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

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

MathSciNet review:
1350931

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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. M. Ash, R. P. Kaufman, and E. Rieders,*The Cantor-Lebesgue property,*Israel J. of Math.,**84**(1993), 179-191. MR**94m:42007****[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. M. Ash and G. V. Welland,*Convergence, uniqueness, and summability of multiple trigonometric series*, Trans. Amer. Math. Soc.**163**(1972), 401-436. MR**45:9057****[C]**R. Cooke,*The Cantor-Lebesgue theorem,*Amer. Math. Monthly**86**(1979), 558-565. MR**81b:42019****[R]**W. Rudin,*Real and Complex Analysis,*3rd ed., McGraw-Hill, New York, 1987. MR**88k:00002****[Z]**A. Zygmund,*Trigonometric Series,*Vol. 2, 2nd rev. ed., Cambridge Univ. Press, New York, 1959. MR**21:6498**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
42A05,
42A50,
42A55

Retrieve articles in all journals with MSC (1991): 42A05, 42A50, 42A55

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