Abstract
Extending the notion of the general monotonicity for sequences to functions, we exploit it to investigate integrability problems for Fourier transforms. The problem of controlling integrability properties of the Fourier transform separately near the origin and near infinity is examined. We then apply the obtained results to the problems of integrability of trigonometric series.
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References
N. Bary, A Treatise on Trigonometric Series. Pergamon, Oxford, 1964.
H. Bateman, A. Erdélyi, Tables of integral transforms, Vol. I. McGraw Hill Book Company, New York, 1954.
E.S. Belinsky, On asymptotic behavior of integral norms of trigonometric polynomials. Metric Questions of the Theory of Functions and Mappings, Nauk. Dumka Kiev 6 (1975), 15–24 (Russian).
P.L. Butzer, R.J. Nessel, Fourier Analysis and Approximation. Birkhäuser Verlag, Basel, 1971.
M. Dyachenko, S. Tikhonov, Convergence of trigonometric series with general monotone coefficients. Comptes Rendus Mathématique, Acad. Sci. Paris, Vol. 345, Is. 3, 1, (2007) 123–126.
J.-P. Kahane, Séries de Fourier absolument convergentes. Springer-Verlag, Berlin-Heidelberg-New York, 1970.
L. Leindler, On the uniform convergence and boundedness of a certain class of sine series. Anal. Math. 27 (2001), 279–285.
E. Liflyand, Fourier transforms of functions from, certain classes. Anal. Math. 19 (1993), 151–168.
E. Liflyand, S. Tikhonov, A concept of general monotonicity and applications. Submitted for publication.
S.A. Telyakovskií, On the behavior near the origin of sine series with convex coefficients. Publ. l’lns. Math., 58(72) (1995), 43–50.
S.A. Telyakovskií, On the behavior of sine series near zero. Makedon. Akad. Nauk. Umet. Oddel. Mat-Tehn. Nauk. Prolozi 21(2000), no. 1–2, 47–54(2002) (Russian).
S. Tikhonov, Trigonometric series with general monotone coefficients. J. Math. Anal. Appl. 326 (2007), 721–735.
S. Tikhonov, Best approximation and moduli of smoothness: computation and equivalence theorems. J. Approx. Theory 153 (2008), 19–39.
E.C. Titchmarsh, Introduction to the theory of Fourier integrals. Oxford, 1937.
R.M. Trigub, A Generalization of the Euler-Maclaurin formula. Mat. Zametki 61 (1997), 312–316 (Russian). — English translation in Math. Notes 61 (1997), 253–257.
R.M. Trigub, E.S. Belinsky, Fourier Analysis and Approximation of Functions. Kluwer, 2004.
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Liflyand, E., Tikhonov, S. (2009). The Fourier Transforms of General Monotone Functions. In: Gustafsson, B., Vasil’ev, A. (eds) Analysis and Mathematical Physics. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-9906-1_18
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DOI: https://doi.org/10.1007/978-3-7643-9906-1_18
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