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Approximate solutions of equations defined by complex spherical multiplier operators. (English) Zbl 1103.41032

It is known that convolution operators on the unit sphere in \(\mathbb R^n\) are multiplier operators with respect to the spherical harmonic decomposition. The paper deals with similar operators on the unit sphere in the multidimensional complex space. These operators are multiplier operators with respect to the decomposition in complex spherical harmonics. Approximate solutions of the corresponding operator equations are studied. The main tools are Sobolev-type spaces, positive definite functions, and spherical interpolation.

MSC:

41A63 Multidimensional problems
47A50 Equations and inequalities involving linear operators, with vector unknowns
32A50 Harmonic analysis of several complex variables
47B38 Linear operators on function spaces (general)
42B15 Multipliers for harmonic analysis in several variables
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
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