Universal series

A series of functions

(1)

by means of which all functions of a given class can be represented in some way or other. For example, there exists a series (1) such that every continuous function on can be approximated by partial sums of this series, , converging uniformly to on .

There exist trigonometric series

(2)

with coefficients tending to zero, such that every (Lebesgue-) measurable function on has an approximation by partial sums of the series (2), converging to almost everywhere.

The given series is called universal relative to approximation by partial sums. One may consider other definitions of universal series. For example, series (1) which are universal relative to subseries or relative to permutations of the terms of (1).

References

[1]	G. Alexits, "Convergence problems of orthogonal series" , Pergamon (1961) (Translated from German)
[2]	N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian)
[3]	A.A. Talalyan, "The representation of measurable functions by series" Russian Math. Surveys , 15 : 5 (1960) pp. 75–136 Uspekhi Mat. Nauk , 15 : 5 (1960) pp. 77–141