Wiener type theorems for Jacobi series with nonnegative coefficients
Authors:H. N. Mhaskar and S. Tikhonov Journal:
Proc. Amer. Math. Soc. 140 (2012), 977-986
MSC (2010):
Primary 33C45, 42C10; Secondary 46E30
Published electronically:
August 31, 2011
MathSciNet review:2869082 Full-text PDF
Abstract: This paper gives three theorems regarding functions integrable on with respect to Jacobi weights and having nonnegative coefficients in their (Fourier-)Jacobi expansions. We show that the -integrability (with respect to the Jacobi weight) on an interval near implies the -integrability on the whole interval if is an even integer. The Jacobi expansion of a function locally in near is shown to converge uniformly and absolutely on ; in particular, such a function is shown to be continuous on . Similar results are obtained for functions in local Besov approximation spaces.
2.Richard
Askey, Smoothness conditions for Fourier series with monotone
coefficients, Acta Sci. Math. (Szeged) 28 (1967),
169–171. MR 0212474
(35 #3345)
3.Richard
Askey, Orthogonal polynomials and special functions, Society
for Industrial and Applied Mathematics, Philadelphia, Pa., 1975. MR 0481145
(58 #1288)
6.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
(95f:41001)
7.A.
A. Konyuškov, Best approximations by trigonometric
polynomials and Fourier coefficients, Mat. Sb. N.S.
44(86) (1958), 53–84 (Russian). MR 0096074
(20 #2571)
15.A.
F. Timan, Theory of approximation of functions of a real
variable, Translated from the Russian by J. Berry. English translation
edited and editorial preface by J. Cossar. International Series of
Monographs in Pure and Applied Mathematics, Vol. 34, A Pergamon Press Book.
The Macmillan Co., New York, 1963. MR 0192238
(33 #465)
J. M. Ash, S. Tikhonov and J. Tung, Wiener's positive Fourier coefficients theorem in variants of spaces, Michigan Math. J. 59, 1 (2010), 143-152. MR 2654143
R. Askey, Orthogonal Polynomials and Special Functions, Regional Conference Series in Applied Mathematics 21, SIAM, Philadelphia, 1975. MR 0481145 (58:1288)
R. A. Askey and G. Gasper, Jacobi polynomial expansions of Jacobi polynomials with non-negative coefficients, Math. Proc. Cambridge Philos. Soc. 70 (1971), 243-255. MR 0296369 (45:5430)
A. A. Konyushkov, Best approximations by trigonometric polynomials and Fourier coefficients (Russian), Mat. Sb. N.S. 44(86) (1958), 53-84. MR 0096074 (20:2571)
S. 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 0236397 (38:4693)
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