On the computation of $K$-functionals

Abstract

Abstract.A new approach to the calculation of the sharp order of a K-functional is suggested. This approach employs the techniques of dyadic spaces. §1. IntroductionThe theory of local polynomial approximation makes it possible to construct dyadic analogs of certain spaces. The dyadic Hardy classes and the dyadic BMO are primary examples of this sort, which have found interesting applications in analysis.In the author's paper (the definition of B λθ p (F ) will be given below). In [1] it was noted that there are at least two reasons for which the study of dyadic spaces is of interest. First, we deal with a new scale of spaces, and the functions belonging to them can be viewed as functions on a graph. Indeed, let us represent the dyadic family F as a tree. Its root vertex corresponds to the d-dimensional unit cube, and 2 d edges emanate from each vertex. Then the "weight" of each vertex is defined in terms of local approximation by polynomials on the cube corresponding to that vertex. This treatment of a space enables us to simplify the proofs, to make them more geometric, and to discard any restrictions on the integrability exponent p. It should be noted that the traditional approach to classical spaces, as developed by Nikol skiȋ and Besov (see, e.g., [2]), is based on integral representations and is not applicable for 0 < p < 1. So, dyadic spaces constitute a useful discretization of classical spaces and provide a useful language for the description of applied algorithms.Second, dyadic B-spaces are closely related to classical B-spaces. Compared to its classical counterpart, the dyadic space B . Thanks to this, many results for B-spaces can be deduced from similar results for dyadic spaces. Our objective in this paper is to show how the techniques of dyadic spaces can be used in classical analysis; this will be done by considering the calculation of a K-functional as an example.A different approach to the calculation of K-functionals was presented in [3]- [5]. The present paper can be regarded as a continuation of the author's paper [1]; it is organized as follows.

Piecewise Polynomial Approximation Methods in the Theory of Nikol’Skiĭ–Besov Spaces

“…The next result (proved in [65]) shows that Theorem 4.15 allows us to take off the restriction for the parameter λ imposed in Theorem 4.14.…”

“…The main part of the mentioned results is contained in [52][53][54][55][56][57][58][59][60][61][62][63][64][65][66]. The author is sincerely grateful to Professor Yu.…”