infinitely differentiable on R function supported on J and extended 2π-periodically, and let f ∈ L 1 2π . Then, for 1 ≤ p < ∞ 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) q ≤ a q + b q , a, b > 0, 0 < q ≤ 1, with q = 1/p, we have Let us estimate the second term in this inequality. From the inequality (4.11), for 1 ≤ p < ∞ we have Since x ∈ J and (s+2kNn)π Nn ∈ I for some k ∈ Z, we have that |I|/4 < x − (s+2kNn)π Nn . Using Lemma 5.2, we obtain s∈κ(I ,n) .
Let p = ∞. From the obvious inequality max s∈K |a s | ≤ max Lemma 5.4. Let I ⊂ R, |I| < 2π, be an interval centered at x 0 , J 1 and J be intervals centered at x 0 such that J ⊂ J 1 ⊂ I, and ζ be an infinitely differentiable on R function supported on I and extended 2π-periodically such that ζ(x) = 1 for all x ∈ J 1 , and let f ∈ L 1 2π . Then, for 1 ≤ p < ∞, we have Proof. Without loss of generality we can assume that |J 1 | = |I|/2 and |J| = |I|/4. Applying Minkowski's inequality for sums (with corresponding modification for p = ∞) k |a k + b k | p 1 2π Since for x ∈ [0, 2π) \ J 1 and (s+2kNn)π Nn ∈ J for some k ∈ Z, it holds that .
Using a similar consideration (with corresponding modification), for p = ∞ we have max s∈κ(J,n) Let now part (c) hold and I be the interval chosen as in that part. Let J 1 and J be intervals centered at x 0 such that J ⊂ J 1 ⊂ I, 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