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On Grasia's criterion for uniform convergence of Fourier series

Part of: Harmonic analysis in one variable

Published online by Cambridge University Press:  09 April 2009

Charles Oehring
Charles Oehring
Affiliation:
Virginia Polytechnic Institute & State University Blacksburg, Virginia 24061-0123, U.S.A.
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Abstract

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Garsia's discovery that functions in the periodic Besov space λ(p-1, p, 1), with 1 < p < ∞, have uniformly convergent Fourier series prompted him, and others, to seek a proof based on one of the standard convergence tests. on one of the standard convergence tests. We show that Lebesgue's test is adequate, whereas Garsia's criterion is independent of other classical critiera (for example, that of Dini-Lipschitz). The method of proof also produces a sharp estimate for the rate of uniform convergence for functions in λ(p-1, p, 1). Further, it leads to a very simple proof of the embedding theorem for these spaces, which extends (though less simply) to λ(α, p, q)

MSC classification

Secondary: 42A20: Convergence and absolute convergence of Fourier and trigonometric series
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 51 , Issue 2 , October 1991 , pp. 305 - 323
Copyright
Copyright © Australian Mathematical Society 1991

References

[1]Aljančič, S., and Tomič, M., ‘Uber den Stetigkeitsmodul von Fourier-Reihen mit monotone Koeffzienten’, Math. Z. 88 (1965), 274–284.CrossRefGoogle Scholar
[2]Bergh, J. and Löfström, J., Interpolation Spaces, (Springer-Verlag, 1976).CrossRefGoogle Scholar
[3]Besov, O. V., ‘Investigation of a family of function spaces in connection with theorems of imbedding and extension’, (Russian), Trudy. Mat. Inst. Steklov. 60 (1961), 42–81.Google Scholar
Amer. Math. Soc. Transl. (2) 40 (1964), 85–126.Google Scholar
[4]Butzer, P. and Berens, H., Semi-groups of Operators and Approximation, (Springer-Verlag, 1967).CrossRefGoogle Scholar
[5]Butzer, P. and Nessel, R., Fourier Analysis and Approximation, Vol. 1 (Academic Press, 1971).Google Scholar
[6]Edwards, R., Fourier Series, a Modern Introduction, Vol. 1, Second Edition (Springer-Verlag, 1979).CrossRefGoogle Scholar
[7]Fournier, J. F. and Self, W. M., ‘Some sufficient conditions for uniform convergence of Fourier series’, J. Math. Anal. App. 126 (1987), 355–374.CrossRefGoogle Scholar
[8]Garsia, A. M., ‘A remarkable inequality and the uniform convergence of Fourier series’, Indiana Univ. Math. J. 25 (1976), 85–102.CrossRefGoogle Scholar
[9]Garsia, A. M., ‘Some combinatorial methods in real analysis’, Proc. Internat. Congress Math. Helsinki (1978), 615–622.Google Scholar
[10]Hardy, G. H., Littlewood, J. E. and Polya, G., Inequalities, (Cambridge University Press, 1952).Google Scholar
[11]Hardy, G. H., and Rogosinski, W. W., Fourier Series, (Cambridge University Press, 1950).Google Scholar
[12]Katznelson, Y., An Introduction to Harmonic Analysis, (Wiley, 1968).Google Scholar
[13]Knopp, K., Theory and Application of Infinite Series, (Blackie, 1951).Google Scholar
[14]Lorentz, G. G., Approximation of Functions, (Holt, Rinehart, and Winston, 1966).Google Scholar
[15]Natanson, I. P., Constructive Function Theory, Vol. I (Ungar, 1964).Google Scholar
[16]Oehring, C. C., ‘Asymptotics of singular numbers of smooth kernels via trigonometric transforms’, J. Math. Anal. App. 145 (1990) 573–605.Google Scholar
[17]Oehring, C. C., ‘Asymptotics of rearranged trigonometric and Walsh-Fourier coefficients of smooth functions’, J. Math. Anal. pp. (to appear).Google Scholar
[18]Pichorides, S. K., ‘On the best values of the constants in the theorems of M. Riesz, Zygmund and Kolmogorov’, Studia Math. 44 (1972) 165–179.CrossRefGoogle Scholar
[19]Potapov, M. K. and Fedorov, V. M., ‘On Jackson theorems for the generalized modulus of smoothness’, Studies in the Theory of Functions of Several Real Variables and the Approximation of Functions, Trudy Mat. Inst. Steklov. 172 (1985), 291–298.Google Scholar
Translated in Proc. Steklov Inst. Mat. (1987), 315–323.Google Scholar
[20]Rivlin, T. J., The Chebyshev Polynomials, (Wiley, 1974).Google Scholar
[21]Stein, E. M., Singular Integrals and Differentiability Properties of Functions, (Princeton University Press, 1970).Google Scholar
[22]Zygmund, A., Trigonometric Series, Vol. I, (Cambridge University Press, 1959).Google Scholar