The modulus of continuity and the best approximation over the dyadic group. (English) Zbl 0774.41026

Tateoka, J.

Acta Math. Hung. 59, No. 1-2, 115-120 (1992).

The connection between the modulus of continuity and the best approximation of functions by Walsh polynomials was studied by C. Watari [Tohoku Math. J., II. Ser. 15, 1-5 (1963; Zbl 0111.265)] for \(L^ p\) space, \(1\leq p<\infty\). A similar result for \(0<p<1\) was obtained by E. A. Storozenko, V. G. Krotov and P. Oswal’d [Math. Sb., n. Ser. 98(140), 395-415 (1975; Zbl 0314.41004)]. On the other hand, direct and converse theorems for the Hardy space \(H^ p\), \(0<p<\infty\), over the \(n\)-dimensional torus were proved by L. Colzani [Ann. Math. Pure Appl., IV. Ser. 137, 207-215 (1984; Zbl 0558.41017)]. In this paper these results for the \(H^ p\) space, \(0<p\leq 1\) and VMO space over the dyadic group are proved.

Reviewer: D.Zarnadze (Tbilisi)

MSC:

41A50	Best approximation, Chebyshev systems
41A17	Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
41A27	Inverse theorems in approximation theory
46A16	Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.)

References:

[1]	L. Colzani, Approximation in Hardy spaces, Ann. Math. Pure Appl., 137 (1984), 207–215. · Zbl 0558.41017 · doi:10.1007/BF01789395
[2]	A. I. Rubinshtein, Moduli of continuity of functions, defined on a zero-dimensional group, Math. Note, 23 (1978), 205–211.
[3]	È. A. Stroženko, V. G. Krotov and P. Oswal’d, Direct and converse theorems of Jackson type in L p spaces, 0<p<1, Math. USSR Sbornik, 27 (1975), 355–374. · Zbl 0372.41004 · doi:10.1070/SM1975v027n03ABEH002519
[4]	M. H. Taibleson, Fourier Analysis on Local Fields (Princeton, 1975). · Zbl 0319.42011
[5]	C. Watari, Best approximation by Walsh polynomials, Tôhoku Math. J., 15 (1963), 1–5. · Zbl 0111.26502 · doi:10.2748/tmj/1178243865