The K-functional and saturated linear approximation processes

It is shown that already the knowledge of the saturation class F(Jt,Lp)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\, {{\mathfrak {F}}}(J_t, L^p) $$\end{document} of a saturated convolution approximation process {Jt}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\, \{J_t\}$$\end{document} on Lp(Rn),1≤p<∞,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\, L^p({{\mathbb {R}}}^n),\, 1\le p< \infty ,$$\end{document} completely determines its norm approximation behavior. This is achieved by using the K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\, K$$\end{document}-functional K(t,f;Lp,F(Jt,Lp))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\, K(t,f;L^p,{{\mathfrak {F}}}(J_t, L^p)) $$\end{document} as a comparison scale, which relates on the one hand the approximation process and on the other appropriate moduli of smoothness. This implies that simultaneously one gets the so-called direct and inverse theorems. There are open problems if the saturation order is slightly perturbed, e.g., by a |logt|λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\, |\log t|^\lambda $$\end{document}-factor, λ>0.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\, \lambda >0.$$\end{document} Proofs are mainly based on the Fourier transformation.


Introduction
The purpose of this paper is to cast a retrospective glance from today's insight at some aspects of the interplay between Fourier analysis and approximation as described in the book by P. L. Butzer and R. J. Nessel [5]. Here the additional ingredient is the use of Peetre's K -functional K (t, f ; L p , F(J t , L p )) defined below. It serves as a basis of comparison in so far as it can be chararacterized by J t f − f p on the one hand and on the other by appropriate moduli of smoothness. Thus one obtains direct and inverse approximation theorems simultaneously. Using maximal functions one also gets some results concerning the a.e. rate of convergence.
In 1947 Jean Favard [9] introduced the concept of saturation into the theory of approximation for 2π -periodic integrable functions. Let us formulate it for the special case of approximation processes of convolution type on L p (R n ). Thus consider where f ∈ L p (R n ), 1 ≤ p < ∞, and k ∈ L 1 (R n ), R n k(y) dy = 1.
One says that the family {J t } t>0 is saturated if there exists a positive function ϕ(t) with lim t→0+ ϕ(t) = 0 such that and if the set F(J t , L p ), defined by contains at least one f ∈ L p , f = 0. One calls ϕ(t) the saturation order (or optimal approximation order) of the process {J t } and F(J t , L p ) its Favard or saturation class.
Before stating a now folklore saturation result, based upon the work of the above authors, let us first give some notation used in the following. Define the Fourier transformation F on the test function space S of smooth, rapidly decreasing functions by its inverse by F −1 . Extend the Fourier transformation in the standard way to the space of tempered distributions S . Denote by M the Fourier transform of M(R n ), the space of bounded measures on R n , by L p the space of Fourier transforms of Here with (ϕ j ) ∞ j=−∞ being a standard smooth partition of unity. Saturation Theorem 1 Let J t be an approximate identity on L p (R n ), 1 ≤ p < ∞, as above and α > 0.
Assume for some μ α ∈ M(R n ) with R n dμ α = 0 that Then J ρ ( f ) is saturated with order ρ −α and has saturation class Typical examples for such approximate identities are the Riesz means the generalized Weierstrass means (Abel-Cartwright means) the integral average occurring in Lebesgue's differentiation theorem For later use note that Here B 1 (0) is the unit ball with the origin as center, |B 1 (0)| its volume. R λ,α 1/t and W α t are saturated with order ϕ(t) = t α , L t with ϕ(t) = t 2 .

The K-functional and norm convergence of saturated linear means
We want to illustrate that already the knowledge of the saturation class may determine the complete norm convergence behavior of the associated approximation process. The main idea is to connect the approximation processes in question and the relevant moduli of smoothnes via Peetre's K -functional, defined as follows The K -functional will be useful in two aspects, it serves as an interface / basis of comparison and its nice calculus can be applied.
We mention that Zamansky [24] takes as basis of comparison the typical means in one dimension whose kernel is Observe the elementary property -see [20, (1.5),(1.6)], [22,Thm. 7] - Let us start by quoting Theorem 5 in [19], proved there for the case L 1 , its extension to L p , 1 < p < ∞, being straightforward.
Theorem 2 Let k ∈ L 1 (R n ) be such that the following two conditions are satisfied: Thus we only need to strengthen the saturation condition a little by adding (b) to relate the K -functional to the norm convergence behavior of the corresponding approximation process. As a first application of Theorem 2 we give a simple deduction of the comparison theorem between the Riesz means (for all ! λ > (n − 1)/2) and the generalized Weierstrass means. Since both means satisfy the above conditions (a) and (b), we obtain Also, this deduction nicely shows the interface property of the K -functional. In these equivalences constants are not specified. It would be nice to obtain good / best possible constants. In this connection we quote a result deduced in [8] for the Fejér means on R.
We mention the following characterization due to Kuznecova and Trigub [14] of the Riesz means (and hence of the corresponding K -functional).
The truncated hypersingular integral involved is a convolution of f with an L 1 -kernel -see [12]. Looking at the integral averaging means L t it is shown in [19,Ex. 7] that the conditions (a) and (b) are satisfied with α = 2, thus regaining a result of Belinskii [1]. Furthermore, applying a result of Ditzian [7], see also [19,Thm. 1], we obtain for 1 ≤ p < ∞ where e j ∈ R n is the j-th unit vector and 2 h denotes the second central difference. If the case p = 1 is excluded one even gets the folklore result This result motivates to look for further characterizations of the above K -functional.
To this end, introduce moduli of smoothness of fractional order α > 0 by and thus α h f converges in the L p -norm. Now follow some relations between K -functionals and moduli of smoothness. First, a result due to Wilmes [22,Thm. 7] gives Concerning the case p = 1, [22,Thm. 7] contains the one-sided estimate For later use observe, if 0 < γ < α < β, then Also if κ, β > α, then for all p, 1 ≤ p < ∞, there holds The second inequality, a Marchaud type inequality, directly follows by [20,Cor. 3.2] when choosing as n-parameter equibounded C 0 -group of operators the group of translations. The inequalities (11) immediately show that, under suitable assumptions, moduli of smoothness of different orders are equivalent.
(b) Note that (1 + | log t|) σ can be replaced by any slowly varying function b, essentially decreasing on (0, 1) and increasing on (1, ∞), with b(t) → ∞ for t → 0+ (for details concerning slowly varying functions see [11]). (c) The equivalence of moduli of smoothness of different orders, as described in Lemma 3, does not hold "near" the saturation order. Ditzian, Sept. 2011 in Barcelona, pointed out that For the proof he combined a sharp Marchaud and a sharp converse Marchaud inequality. Proceeding in the same way and using [18, Theorem 2.1], we obtain for the example t α | log t| σ , α > 0, σ ≥ 0, 1 < p < ∞, and t → 0+ that Intermediate résumé: On L p (R n ), 1 < p < ∞, one can handle the relationship between approximation order and smoothness of the involved function relatively satisfactory (though there may occur technical problems in the fractional case). On L 1 (R n ) one gets good results as long as the approximation order is "sufficiently away" from the saturation order. This follows from (13) when in the case α = 2k + 1, k ∈ N 0 , one additionally uses (4). Excluding the case α = 2k, k ∈ N, one is faced with the problem what happens precisely for approximation orders "near" the saturation order. Let us restrict ourselves to the specific example of the Abel-Poisson means ( α = 1) and the approximation order t(1 + | log t|) σ , σ > 0. This certainly cannot be handled with the aid of (6).
For C 2π -functions V. V. Zhuk 1967 (see [21, p. 387]) has given a characterization of the approximation behavior of the Fejér means, which carries over to the L 1 (R)case (follow the hint in [21, p. 387]) Here F(x) = ∞ x f (u) du and F symbolically denotes the Hilbert transform of F (actually one has to consider the central difference 2 h F ∈ L 1 (R) whose Hilbert transform exists as an L 1 (R)-function). There is the question if one can split up (14) in two parts: (a) one taking care of the non-smooth functions, (b) the other, say, of functions satisfying a Lipschitz condition.
(b) If one looks at smoother functions f , say ω 1 (t, f ) L 1 (R) t δ for some δ, 0 < δ < 1, then its Hilbert transformf also satisfies ω 1 (t,f ) L 1 (R) t δ . In this case the second term on the right hand side of (14) can be simplified. Since for h > 0 we obtain by the convolution theorem In view of (5) for α = 1 and (15) it would be nice to prove the following Conjecture. Suppose f ∈ L 1 (R) satisfies some Hölder condition, say ω 1 (t, f ) L 1 (R) t δ , 0 < δ < 1, then Concerning an extension of Zhuk's characterization of the K -functional to several dimensions one should have in mind the characterization of the saturation class of the Abel-Poisson means (see [6]) where R j is the j-th Riesz transform, e j being the standard j-th unit vector. Thus, it is near at hand to examine if the following holds. Conjecture. .
Here Here, m may be chosen large (cf. (5)), thus giving a behavior of at t = 1 as smooth as we want with where g t,α ∈ L p (R n ).