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A criterion for weak generalized localization in the class \(L_1\) for multiple Fourier trigonometric series from the point of view of isometric transformations. (English. Russian original) Zbl 1024.42004

Math. Notes 71, No. 4, 464-476 (2002); translation from Mat. Zametki 71, No. 4, 508-521 (2002).
Summary: We study the problem of the variation (if any) of the sets of convergence and divergence everywhere or almost everywhere of a multiple Fourier series (integral) of a function \(f\in L_p\), \(p\geq 1\), \(f(x)= 0\), on a set of positive measure \({\mathfrak A}\subset \mathbb{T}^N= [-\pi,\pi)^N\), \(N\geq 2\), depending on the rotation of the coordinate system, i.e., depending on the element \(\tau\in{\mathcal F}\), where \({\mathcal F}\) is the rotation group about the origin in \(\mathbb{R}^N\). This problem has been reduced to the study of the change in the geometry of the sets \(\tau^{-1}({\mathfrak A})\cap \mathbb{T}^N\) (where \(\tau^{-1}\in{\mathcal F}\) satisfies \(r^{-1}\cdot\tau=1\)) and \(\mathbb{T}^N\setminus\text{supp}(f\circ\tau)\) depending on the “rotation”, i.e., on \(\tau\in{\mathcal F}\). In the present paper, we consider two settings of this problem (depending on the sense in which the Fourier series of the function \(f\circ\tau\) is understood) and give (for both cases) possible solutions of the problem in the class \(L_1(\mathbb{T}^N)\), \(N\geq 2\).

MSC:

42B05 Fourier series and coefficients in several variables
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