## Abstract

If {φ_{n}} is a lacunary sequence of integers, and if for each n, c_{n}(x) and c_{-n}(x) are trigonometric polynomials of degree n, then (c_{n}(x)} must tend to zero for almost every x whenever {c_{n}(x)e^{iφnx} + c_{-n}(-x)e-^{iφnx}} does. We conjecture that a similar result ought to hold even when the sequence {φ_{n}} has much slower growth. However, there is a sequence of integers {n_{j}} and trigonometric polynomials {P_{j}} such that {e^{injx} - P_{j}(x)} tends to zero everywhere, even though the degree of P_{j} does not exceed n_{j} - j for each j. The sequence of trigonometric polynomials {√nsin^{2n} 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 4^{n}(_{n}^{2n})^{-1} sin^{2n} (f ), while the smallest one "near" x = 0 is unknown.

Original language | English |
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Pages (from-to) | 219-228 |

Number of pages | 10 |

Journal | Proceedings of the American Mathematical Society |

Volume | 125 |

Issue number | 1 |

DOIs | |

State | Published - 1997 |

## Keywords

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