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THE SETS OF CONVERGENCE IN MEASURE OF MULTIPLE ORTHOGONAL FOURIER SERIES
Published online by Cambridge University Press: 20 July 2005
Abstract
Let $\{ \vf_k(x),\,k=1,2,\ldots\}$ be an arbitrary orthonormal system on [0,1] that is uniformly bounded by a constant $M$. Let $T$ be a subset of [0,1]$^2$ such that the Fourier series of all Lebesgue integrable functions on [0,1]$^2$ with respect to the product system $\{ \vf_k(x) \vf_l(y),\,k, l=1,2,\ldots\}$ converge in measure by squares on $T$. The following problem is studied. How large may the measure of $T$ be?
A theorem is proved that implies that for each such system, there is $\mu_2T \leq 1-M^{-4} $ (for the $d$-fold product systems, $\mu_dT \,{\leq}\, 1-M^{-2d}$, d\,{\geq}\, 2$). This estimate is sharp in the class of all such product systems.
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- Notes and Papers
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- The London Mathematical Society 2005