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

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.)
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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
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