# PREPRINT 07/2005

S.V. Kislyakov and N.Ya. Kruglyak

## STABILITY OF APPROXIMATION UNDER SINGULAR INTEGRALS, AND CALDER\'ON--ZYGMUND TYPE DECOMPOSITIONS

This preprint was accepted March 20, 2005

ABSTRACT:
Let $X$ be a space of smooth'' functions on $\Bbb R^n$ (like
$\Lip_{\al}$, or $W^{(1)}_p$, or BMO). It is fairly well known
that a specific Calder\'on--Zygmund type algorithm can be used
to exhibit an element $u$ of almost optimal $L^1$-approximation
of a given function $f\in L^1$ by the ball of a fixed radius in
$X$. We show that for most of the singular integral operators
$T$ the function $Tu$ also approximates $Tf$ in the same
$L^1$-optimal sense provided $Tf$ is integrable.

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