STRONG APPROXIMATION BY DIRICHLET INTEGRALS IN L-LAMBDA(R)-NORM, 1-LESS-THAN-LAMBDA-LESS-THAN-INFINITY

[摘要] Let f be a real valued function which belongs to L(r) = L(r)(-infinity, infinity) for some 1 less than or equal to r < infinity. We consider the cosine transform ($) over cap f(c), sine transform ($) over cap f(s), (complex) Fourier transform ($) over tilde f, and Hilbert transform f of f. We study the strong approximation of order p, 0 < p < infinity, of f and ($) over tilde f by their Dirichlet integrals, respectively. We prove that the saturation class in L(lambda)-norm is the Lizorkin-Triebel space F(lambda,)(alpha)p, where alpha = 1/p, 2 less than or equal to p < infinity, and 1 < lambda < infinity. To this effect, we introduce several so-called Littlewood-Paley functions and make use of a number of equivalence theorems. Our machinery is also appropriate to characterize the saturation class concerning the strong approximation of order p of a periodic function f is an element of L(2 pi)(1) = L(1)(-pi,pi) by the partial sums of its Fourier series in L(2 pi)(lambda)-norm, where again 2 less than or equal to p < infinity and 1 < lambda < infinity. (C) 1994 Academic Press, Inc.

[发布日期] 1994-11-01