A Cantor-Lebesgue theorem with variable ``coefficients' |
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Authors: | J. Marshall Ash Gang Wang David Weinberg |
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Affiliation: | Department of Mathematics, DePaul University, Chicago, Illinois 60614-3504 ; Department of Mathematics, DePaul University, Chicago, Illinois 60614-3504 ; Department of Mathematics, Texas Tech University, Lubbock, Texas 79409-1042 |
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Abstract: | If is a lacunary sequence of integers, and if for each , and are trigonometric polynomials of degree then must tend to zero for almost every whenever does. We conjecture that a similar result ought to hold even when the sequence has much slower growth. However, there is a sequence of integers and trigonometric polynomials such that tends to zero everywhere, even though the degree of does not exceed for each . The sequence of trigonometric polynomials tends to zero for almost every , although explicit formulas are developed to show that the sequence of corresponding conjugate functions does not. Among trigonometric polynomials of degree with largest Fourier coefficient equal to , the smallest one ``at' is while the smallest one ``near' is unknown. |
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Keywords: | Cantor Lebesgue theorem conjugate trigonometric series lacunary trigonometric series Plessner's theorem trigonometric polynomials |
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