A Cantor-Lebesgue theorem

with variable ``coefficients''

Authors:
J. Marshall Ash, Gang Wang and David Weinberg

Journal:
Proc. Amer. Math. Soc. **125** (1997), 219-228

MSC (1991):
Primary 42A05; Secondary 42A50, 42A55

MathSciNet review:
1350931

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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.

**[AKR]**J. Marshall Ash, Eric Rieders, and Robert P. Kaufman,*The Cantor-Lebesgue property*, Israel J. Math.**84**(1993), no. 1-2, 179–191. MR**1244667**, 10.1007/BF02761699**[AWa]**J. M. Ash and G. Wang,*One and two dimensional Cantor-Lebesgue type theorems,*Trans. Amer. Math. Soc. (to appear). CMP**96:03****[AWe]**J. Marshall Ash and Grant V. Welland,*Convergence, uniqueness, and summability of multiple trigonometric series*, Trans. Amer. Math. Soc.**163**(1972), 401–436. MR**0300009**, 10.1090/S0002-9947-1972-0300009-X**[C]**R. L. Cooke,*The Cantor-Lebesgue theorem*, Amer. Math. Monthly**86**(1979), no. 7, 558–565. MR**542767**, 10.2307/2320583**[R]**Walter Rudin,*Real and complex analysis*, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR**924157****[Z]**A. Zygmund,*Trigonometric series. 2nd ed. Vols. I, II*, Cambridge University Press, New York, 1959. MR**0107776**

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Additional Information

**J. Marshall Ash**

Email:
mash@math.depaul.edu

**Gang Wang**

Affiliation:
(J. M. Ash and G. Wang) Department of Mathematics, DePaul University, Chicago, Illinois 60614-3504

Email:
gwang@math.depaul.edu

**David Weinberg**

Affiliation:
(D. Weinberg)Department of Mathematics, Texas Tech University, Lubbock, Texas 79409-1042

Email:
weinberg@math.ttu.edu

DOI:
https://doi.org/10.1090/S0002-9939-97-03568-5

Keywords:
Cantor Lebesgue theorem,
conjugate trigonometric series,
lacunary trigonometric series,
Plessner's theorem,
trigonometric polynomials

Received by editor(s):
July 27, 1995

Additional Notes:
The research of J. M. Ash and G. Wang was partially supported by a grant from the Faculty and Development Program of the College of Liberal Arts and Sciences, DePaul University

Communicated by:
Christopher D. Sogge

Article copyright:
© Copyright 1997
American Mathematical Society