Nikolskii-type inequalities in connection with regular spectral measures

Abstract

There are counterparts for polynomials in several variables as well as for entire functions of exponential type, also connected with the names of D. Jackson, M. Plancherel, G. Polya, G. Szeg6, A. Zygmund, for all the details and comprehensive bibliographical comments see [8, 15, 16]. The aim of the present note is to construct a framework which enables one to consider inequalities of type (1.2) within a general class of orthogonal expansions in Banach spaces. To this end, Section 2 develops a multiplier concept for Banach spaces Xwhich are admissible with respect to a regular spectral measure E for some appropriate Hilbert space H. This continues and extends slightly our approach given in [4] which already turned out to be quite useful in connection with other generalizations (cf. [16] and the literature cited there). Let us mention that the classical situation (1.2) is covered for 1 <p=< c~ with H = L ~ and X=LP~, 1 < p < ~ , or X = L ~ =(L~)*, respectively. Section 3 defines polynomials in this general frame and sets up de la Vallre Poussin (or delayed) means (see Theorem 2), a basic tool in the treatment of problems (such as (1.2)) coneemed with functions having compact spectra. Section 4 first deals with interpolation of admissible Banach spaces. Here we only discuss the real method of Lions-Peetre but other constructions such as the complex method of Calderon may be considered as well (see [9, 18] for details). This is then used in Theorem 3 to derive general Nikolskii-type inequalities. Finally, Section 5 is concerned with some applications to illustrate the wide applicability of the general results obtained. While Section 5.1 considers trigonometric polynomials in several variables for different scales of admissible Banach spaces, Section 5.2 treats the (eontinuous)