Sets of uniqueness for spherically convergent multiple trigonometric series |
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Authors: | J Marshall Ash Gang Wang |
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Institution: | Mathematics Department, DePaul University, Chicago, Illinois 60614 ; Mathematics Department, DePaul University, Chicago, Illinois 60614 |
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Abstract: | A subset of the -dimensional torus is called a set of uniqueness, or -set, if every multiple trigonometric series spherically converging to outside vanishes identically. We show that all countable sets are -sets and also that sets are -sets for every . In particular, , where is the Cantor set, is an set and hence a -set. We will say that is a -set if every multiple trigonometric series spherically Abel summable to outside and having certain growth restrictions on its coefficients vanishes identically. The above-mentioned results hold also for sets. In addition, every -set has measure , and a countable union of closed -sets is a -set. |
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Keywords: | Abel summation Baire category Fourier series generalized Laplacian Green's function $H^{J}$ sets multiple trigonometric series set of uniqueness spherical convergence subharmonic function uniqueness |
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