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Smoothness of functions and Fourier coefficients: a functional analyst's approach

Published online by Cambridge University Press:  24 October 2008

P. Wojtaszczyk
P. Wojtaszczyk
Affiliation:
IMPAN, Warsaw, Poland
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Abstract

We give new proofs of transference theorems which allow the transfer of the bad behaviour of Fourier coefficients from one complete orthonormal system to the other. We also present some results on Carleman-type singularities.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 103 , Issue 1 , January 1988 , pp. 117 - 126
Copyright
Copyright © Cambridge Philosophical Society 1988

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References

REFERENCES

[1]Bari, N. K.. Trigonometric Series (Fizmatgiz, Moscow, 1961) (in Russian).Google Scholar
[2]Bennett, G.. Lectures on matrix transformations of lp spaces, in Notes on Banach Spaces (University of Texas Press, 1980).Google Scholar
[3]Bernstein, S.. Sur la convergence absolue des séries trigonometriques. C. R. Acad. Sci. Paris 158 (1914), 1661–1663.Google Scholar
[4]Bočkariov, S. V.. A method of averaging in the theory of orthogonal series and some problems in the theory of bases. Trudy Steklov Inst. 146 (1978) (in Russian).Google Scholar
[5]Bourgain, J.. New Banach space properties of the disc algebra and H ∞. Acta Mathematica 152 (1984), 1–48.CrossRefGoogle Scholar
[6]Ciesielski, Z.. Properties of the orthonormal Franklin system II. Studia Math. 27 (1966), 289–323.CrossRefGoogle Scholar
[7]Fredholm, I.. Sur une classe d'équations fonctionelles. Acta Math. 27 (1903), 365–390.CrossRefGoogle Scholar
[8]Gilbert, J. E. and Leih, T.. Factorisation, tensor products and bilinear forms in Banach space theory, in Notes in Banach Spaces (University of Texas Press, 1980).Google Scholar
[9]Gohberg, J. C. and Krein, M. G.. Introduction to the theory of linear self-adjoint operators (Nauka, Moscow, 1965) (in Russian).Google Scholar
[10]Grothendieck, A.. Résumé de la théorie métrique des produits tensoriels topologiques. Bol. Soc. Mat. São-Paulo 8 (1956), 1–79.Google Scholar
[11]Gulisashvili, A. B.. On singularities of summable functions. Zapiski Naucz. Sem. LOMI, vol. 113 (Leningrad, 1981), 76–95 (in Russian).Google Scholar
[12]Kahane, J.-P.. Séries de Fourier absolument convergentes (Springer-Verlag, 1970).CrossRefGoogle Scholar
[13]Kashin, B. S.. On coefficients of developments of one class of functions with respect to complete systems. Sibirsk. Mat. J. 18 (1977), 122–131 (in Russian).Google Scholar
[14]Kashin, B. S. and Saakjan, A. A.. Orthogonal Series (Nauka, Moscow, 1984) (in Russian).Google Scholar
[15]Lindenstrauss, J. and Pelczyński, A.. Absolutely summing operators in ℒp spaces and their applications. Studia Math. 29 (1968), 275–326.CrossRefGoogle Scholar
[16]Maurey, B.. Théorèmes de factorisation pour les opérateurs linéaires à valeurs dans les espaces Lp. Astérisque 11 (1974).Google Scholar
[17]Mitiagin, B. S.. Absolute convergence of the series of Fourier coefficients. Dokl. AN SSSR 157 (1964), 1047–1050 (in Russian).Google Scholar
[18]Olevskii, A. M.. On singularities of Carleman type for complete orthonormal systems. Sibirsk. Mat. J. 8 (1967), 807–826 (in Russian).Google Scholar
[19]Olevskii, A. M.. Fourier Series with Respect to General Orthogonal Systems (Springer-Verlag, 1975).CrossRefGoogle Scholar
[20]Saburova, T. N.. On embedding theorems for some classes of continuous functions. Vestnik M.G.U. series 1, Math. Mech. No. 3 (1971).Google Scholar
[21]Szasz, O.. Über den Konvergenzexponenten der Fourierischen Reihen gewisser Funktionenklassen. S.-B. Math.-Phys. Kl: 1922, 135–150.Google Scholar
[22]Zygmund, A.. Trigonometric Series (Cambridge University Press, 1959).Google Scholar