Absolute convergence of series of Fourier coefficients

McLaughlin, James R.

doi:10.1090/S0002-9947-1973-0336203-2

Absolute convergence of series of Fourier coefficients
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by James R. McLaughlin PDF

Trans. Amer. Math. Soc. 184 (1973), 291-316 Request permission

Abstract:

In this article the author unifies and generalizes practically all known sufficiency results for absolute convergence of series of Fourier coefficients that are given in terms of the integrated modulus of continuity, best approximation, or bounded pth variation. This is done for the trigonometric, Walsh, Haar, Franklin, and related systems as well as general orthonormal systems. Many of the original proofs of previous results relied upon special properties of the trigonometric, Haar, and other systems and were done independently of one another. Also, several authors have proved results which at the time they believed to be generalizations of past results, but are, in fact, corollaries of them. The present author will expose underlying principles and illustrate their usefulness.

References

G. Alexits, Konvergenzprobleme der Orthogonalreihen, Verlag der Ungarischen Akademie der Wissenschaften, Budapest, 1960 (German). MR 0162091
N. K. Bari, Trigonometricheskie ryady, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1961 (Russian). With the editorial collaboration of P. L. Ul′janov. MR 0126115
S. L. Bljumin and B. D. Kotljar, Hilbert-Schmidt operators and the absolute convergence of Fourier series, Izv. Akad. Nauk SSSR Ser. Mat. 34 (1970), 209–217 (Russian). MR 0306979
S. V. Bočkarev, Absolute convergence of Fourier series in complete orthogonal systems, Uspehi Mat. Nauk 27 (1972), no. 2(164), 53–76 (Russian). MR 0394024
S. V. Bočkarev, The absolute convergence of Fourier series, Dokl. Akad. Nauk SSSR 202 (1972), 995–997 (Russian). MR 0294980
H. E. Chrestenson, A class of generalized Walsh functions, Pacific J. Math. 5 (1955), 17–31. MR 68659, DOI 10.2140/pjm.1955.5.17
Z. Ciesielski, Properties of the orthonormal Franklin system, Studia Math. 23 (1963), 141–157. MR 157182, DOI 10.4064/sm-23-2-141-157
Z. Ciesielski, Properties of the orthonormal Franklin system. II, Studia Math. 27 (1966), 289–323. MR 203349, DOI 10.4064/sm-27-3-289-323
Z. Ciesielski and J. Musielak, On absolute convergence of Haar series, Colloq. Math. 7 (1959), 61–65. MR 109980, DOI 10.4064/cm-7-1-61-65
N. J. Fine, On the Walsh functions, Trans. Amer. Math. Soc. 65 (1949), 372–414. MR 32833, DOI 10.1090/S0002-9947-1949-0032833-2
B. I. Golubov, On Fourier series of continuous functions with respect to a Haar system, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 1271–1296 (Russian). MR 0172057
B. I. Golubov, Functions of bounded $p$-variation, Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968), 837–858 (Russian). MR 0235080
B. I. Golubov, A certain class of complete orthogonal systems, Sibirsk. Mat. Ž. 9 (1968), 297–314 (Russian). MR 0225083
B. I. Golubov and A. I. Rubinšteĭn, A class of convergence systems, Mat. Sb. (N.S.) 71 (113) (1966), 96–115 (Russian). MR 0200665
I. I. Hirschman Jr., On multiplier transformations, Duke Math. J. 26 (1959), 221–242. MR 104973
Masako Izumi and Shin-ichi Izumi, On absolute convergence of Fourier series, Ark. Mat. 7 (1967), 177–184 (1967). MR 221195, DOI 10.1007/BF02591034

Best approximation by trigonometric polynomials and Fourier coefficients

44

20

James R. McLaughlin, Haar series, Trans. Amer. Math. Soc. 137 (1969), 153–176. MR 236605, DOI 10.1090/S0002-9947-1969-0236605-8
James R. McLaughlin, Integrated orthonormal series, Pacific J. Math. 42 (1972), 469–475. MR 313703, DOI 10.2140/pjm.1972.42.469
C. W. Onneweer, Absolute convergence of Fourier series on certain groups, Duke Math. J. 39 (1972), 599–609. MR 316976, DOI 10.1215/S0012-7094-72-03965-8

A remarkable series of orthogonal functions

34

J. J. Price, Certain groups of orthonormal step functions, Canadian J. Math. 9 (1957), 413–425. MR 87792, DOI 10.4153/CJM-1957-049-x

Absolute convergence of orthogonal series

2

S. B. Stečkin, On absolute convergence of orthogonal series. I, Mat. Sbornik N.S. 29(71) (1951), 225–232 (Russian). MR 0043247
Otto Szász, Quasi-monotone series, Amer. J. Math. 70 (1948), 203–206. MR 22917, DOI 10.2307/2371946
T. Š. Tevzadze, The absolute convergence of Fourier series, Sakharth. SSR Mecn. Akad. Moambe 57 (1970), 525–528 (Russian, with Georgian and English summaries). MR 0270049
P. L. Ul′janov, On Haar series, Mat. Sb. (N.S.) 63 (105) (1964), 356–391 (Russian). MR 0161086
N. Ja. Vilenkin, On a class of complete orthonormal systems, Amer. Math. Soc. Transl. (2) 28 (1963), 1–35. MR 0154042, DOI 10.1090/trans2/028/01
N. Ya. Vilenkin, Supplement to “Theory of orthogonal series”, Amer. Math. Soc. Transl. (2) 17 (1961), 219–250. MR 0124685, DOI 10.1090/trans2/017/08
Chinami Watari, Best approximation by Walsh polynomials, Tohoku Math. J. (2) 15 (1963), 1–5. MR 147816, DOI 10.2748/tmj/1178243865

The quadratic variation of a function and its Fourier coefficients

3

Ingemar Wik, Criteria for absolute convegence of Fourier series of functions of bounded variation, Trans. Amer. Math. Soc. 163 (1972), 1–24. MR 285851, DOI 10.1090/S0002-9947-1972-0285851-6
L. C. Young, An inequality of the Hölder type, connected with Stieltjes integration, Acta Math. 67 (1936), no. 1, 251–282. MR 1555421, DOI 10.1007/BF02401743
A. Zygmund, Trigonometric series: Vols. I, II, Cambridge University Press, London-New York, 1968. Second edition, reprinted with corrections and some additions. MR 0236587