Wiener type theorems for Jacobi series with nonnegative coefficients
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- by H. N. Mhaskar and S. Tikhonov PDF
- Proc. Amer. Math. Soc. 140 (2012), 977-986 Request permission
Abstract:
This paper gives three theorems regarding functions integrable on $[-1,1]$ with respect to Jacobi weights and having nonnegative coefficients in their (Fourier–)Jacobi expansions. We show that the $L^p$-integrability (with respect to the Jacobi weight) on an interval near $1$ implies the $L^p$-integrability on the whole interval if $p$ is an even integer. The Jacobi expansion of a function locally in $L^\infty$ near $1$ is shown to converge uniformly and absolutely on $[-1,1]$; in particular, such a function is shown to be continuous on $[-1,1]$. Similar results are obtained for functions in local Besov approximation spaces.References
- J. M. Ash, S. Tikhonov, and J. Tung, Wiener’s positive Fourier coefficients theorem in variants of $L^p$ spaces, Michigan Math. J. 59 (2010), no. 1, 143–151. MR 2654143, DOI 10.1307/mmj/1272376029
- Richard Askey, Smoothness conditions for Fourier series with monotone coefficients, Acta Sci. Math. (Szeged) 28 (1967), 169–171. MR 212474
- Richard Askey, Orthogonal polynomials and special functions, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1975. MR 0481145
- Richard Askey and George Gasper, Jacobi polynomial expansions of Jacobi polynomials with non-negative coefficients, Proc. Cambridge Philos. Soc. 70 (1971), 243–255. MR 296369, DOI 10.1017/s0305004100049847
- Ralph Philip Boas Jr., Entire functions, Academic Press, Inc., New York, 1954. MR 0068627
- Ronald A. DeVore and George G. Lorentz, Constructive approximation, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 303, Springer-Verlag, Berlin, 1993. MR 1261635
- A. A. Konyuškov, Best approximations by trigonometric polynomials and Fourier coefficients, Mat. Sb. N.S. 44(86) (1958), 53–84 (Russian). MR 0096074
- Tom Koornwinder, Jacobi polynomials. II. An analytic proof of the product formula, SIAM J. Math. Anal. 5 (1974), 125–137. MR 385198, DOI 10.1137/0505014
- G. G. Lorentz, Fourier-Koeffizienten und Funktionenklassen, Math. Z. 51 (1948), 135–149 (German). MR 25601, DOI 10.1007/BF01290998
- H. N. Mhaskar, Polynomial operators and local smoothness classes on the unit interval, J. Approx. Theory 131 (2004), no. 2, 243–267. MR 2106540, DOI 10.1016/j.jat.2004.10.002
- R. E. A. C. Paley, On Fourier series with positive coefficients, J. London Math. Soc. 7 (1932), 205–208.
- Harold S. Shapiro, Majorant problems for Fourier coefficients, Quart. J. Math. Oxford Ser. (2) 26 (1975), 9–18. MR 372515, DOI 10.1093/qmath/26.1.9
- Gábor Szegő, Orthogonal polynomials, 4th ed., American Mathematical Society Colloquium Publications, Vol. XXIII, American Mathematical Society, Providence, R.I., 1975. MR 0372517
- Sergey Tikhonov, Characteristics of Besov-Nikol′skiĭ class of functions, Electron. Trans. Numer. Anal. 19 (2005), 94–104. MR 2149272
- A. F. Timan, Theory of approximation of functions of a real variable, A Pergamon Press Book, The Macmillan Company, New York, 1963. Translated from the Russian by J. Berry; English translation edited and editorial preface by J. Cossar. MR 0192238
- Stephen Wainger, A problem of Wiener and the failure of a principle for Fourier series with positive coefficients, Proc. Amer. Math. Soc. 20 (1969), 16–18. MR 236397, DOI 10.1090/S0002-9939-1969-0236397-8
- A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776
Additional Information
- H. N. Mhaskar
- Affiliation: Department of Mathematics, California State University, Los Angeles, California 90032
- Email: hmhaska@calstatela.edu
- S. Tikhonov
- Affiliation: ICREA and Centre de Recerca Matemàtica, Campus de Bellaterra, Edifici C, 08193 Bellaterra (Barcelona), Spain
- Email: stikhonov@crm.cat
- Received by editor(s): February 27, 2010
- Received by editor(s) in revised form: December 19, 2010
- Published electronically: August 31, 2011
- Additional Notes: The research of the first author was supported in part by grant DMS-0908037 from the National Science Foundation and grant W911NF-09-1-0465 from the U.S. Army Research Office.
The research of the second author was supported in part by grants MTM2008-05561-C02-02/MTM, 2009 SGR 1303, RFFI 09-01-00175, and NSH3252.2010.1. - Communicated by: Walter Van Assche
- © Copyright 2011 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 140 (2012), 977-986
- MSC (2010): Primary 33C45, 42C10; Secondary 46E30
- DOI: https://doi.org/10.1090/S0002-9939-2011-10964-X
- MathSciNet review: 2869082