Union of sets of divergence of functional series

Abstract

Listen

Translate article

The development of the theory of functional series gave new methods for obtaining, for example, such classical results as the examples of A. N. Kolmogorov of almost everywhere and everywhere divergent Fourier trigonometric series. Using a result of Katznelson (see [i], [2, pp. 55-61]), it is easy to deduce the existence of almost everywhere divergent Fourier trigonometric series from such a general theorem as the theorem of Bochkarev (see [3], [4, p. 67]) on the existence of a Fourier series, divergent on a set of positive measure, for any jointly bounded orthonormal system. And by drawing on additional results on the divergence of Fourier trigonometric series on sets of measure zero, one can also prove the existence of everywhere divergent Fourier trigonometric series. The essence of the method of Y. Katznelson, subsequently refined and extended by Sh. V. Kheladze to bounded multiplicative Vilenkin systems (see [5-7]), involves the possibility of constructing for any finite or countable collection of Fourier trigonometric series a Fourier trigonometric series, divergent at each point where at least one of the series of the collection is divergent. Here one makes essential use of the specifics of trigonometric series (in the case of Sh. V. Kheladze, of bounded multiplicative Vilenkin systems). In See. i of the present paper we investigate whether an analogous result can be obtained for general functional series, if the new series is taken as a sum with rapidly decreasing coefficients of given series. It turns out that in this way one can construct series, which, up to sets of measure zero, diverge at precisely those points at which at least one of the given series diverges, while it is shown that the ex_ceptional set of measure zero can be a continuum for any coefficients.