On Marcinkiewicz-Zygmund Type Inequalities for Irregular Knots in Lp-Spaces, 0 p ≦ +∞
✍ Scribed by K. V. Runovskii; H. J. Schmeisser
- Publisher
- John Wiley and Sons
- Year
- 1998
- Tongue
- English
- Weight
- 432 KB
- Volume
- 189
- Category
- Article
- ISSN
- 0025-584X
No coin nor oath required. For personal study only.
✦ Synopsis
An equivalence of a discrete norm and a continuous norm of a trigonometric polynomial is proved for the cme of irregular knots in L, -spaces, where 0 < p 5 +m.
1. Introduction
Theorems on equivalent norms of trigonometric polynomials in certain metrics have many applications in the modern theory of functions (cf., e. g., [ll]). The problem of equivalence of continuous and discrete norms of one variable trigonometric polynomials in the case of regular knots and L, -spaces, where 1 5 p 5 co , was investigated by J. MARCINKIEWICZ and A. ZYGMUND ([12, pp. 46, 54 I).
The main purpose of this article is to establish the analogies of Marcinkiewicz-Zygmund type theorems for irregular knots in general case 0 < p 5 +m. The interest in the case 0 < p < 1 stems from approximation theory on the one hand and from the theory of function spaces on the other hand (cf. [6], [lo]). In particular, it turned out that in various respects p = 1 is not a natural lower bound for admissible parameters. Key inequalities (JACKSON, BERNSTEIN, NIKOLSKIJ) are valid for 0 < p c 1 , too (cf.
1991 Mathematics Subject Classification. Keywords and phrases.