View source for Universal series

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A [[Series|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Alexits,   "Convergence problems of orthogonal series" , Pergamon  (1961)  (Translated from German)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.K. [N.K. Bari] Bary,   "A treatise on trigonometric series" , Pergamon  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  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</TD></TR></table>