Convergence multipliers

for a series $\sum_{n=0}^\infty u_n(x)$ of functions

Numbers $\lambda_n$, $n=0,1,\ldots,$ such that the series $\sum_{n=0}^\infty\lambda_nu_n(x)$ converges almost-everywhere on a measurable set $X$, where the $u_n(x)$ are numerical functions defined on $X$.

For example, for the trigonometric Fourier series of a function from $L_1$, the numbers $\lambda_n=1/\ln n$, $n=2,3,\ldots,$ are convergence multipliers ($\lambda_0$ and $\lambda_1$ can be chosen arbitrarily), i.e. if $f\in L_1[-\pi,\pi]$ and if

$$\frac{a_0}{2}+\sum_{n=1}^\infty a_n\cos nx+b_n\sin nx$$

is its trigonometric Fourier series, then the series

$$\sum_{n=2}^\infty\frac{a_n\cos nx+b_n\sin nx}{\ln n}$$

converges almost-everywhere on the whole real line. If $f\in L_p[-\pi,\pi]$, $p>1$, then its trigonometric Fourier series itself converges almost-everywhere (see Carleson theorem).