## A Method of Averaging in the Theory of Orthogonal Series and Some Problems in the Theory of Bases, Issue 146 |

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### Contents

Chapter I Averaging Over Independent Arrangements of Signs | 5 |

Averaging Over Shifts and Singular Functions | 37 |

Averaging Over Supports of 8 Functions | 67 |

Chapter IV Some Problems in the Theory Bases | 75 |

### Common terms and phrases

absolute convergence absolutely continuous function arbitrary complete orthonormal arbitrary orthonormal systems arrangements of signs averaging over shifts bounded complete orthonormal bounded orthonormal systems bounded variation Chapter coefficients of functions complete orthonormal system continuum of singular converge absolutely convergence almost everywhere convergence of Fourier converges uniformly converse assertion defined Dirichlet kernels dyadic rational exists a function Fatou-Lebesgue theorem Fn(x Fourier coefficients Fourier-Haar series Franklin system function f(t functions of bounded Fx(t Haar functions Haar system interval Lebesgue functions Littlewood conjecture lower estimate modulus of continuity natural number norm obtained orthogonal series polynomial positive constant positive measure proof of Theorem Schauder functions Schauder system sequence of integers sequence of numbers series of functions shifts and singular singular functions supports of 6-functions Suppose system of functions system of uniformly system pn Szasz Theorem 1.1 transfer theorem trigonometric system uniformly bounded functions Walsh system Zygmund problem

### Popular passages

Page 3 - This contradicts the fact that A is not a cylinder. The theorem is now proved. Naturally the question arises as to whether a non-recursive m-degree that is not indecomposable contains a countable number of pairwise incomparable 1 -degrees. For re m-degrees a positive answer to this question was obtained by the author in [4] . Here we give a more modest result illustrating the basic idea of the construction of such an antichain. THEOREM 2. If S is a re non-recursive set and not a cylinder, then sets...