Mean Periodic Functions on Multidimensional Domains
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The chapter consists of recent developments in the theory of mean periodic functions on domains in ℝ n , n≥2. The first circle of questions is connected with the John theorem on global uniqueness for integrals of a function f over spheres of radius 1 when supp f is disjoint from |x|≤1. Similar but much more complex results concerning mean periodic functions are described. In particular, the exact dependence between the order of smoothness of functions satisfying John-type conditions and the set of nonzero coefficients in their Fourier expansions with respect to spherical harmonics is obtained. Further topics include analogues of the Taylor and Laurent expansions for mean periodic functions, some mean periodic extendability results, and the study of the asymptotic behavior of mean periodic functions. The final section is devoted to the problem of approximation on domains in ℝ n of solutions of a convolution equations by exponential solutions. Hörmander’s approximation theorem shows that in the case of convex domains this question is solved positively. Some analogues of Hörmander’s result for domains without the convexity condition are presented.
KeywordsPeriodic Function Convergence Theorem Helmholtz Equation Nonzero Function Laurent Expansion
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