View source for Universal series

A [[Series|series]] of functions

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095740/u0957401.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095740/u0957402.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095740/u0957403.png" /> can be approximated by partial sums of this series, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095740/u0957404.png" />, converging uniformly to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095740/u0957405.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095740/u0957406.png" />.

There exist trigonometric series

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095740/u0957407.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

with coefficients tending to zero, such that every (Lebesgue-) measurable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095740/u0957408.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095740/u0957409.png" /> has an approximation by partial sums of the series (2), converging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095740/u09574010.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095740/u09574011.png" /> or relative to permutations of the terms of (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>