C A ] 1 M ar 2 01 9 Smoothness of functions vs . smoothness of approximation processes

We provide a comprehensive study of interrelations between different measures of smoothness of functions on various domains and smoothness properties of approximation processes. Two general approaches to this problem have been developed: the first based on geometric properties of Banach spaces and the second on Littlewood-Paley and Hörmander type multiplier theorems. In particular, we obtain new sharp inequalities for measures of smoothness given by the K-functionals or moduli of smoothness. As examples of approximation processes we consider best polynomial and spline approximations, Fourier multiplier operators on T, R, [−1, 1], nonlinear wavelet approximation, etc.


Introduction
The fundamental problem in approximation theory is to find for a complicated function f in a quasinormed space X a close-by, simple approximant P n from a subset of X such that the error of approximation f − P n X can be controlled by a specific majorant. In many cases, this problem is solved completely and necessary and sufficient conditions are given in terms of smoothness properties of either the function f or approximants P n of f .
We illustrate this by considering the well-known case of approximation of periodic functions by trigonometric polynomials on T = [0, 2π]. If f ∈ L p (T), 1 ≤ p ≤ ∞, and 0 < α < r, for the best approximant T * n and the modulus of smoothness ω r (f, t) p , the following conditions are equivalent: See [St], [BS], and [DL,Ch. 7]; for functions on T d see [Jo]. Let us also mention earlier results by Salem and Zygmund [SZ], Zamansky [Za], and Civin [C]. Similar results in the case of approximation by algebraic polynomials of functions on [−1, 1] can be found in [DT,Ch. 8] and [BJS]. Equivalence (i 1 ) ⇔ (i 2 ) easily follows from the classical Jackson and Bernstein approximation theorems, see, e.g., [DL,Ch. 7], given by E n (f ) p ω r (f, 1/n) p 1 n r n k=0 (k + 1) r−1 E k (f ) p , 1 ≤ p ≤ ∞, or their sharper versions for 1 < p < ∞, see, e.g., [DDT], 1 n r n k=0 (k + 1) rτ −1 E k (f ) τ p 1 τ ω r (f, 1/n) p 1 n r n k=0 (k + 1) rθ−1 E k (f ) θ where E n (f ) p is the error of the best approximation, τ = max(p, 2) and θ = min(p, 2). The equivalence (i 2 ) ⇔ (i 3 ) follows from the inequalities The left-hand side estimate is a corollary of the well-known Nikolskii-Stechkin inequality T (r) n p n r ω r (T n , 1/n) p . The right-hand side estimate was proved in [ZN]. Jackson and Bernstein approximation theorems as well as the corresponding equivalence (i 1 ) ⇔ (i 2 ) are known to be true in various settings. Surprisingly enough the results involving the smoothness of approximation processes given in the strong form, i.e., similar to inequalities (1.1), or, even in the weak form, i.e., similar to equivalence (i 2 ) ⇔ (i 3 ), are much less known in the literature. It is clear that such results provide additional information on smoothness properties of approximants and, therefore, they are useful for applications. As an example, we mention that the smooth function spaces (Lipschitz, Sobolev, Besov) can be characterize in terms of smoothness of approximation processes.
The main goal of this paper is to present a thoughtful study of interrelations between smoothness properties of functions on various domains and smoothness properties of approximation processes. In particular, we extend inequalities (1.1) as follows: for f ∈ L p (T), 1 < p < ∞, where T * 2 k stands for the best approximants, partial sums of the Fourier series, de la Vallée Poussin means, Fejér means, etc.
In the general form, our main results state that for f ∈ X where the parameters τ and θ are related to geometry of the space X, and, in particular, for X = L p , 0 < p ≤ ∞, are given by τ = max(p, 2), 1 < p < ∞, ∞, otherwise , θ = min(p, 2), p < ∞, 1, p = ∞ .
Here Y is a smooth function space (Sobolev or Besov spaces), P n (f ) is a suitable (linear or non-linear) approximation method, and Ω(f, 2 −nα , L p , Y ) is some measure of smoothness related to the spaces L p and Y . It is worth mentioning that the classical modulus of smoothness is equivalent to the K-functional for a couple (L p , W r p ), namely, K(f, t; L p (T), W r p (T)) p ≍ ω r (f, t), see, e.g., [DL,p. 177]. Therefore, as a measure of smoothness it is natural to consider the K-functional K(f, 2 −nα , L p , Y ) in the case 1 ≤ p ≤ ∞ and either an appropriate modulus of smoothness or a realization of the K-functional for any 0 < p ≤ ∞.
The rest of the paper is organized as follows. In Section 2, we deal with general approximation processes {P 2 n (f )} and abstract measures of smoothness Ω(f, t) X in the metric space X. We prove that where λ is a parameter related to the geometry of X. Let us emphasize that this result holds under very mild conditions on the approximants P 2 n (f ). Moreover, these inequalities easily imply the results similar to those given in the equivalence (i 2 ) ⇔ (i 3 ).
In Section 3, we consider general (Banach) spaces and investigate smoothness properties of the best approximants. Using again geometric properties of X (more complicated than those given in Section 2), we obtain inequalities (1.2) for appropriate θ and τ .
Section 4 studies the smoothness properties of Fourier means of functions from L p,w (D). Our approach is based on Littlewood-Paley-type inequalities and Hörmander's type multiplier theorems. In particular, inequalities (1.2) are obtained for a wide class of Fourier multiplier operators, which includes partial sums of Fourier series, de la Valée Poussin means, Fejér means, Riesz means, etc.
In Sections 5-8, we illustrate our main results obtained in Sections 2-4 by several important examples. In particular, in Section 5, we investigate relationship between smoothness of periodic functions on T d and smoothness of the best trigonometric approximants, various Fourier means, and smoothness of interpolation operators. Moreover, we consider approximations in Hardy spaces H p (D), 0 < p ≤ 1, and smooth (Lipschitz, Sobolev) spaces. Section 6 is devoted to approximation processes on R d . In this case, we study smoothness properties of band-limited functions that approximate functions from L p (R d ).
In Section 7, we deal with functions on L p,w [−1, 1], where w is the Jacobi weight. In particular, we study smoothness properties of algebraic polynomials and splines of the best approximation and consider some Fourier means related to Fourier-Jacobi series.
In Section 8, we show that the results of Sections 2 and 3 can be applied to study smoothness properties of non-linear approximation processes. As examples, we treat non-linear wavelet approximation and splines with free knots.
Finally, in Section 9, we study the optimality of inequalities (1.2), showing that the parameters τ and θ cannot be improved in general. Moreover, we define function classes such that the right-hand side and the left-hand side sums in (1.2) (with appropriate values of τ and θ) are equivalent to the corresponding modulus of smoothness.
Throughout the paper, we use the notation F G, with F, G ≥ 0, for the estimate F ≤ C G, where C is a positive constant independent of the essential variables in F and G (usually, f , δ, and n). If F G and G F simultaneously, we write F ≍ G and say that F is equivalent to G.
Acknowledgements. The first author was partially supported by DFG project KO 5804/1-1. The second author was partially supported by MTM 2017-87409-P, 2017 SGR 358, and by the CERCA Programme of the Generalitat de Catalunya. Part of the work was done during the visit of the second author to the Isaac Newton Institute for Mathematical Sciences, EPSCR Grant no EP/K032208/1.

General approximation processes and measures of smoothness
For a fixed positive λ, we consider a metric space (X, ρ) with the metric ρ : where the functional · X : X → R + is such that for all f, g ∈ X the following properties hold: Note that the metric · X = ρ(f, 0) is not a norm in general since the homogeneity property is not assumed.
Let us consider the following functional, which to some extend, plays a role of a measure of smoothness (abstract modulus of smoothness) which satisfies the following conditions: for any f, g ∈ X and δ > 0, where C j = C j (X, λ), j = 1, 2, 3.
As an approximation tool, we consider the family of operators P n : X → X, n ∈ N, such that the following two properties hold: for any f ∈ X and n ∈ N, Inequality (2.5) trivially holds when P n (f ) is a best approximant to f in X or P n (f ) is such that P n (P 2n (f )) = P n (f ), for example, take a de la Vallée Poussin-type operator or a projection operator. The second inequality is the Jackson-type theorem.
Theorem 2.1. Let f ∈ X and n ∈ N. Then where the left-hand side inequality holds if we assume only (2.2), (2.3), and (2.6).
Note that in the case of the Banach space X, a similar result for K-functionals and holomorphic semi-groups was obtained in [BB,Lemmas 3.5.4 and 3.5.5].
Let us prove the right-hand side inequality. Denote A typical example when Remark 2.1 can be applied is considering the partial sums of Fourier series P n (f ) = S n (f ) in the case X = L p (T), p = 1, ∞, and ξ(t) = log(t + 1); for details see Corollary 5.2.
Proof. The proof follows from (2.7) and the simple fact that (2.12) is equivalent to see, e.g., [Ti04].
For a given modulus of continuity ω, we define the function class The next corollary provides sharpness of Theorem 2.1.
Corollary 2.3. Let f ∈ Ξ ω and ω satisfy (2.12). Then, for large enough n ∈ N, Proof. First, we prove that The part in (2.15) is given by (2.7). To show the part , we note that by (2.12) and monotonicity of ω, for any m < n, we have Then, using (2.4), (2.3), (2.2), and (2.6) and choosing large enough m ∈ N, we derive To prove the second equivalence in (2.14), we note the part follows from the right-hand side inequality of (2.7) and (2.15) while the part follows from (2.13), the left-hand side inequality in (2.7), and (2.15), Remark 2.2. Corollaries 2.2 and 2.3 imply that if ω(δ) = δ α , α > 0, then, for any f ∈ X and n ∈ N, we have If, in addition, f ∈ Ξ ω , then The results of Remark 2.2 can be extended to Besov-type spaces. For a given modulus of smoothness Ω, s > 0, and 0 < q ≤ ∞, we define the Besov-type space as follows: with the usual modification in the case q = ∞.
We have the following characterization of B s X,q .
Corollary 2.4. Let s > 0 and 0 < q ≤ ∞. We have Proof. The proof easily follows from Theorem 2.1 and the Hardy-type inequality where A ν ≥ 0 and s, q > 0.

K-functionals and smoothness of best approximants
Let (X, Y ) be a couple of normed function spaces with (semi-)norms · X and · Y respectively and Y ⊂ X. The Peetre K-functional for this couple is given by for any f ∈ X and t > 0. Let {G n } ∞ n=1 be a family of subsets of Y such that: The best approximation of f ∈ X by elements from G n is given by Moreover, we suppose that the family {G n } is such that Jackson and Bernstein type inequalities are valid. Namely, there are positive constants c 1 , c 2 , and α such that for any n ∈ N we have The latter condition implies that, for every g ∈ G n , Clearly, if G n is a linear space, then (3.3) and (3.4) are equivalent. It is also plain to see that the Jackson-type inequality (3.2) implies the direct approximation theorem given by Our main goal in this section is to obtain inequalities for K(f, t; X, Y ) in terms of the best approximation of f by elements from G n .
In what follows, we denote by P n (f ) an element of the best approximation of f ∈ X by functions from G n (assuming it exists), i.e., An element of the near best approximation of f ∈ X by functions from G n is denoted by Q n (f ), i.e., there exists a constant c > 0 independent of f and n such that One of our main tools is the realization of K-functional given by Clearly, K(f, n −α ; X, Y ) ≤ R(f, n −α ; X, G n ), f ∈ X, n ∈ N, but for applications it is important to know when The next proposition describes such cases. (i) for every f ∈ X and n ∈ N, Even though Proposition 3.1 in this form was not mentioned in [HI], its proof easily follows from [HI,Theorem 2.2] taking into account that by (3.5), for the near best approximation Q n (f ), we have for any f ∈ X and n ∈ N.
Remark 3.1. It follows from [HI,Theorem 2.2] that under conditions of Proposition 3.1, assertions (i) and (ii) are equivalent to the following conditions: (iii) for every f ∈ X and n ∈ N, (iv) for every g ∈ G n and n ∈ N, The next theorem is the main result of this section.
(A) Suppose that there exist positive constants A and τ such that any g ∈ G n . Then, for any n ∈ N, we have (B) Suppose that there exist positive constants B and θ such that for all g ∈ G n . Then, for any n ∈ N, we have Proof. (A) Using the representation (3.12) Next, by Hardy's inequality and Bernstein's inequality (3.3), we obtain Using (3.8) with g = P 2 k−1 (f ) and n = 2 k , we get Thus, combining (3.14) and (3.15) and taking into account that Finally, combining (3.12) and (3.16) and using Proposition 3.1, we obtain (3.9).
(B) By the definition of the K-functional, we have Thus, to prove (3.11) it is enough to show that Thus, (3.18) and (3.19) yield (3.17), completing the proof.
Remark 3.2. (i) It follows from the proof of Theorem 3.1 that conditions (3.8) and (3.10) can be replaced by the following weaker conditions (ii) Note that by triangle inequality, estimate (3.10) is always valid with θ = B = 1.
We finish this section by discussing optimal parameters τ and θ in Theorem 3.1 for the important case X = L p .
Proposition 3.2. ([LS, Theorem 1], see also [S86, S90]) Let X be an abstract L p space with 1 < p < ∞, i.e. let X be a Banach lattice for which x + y p = x p + y p , whenever x, y ∈ X and min(x, y) = 0. Let M be a closed convex nonempty subset of X, dimX ≥ 2. Then there exist positive constants A p and B p such that if m is a best approximation in M to an element x ∈ X, then Using Theorem 3.2 and Proposition 3.2, we obtain the following result.
Theorem 3.2. Let inequalities (3.2), (3.3), and (3.7) are valid for X = L p , 1 < p < ∞. Then, for any f ∈ L p and n ∈ N, we have In Section 9, we will see that the parameters τ and θ in Theorem 3.2 are optimal.

Smoothness of Fourier multiplier operators
4.1. Realization and Littlewood-Paley-type inequality. First we introduce basic notations and collect auxiliary results. We follow the discussion in the paper [DDT].
We assume that Q(D) is a self-adjoint operator, that is, Examples of such operators and matching spaces are: We define In what follows, we suppose that λ k ≍ k σ for some positive σ > 0. Note that in the example above σ = 2 except for the eigenvalues of −∆ + |x| 2 where σ = 1 (see [Di98]).
As usual, we define the In this section, we consider approximation processes, which are defined by means of the Fourier multiplier operator T µ given by We will use the following assumption related to a Hörmander-Mikhlin-type theorem.
It is clear that under Assumption 4.1, the de la Vallée Poussin-type operator Here and in what follows, we assume that Moreover, the following realization result (see [Di98,Theorem 7.1]) holds: The following Littlewood-Paley-type theorem plays a crucial role in our further study.  [DDT,Theorem 3.1].) Let f ∈ L p,w (D), 1 < p < ∞, and Assumption 4.1 be satisfied, then If, in addition, γ > 0, then

Smoothness of the de la Vallée Poussin means in
Proof. Denote α = σγ and Then (4.9) By (4.6), we have Let us estimate the first sum J 1 . By (4.3), we have Using the fact that and, therefore, Dealing with J 3 , we observe that θ k (η 2 n f ) = η 2 n (θ k (f )). Then where in the last estimate we used (4.10) with θ k (f ) in place of f .

General Fourier multiplier operators.
In this subsection, we extend Theorem 4.2 considering general Fourier multiplier operators given by Together with the operator Ψ n , additionally assuming that ψ(x) = 0 for all x ∈ [0, 2 −m ] for some m ∈ Z + , we will also use the operator which plays a role of the inverse operator to Ψ n .
Theorem 4.3. Suppose that the conditions of Theorem 4.2 are satisfied.
(A) Let the operators Ψ 2 n be such that, for any f ∈ L p,w (D) and n ∈ N, where the constant C does not depend on f and n. Then and the operators Ψ 2 n are such that, for any f ∈ L p,w (D) and n ∈ N, where the constant C does not depend on f and n. Then Proof. To prove inequality (4.15), it is enough to note that by (4.14) one has Thus, (4.7) clearly implies (4.15).
To show (4.17), we note that by (4.16), we have This and (4.8) imply completing the proof.
Example. Many classical Fourier means are covered by Theorem 4.3. In particular, these cases include the following operators Ψ n f ∼ n k=0 ψ k n A k f : 1) Partial sums of Fourier series, the case ψ(x) = χ [0,1] (x); 2) Fejér means that are generated by the function ψ(x) = (1 − x) + ; 3) More generally, Riesz means for which ψ(x) = (1 − x α ) δ + , α, δ > 0; 4) Rogosinskii means that are generated by The precise formulation of the corresponding results in the periodic case will be given in Corollary 5.1.

Smoothness of best approximants.
In this subsection, we give analogues of Theorems 2.1 and 3.2 for best trigonometric approximants in L p (T d ) spaces. We recall some basic notations. Denote the set of all trigonometric polynomials of degree at most n by T n = span {e i(k,x) : |k| ≤ n}, where |k| = (k 2 1 + · · · + k 2 d ) 1/2 . The best approximation by trigonometric polynomials is given by As above, by P n (f ) we denote the best approximant of a function f in L p (T d ), that is, where P n (f ) ∈ T n .
In what follows, we will use the well-known Jackson type inequality, see, e.g., [Tim] and [SO]: where ω r (f, h) p is the classical modulus of smoothness, , and the constant C does not depend on f and n.
We will also need the following Stechkin-Nikolskii-type inequality (see [KT,Theorem 3.2]), which states that, for any n ∈ N and 0 < δ ≤ π/n, where the constants in this equivalence are independent of T n and δ. Here the homogeneous Sobolev norm is given by Using Theorem 2.1 with X = L p (T d ), 0 < p ≤ ∞, and Ω(f, δ) X = ω r (f, δ) Lp(T d ) for some r ∈ N, one can easily verify that properties (2.1)-(2.6) are valid. Therefore, applying Stechkin-Nikolskii-type inequality (5.2), we obtain the following result.
5.2. The case of Fourier multiplier operators. In this subsection, we give an analogue of Theorem 4.3 in the case D = T d . We start by recalling the multiplier theorem (Assumption 4.1) and the Littlewood-Paley-type theorem in L p (T d ) for 1 < p < ∞.
Concerning Assumption 4.1, the well-known Mikhlin-Hörmander multiplier theorem (see [GrI,p. 224]) states that the condition and f (k) = 1 (2π) d T d f (y)e −i(k,y) dy. We define the de la Vallée Poussin-type multiplier operator by and similarly to (4.5), we set An analogue of the Littlewood-Paley theorem in the case D = T d is given by the following two inequalities, see, e.g., [DDT,Theorem 4.1] or [GrII,Ch. 6]: for Let us consider the Fourier means given by where the function ψ : We derive the following analogue of Theorem 4.3 in the case D = T d .
(the Wiener class of absolutely convergent Fourier integrals), then the operators {Ψ n } are uniformly bounded in L p (T d ) for all 1 ≤ p ≤ ∞, see, e.g., [SW, Ch. VII]. Various useful conditions to insure that ψ ∈ A(R d ) can be found in the survey [LST], see also [TB,Ch. 4 and 6].
(ii) Concerning the uniform boundedness of { Ψ n }, one can use following version of 1 f -Wiener theorem (see [Lö,p.102]): Let f ∈ A(R d ). If f (x) = 0 on a closed bounded set V ⊂ R d , then 1 f (x) is extendable to a function in A(R d ), i.e., there exists a function g ∈ A(R d ) such that f (x) ≡ g(x) on V .
(iii) To verify the uniform boundedness of {Ψ n } and { Ψ n } in L p (T d ) for 1 < p < ∞, one can use the Mikhlin-Hörmander multiplier condition (5.7), which is less restrictive than the conditions given in parts (i) and (ii) of this remark.
(iv) Under conditions of Theorem 5.3, we have that for any and As examples, let us consider the following approximation processes: 1) the ℓ q -partial Fourier sums 2) the de la Vallée Poussin-type means 3) the Riesz spherical means Corollary 5.1. Let f ∈ L p (T d ), 1 < p < ∞, α > 0, τ = max(2, p), and θ = min(2, p). Then It is enough to note that these means are uniformly bounded in L p (T d ), 1 < p < ∞, see, e.g., [SW, Ch. VII] and [We], and to apply the Mikhlin-Hörmander multiplier condition to show that the corresponding inverse operators { Ψ n } are also uniformly bounded in L p (T d ).
Remark 5.2. In the univariate case of the Fejér means T 2 k f = R 1,1 2 k f , the righthand side of inequality (5.8) was obtained earlier by Zhuk and Natanson in [ZN].
Note that for α ∈ N and 1 < p < ∞ inequality (5.8) can be equivalently written as follows We give its analogue for the cases p = 1, ∞.

Inequalities in the
where c k = c k (f ) are the Taylor coefficients of f . Then, the realization result is given as follows (see [KT,Sec. 11]): f − η 2 n f Hp + 2 −αn (η 2 n f ) (α) Hp ≍ ω α (f, 2 −n ) Hp . Using the scheme of the proof of Theorem 4.2 and the Littlewood-Paley theorem in the Hardy spaces H p (D), 0 < p ≤ 1, see, e.g., [GrII,Ch. 6], we obtain the following result.
(ii) Inequalities (5.11) and (5.12) are also valid if we replace the de la Vallée Poussin means η 2 k f by the corresponding means Ψ 2 k f with the properties similar to those indicated in Theorem 4.3.
Remark 5.4. Using the well-known facts about simultaneous approximation of functions and their derivatives in L p (T), see, e.g. [CF] and [DL,Ch.7,Theorem 2.7], it is not difficult to obtain analogues of Theorems 5.5 and 5.6 in the Sobolev spaces W r p (T), 1 ≤ p ≤ ∞, and r ∈ N, cf. Remark 5.1 (iv).

Interpolation operators.
In the above sections, we deal with polynomials of the best approximation and Fourier means. It turns out that Theorem 2.1 can be also applied for interpolation operators. As an example, let us consider an interpolation analogue of the de la Vallée Poussin means: Recall some basic properties of V n f (see [Sz]).
Proposition 5.1. The following assertions hold: (4) for all f ∈ C(T) and r, n ∈ N, we have Thus, noting that V n (V 2n f ) = V n f and using Theorem 2.1, Proposition 5.1, and the Nikolskii-Stechkin-type inequality (5.2), we derive the following result.

Smoothness of best approximants.
In what follows, the class of bandlimited functions B σ p , 1 ≤ p ≤ ∞, σ > 0, is given by . In contrast to the periodic case, we do not know analogues of Jackson and Nikolskii-Stechkin inequalities type inequalities for the full range of parameters d ∈ N and 0 < p < 1. Because of this, we restrict ourselves to the case 1 ≤ p ≤ ∞.
Theorem 6.3. Let f ∈ L p (R d ), 1 < p < ∞, α > 0, τ = max(2, p), and θ = min(2, p). then An analogue of Corollary 5.1 on R d , namely, inequality (5.8) holds for the following Fourier means: 1) the ℓ q -Fourier means given by 2) the de la Vallée Poussin-type means η n f (x); 3) the Riesz spherical means R β,δ n given by for β > 0 and δ > d| 1 p − 1 2 | − 1 2 . At the same time, an analogue of Corollary 5.2 on R d is valid only for the de la Vallée Poussin-type means and the Riesz spherical means. Namely, for any f ∈ L p (R d ), p = 1, ∞, and α ∈ N, we have where T 2 k f = η 2 k f or R β,δ 2 k f with δ > (d − 1)/2. Finally in this section, we give a characterization of the classical Besov spaces B s p,q (R d ) in terms of best approximants and Fourier means. Using Theorems 6.1, 6.2 and 6.3 and the same arguments as in Corollary 2.4, we derive Corollary 6.1. Let 1 < p < ∞, 0 < q ≤ ∞, and 0 < s < α. We have where P 2 k (f ) stands for the best approximants or the Fourier means Ψ 2 k f with the properties given in Theorem 6.3. In the case p = 1 or ∞ and α ∈ N, s < α, we have where P 2 k (f ) stands for the best approximants, the de la Vallée Poussin-type means η n f (x), or the Riesz spherical means R β,δ n with δ > (d − 1)/2. Note that a similar assertion for the Gauss-Weierstrass semi-group W t f (x) = (4πt) d/2 R d e − |x−y| 2 4t f (y)dy = (e −t|ξ| 2 f (ξ))(x), t > 0, was obtained in [BB,Theorem 3.4.6,p. 198] and [Tr,Section 1.13.2,.

Smoothness of approximation processes on [−1, 1]
7.1. Sharp inequalities for algebraic polynomials. Let L w,p = L p ([−1, 1]; w), 0 < p ≤ ∞, be the space of all functions f with the finite (quasi-)norm Further, let P n be the set of all algebraic polynomials of degree at most n. As usual, the error of the best approximation of a function f ∈ L w,p by algebraic polynomials is defined as follows: . Then the corresponding K-functional is given by (4.1) with σ = 2 and D = [−1, 1].
Recall that by (4.4) and [DD, Section 6], we have 1] , where the de la Vallée Poussin means η n f are given by Thus, using Theorem 3.2, Theorem 4.2, and the needed facts from [DD, Section 6], we obtain the following result.
Denote by S m,n the set of all spline functions of degree m − 1 with the knots t j = t j,n = j/n, j = 0, . . . , n, i.e., S ∈ S m,n if S ∈ C m−2 [0, 1] and S is some algebraic polynomial of degree m − 1 in each interval (t j−1 , t j ), j = 1, . . . , n. Let be the best approximation of a function f by splines S ∈ S m,n in L p [0, 1]. The Jackson type inequality is given by ([Os, Theorem 1], see also [DL,Ch. 12,p. 379 where f ∈ L p [0, 1], 0 < p ≤ ∞, n ∈ N, and is the modulus of smoothness of order r ∈ N. Note that any spline S n ∈ S r,n can be represented (see [Os]) as follows: where P ∈ P r−1 , x + = x if x ≥ 0 and x + = 0 if x < 0. Moreover, one has where C is a positive constant that depends only on r and p. Inequalities (7.4) were proved in [HY,Lemma 2.1] (see also [Hu]) in the case 1 ≤ p < ∞. It is easy to see that the same also holds in the case 0 < p < 1.
It is important to mention that (7.4) implies that for any S n ∈ S r,n , n, r ∈ N, one has where V (f ) p denotes the p-variation of the function f , that is, In its turn, (7.5) implies the following analogue of the Bernstein inequality Moreover, by (7.3) and (7.5), for any S n ∈ S r,n , n, r ∈ N, such that f − S n Lp[0,1] = E r,n (f ) p , we have The above results allow us to apply Theorem 2.1 to obtain the following result.
Theorem 7.4. Let f ∈ L p [0, 1], 1 < p < ∞, r, n ∈ N, and τ = max(2, p), θ = min(2, p). Then 8. Nonlinear methods of approximation 8.1. Nonlinear wavelet approximation. We restrict ourselves to the case of compactly supported biorthogonal wavelets and follow the discussion in [De,Section 7]. Let ϕ and ϕ be two refinable compactly supported functions and let ψ and ψ be their corresponding wavelets. Suppose that ϕ and ϕ are in duality as follows where δ jk is the Kronecker delta. Then each function f ∈ L p (R) has the following wavelet decomposition: see, e.g., [CDF] and [Da]. In the above formula, D is the set of all dyadic intervals in R, I denotes the dyadic cube I = 2 −k (j + [0, 1]) associated with j, k ∈ Z and where Λ ⊂ D is a set of dyadic intervals of cardinality #Λ ≤ n. Thus Σ w n is the set of all functions which are a linear combination of n wavelet functions. We define Let B r p,q (R), r > 0, 0 < p, q ≤ ∞, be the classical Besov spaces. The Jackson and Bernstein type inequalities are given in the following two propositions (see [CDH,Corollary 4.1 and Theorem 4.3]).
Proposition 8.1. Let 1 < p < ∞, r > 0, and f ∈ L p (R), 1/γ = r + 1/p. If ψ has m vanishing moments with m > r and ψ is in B ρ γ,q (R) for some q > 0 and some ρ > r, then We will also use the fact that there exists (see for details [CDH]). This realization result in particular implies the Nikolskii-Stechkin-type inequality K S, n −r ; L p (R), B r γ,γ (R) ≍ n −r |S| B r γ,γ (R) , S ∈ Σ w n . Thus, using Theorem 3.2, Propositions 8.1 and 8.2, we obtain the following result.
As a corollary, we obtain the characterization of the Besov space B r X,q (interpolation space) given in (2.16) with X = L p (R) and Ω(f, 2 −k ) X = K(f, 2 −rk , L p (R), B r γ,γ (R)).
Corollary 8.1. Under conditions of Proposition 8.1, if 0 < σ < r and 0 < q ≤ ∞, then Free knot piecewise polynomial approximation. Let r ∈ N be fixed and for each n = 1, 2, . . . , let Σ r,n be the space of piecewise polynomials of degree r with n pieces on [0, 1]. That is, for each element S ∈ Σ r,n there is a partition Λ of [0, 1] consisting of n disjoint intervals I ⊂ [0, 1] and polynomials P I ∈ P r such that S = I∈Λ P I χ I .
For each 0 < p < ∞, we define the error of the best approximation by Recall the well-known Jackson-type inequality (see [Pe,Theorem 2.3]).

Optimality
In the previous sections, we derived the following inequalities: Y is an appropriate smooth function space, and P n (f ) is a suitable approximation method. In this section, we show that the parameters θ and τ are optimal. For this, we restrict ourselves to the case of D = T and approximation of periodic L p -functions by S n (f ), the n-th partial sums of the Fourier series of f , and the de la Vallée Poussin means η n f .
Recall that if f ∈ L p (T), 1 < p < ∞, then inequality (9.1) in particular implies If f ∈ L p (T), p = 1, ∞, and P n (f ) = η n f, estimate (9.1) can be written by 9.1. Optimality of (9.1) in the case 1 < p < ∞. In this subsection, we deal with not only sharpness of the parameters τ = max(2, p) and θ = min(2, p) but we also show that for the classes of functions with lacunary and general monotone Fourier coefficients, inequality (9.1) becomes an equivalence with τ = θ = 2 and τ = θ = p, respectively.
We start with lacunary series and first give a simple proof of Zygmund's theorem in L p , 1 < p < ∞, based on the Littlewood-Paley technique given in Section 4.1. We deal with the general case of functions represented by For convenience, we suppose that the dimension d k = 1 for all k ∈ Z + .
We will say that the Fourier expansion of f ∈ L p,w (D) is lacunary, , A k f = 0 for k = 2 j , j ∈ Z + . Let us first derive an analogue of Zygmund's theorem.
Remark 9.1. As an example of the system {ψ k } in Lemma 9.1, one can take the trigonometric system, the Walsh system, systems of the Chebyshev polynomials and, more generally, the system of normalized Jacobi polynomials for specific range of parameters α, β > −1 indicated in [ABD].
Proof. Using the realization result (4.4) and Lemma 9.1, we get In particular, for the classical Fourier series on D = T we obtain where 1 < p < ∞ and α > 0; cf. (9.2).
Remark 9.2. It is clear that (9.5) gives the sharpness of the parameter θ for p ≥ 2 and τ for p ≤ 2 in inequality (9.2).

Lp(T)
2−ε 1 2−ε with some ε ∈ (0, 2) does not hold for f (x) = ∞ n=1 a 2 n cos 2 n x ∈ L p , where a 2 n = n −1/(2−ε) . Now, let us consider the case of the classical Fourier series with general monotone coefficients. In what follows, we say (see [Ti07]) that a (complex) sequence {d n } is general monotone, written where C does not depend on n. Note that any monotone (quasi-monotone) sequences are general monotone. We denote by GM the class of integrable functions such that f (x) ∼ ∞ n=1 (a n cos nx + b n sin nx) with {a n }, {b n } ∈ GM. Theorem 9.2. Let f ∈ L p (T) ∩ GM , 1 < p < ∞, and α > 0. Then (9.7) ω α (f, 2 −n ) Lp(T) ≍ completing the proof.
Remark 9.3. Similarly to Remark 9.2, equivalence (9.7) provides the sharpness of the parameter θ for p ≤ 2 and τ for p ≥ 2 in (9.2). 9.2. Optimality of the right-hand inequality in (9.1) for p = 1 and p = ∞. We start by obtaining two simple results for lacunary Fourier series.
Theorem 9.3. Let f ∈ L 1 (T) ∩ Λ and α > 0. Then Proof. The proof repeats the one of Theorem 9.1 since by Zygmund's theorem (see [GrI,Theorem 3.7.4]), we have Proof. By Stechkin's theorem, see [GrI,Theorem 3.7.6], we have Note that Theorem 9.4 shows that in the case p = ∞, the right-hand inequality (9.1) is sharp for θ = 1, in other words this inequality cannot be improved for some θ > 1 in the general case. At the same time, we remark that Theorem 9.3 only shows that in the case p = 1, the right-hand inequality (9.1) is sharp for θ = 2, that is, (9.1) cannot be sharpen with any θ > 2. Now we show that (9.1) is in fact sharp for θ = 1.
Theorem 9.5. Let α ∈ N. Then for any q > 1 there exists a function f ∈ L 1 (T) such that is not valid with a constant C independent of n and f .
Proof. We will use the following well-known Kolmogorov's estimates for the L 1norms of trigonometric series: (9.9) π 0 ∞ k=1 a k cos kx dx ∞ k=1 k|∆ 2 a k |, where ∆ 2 a k = a k+2 − 2a k+1 + a k . Inequality (9.9) was obtained in [K], see also [Te]; for inequality (9.10) see [Te]. We will also need the following estimate for the error of the best approximation given by (see [Ge,Lemma 2 a k sin kx ∈ L 1 (T).
9.3. Optimality of the left-hand inequality in (9.1) for p = 1 and p = ∞. In this subsection, we show that the left-hand inequality in (9.1) cannot be improved in general. In particular, for p = 1 or p = ∞, the following inequality is not valid for any q > 0 (9.14) Theorem 9.6. Let p = 1 or ∞ and α ∈ N. Then for any q > 0 there exists a function f ∈ L p (T) such that inequality (9.14) is not valid with a constant C independent of n and f .
If α is even, we obviously have For odd α, using Bernstein's inequality, we derive m α cos mx log γ (m + 1)
At the same time, by (9.16) and (9.17), we derive The last two formula imply that inequality (9.14) is not valid in the case p = ∞. Now, let us consider the case p = 1. We put a m cos mx, a m = 1 log γ (m + 1) , γ > 1.
Recall that if a convex sequence {a m } is the sequence of cosine Fourier coefficients of an even function f ∈ L 1 (T), then using Theorem 1 from [Al], we have (9.18) ω α (f, 2 −n ) L 1 (T) 1 2 αn 2 n m=1 m α−1 a m 1 n γ .