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Journal of the London Mathematical Society 2005 72(1):239-257; doi:10.1112/S0024610705006678
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© The London Mathematical Society

The Sets of Convergence in Measure of Multiple Orthogonal Fourier Series

Rostom Getsadze

Department of Mathematics, University of Umeå S-901 87, Umeå, Sweden rostom.getsadze{at}math.umu.se

Received 10 May 2004.

Let {{varphi}k(x), k = 1, 2, ...} 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 {{varphi}k(x) {varphi}l(y), k, l = 1,2,...} 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

µ2T ≤ 1 – M–4

(for the d-fold product systems, µdT ≤ 1 – M–2d, d≥2). This estimate is sharp in the class of all such product systems.


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