A cantor-lebesgue theorem with variable "coefficients"

J. Marshall Ash, Gang Wang, David Weinberg

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

If {φn} is a lacunary sequence of integers, and if for each n, cn(x) and c-n(x) are trigonometric polynomials of degree n, then (cn(x)} must tend to zero for almost every x whenever {cn(x)eiφnx + c-n(-x)e-iφnx} does. We conjecture that a similar result ought to hold even when the sequence {φn} has much slower growth. However, there is a sequence of integers {nj} and trigonometric polynomials {Pj} such that {einjx - Pj(x)} tends to zero everywhere, even though the degree of Pj does not exceed nj - j for each j. The sequence of trigonometric polynomials {√nsin2n x/2} tends to zero for almost every x, although explicit formulas are developed to show that the sequence of corresponding conjugate functions does not. Among trigonometric polynomials of degree n with largest Fourier coefficient equal to 1, the smallest one "at" x = 0 is 4n(n2n)-1 sin2n (f ), while the smallest one "near" x = 0 is unknown.

Original languageEnglish
Pages (from-to)219-228
Number of pages10
JournalProceedings of the American Mathematical Society
Volume125
Issue number1
DOIs
StatePublished - 1997

Keywords

  • Cantor lebesgue theorem
  • Conjugate trigonometric series
  • Lacunary trigonometric series
  • Plessner's theorem
  • Trigonometric polynomials

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