Universal series

A series of functions

$$\sum_{i=1}^\infty\phi_i(x),\quad x\in[a,b],\tag{1}$$

by means of which all functions of a given class can be represented in some way or other. For example, there exists a series \ref{1} such that every continuous function $f$ on $[a,b]$ can be approximated by partial sums of this series, $\sum_{i=1}^{n_k}\phi_i(x)$, converging uniformly to $f(x)$ on $[a,b]$.

There exist trigonometric series

$$\frac{a_0}{2}+\sum_{i=1}^\infty(a_i\cos ix+b_i\sin ix),\tag{2}$$

with coefficients tending to zero, such that every (Lebesgue-) measurable function $f$ on $[0,2\pi]$ has an approximation by partial sums of the series \ref{2}, converging to $f(x)$ 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 \ref{1} which are universal relative to subseries $\sum_{k=1}^\infty\phi_{i_k}(x)$ or relative to permutations of the terms of \ref{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