Generalized "second Ritt theorem" and explicit solution of the polynomial moment problem

Details Generalized "second Ritt theorem" and explicit solution of the polynomial moment problem Generalized "second Ritt theorem" and explicit solution of the polynomial moment problem

2009-08-18

arxiv.org/abs/0908.2508v3

Mathematics

Dynamical Systems

A new version with some corrections, alterations, and additions

Scientific paper

In the recent paper arXiv:0710.4085 was shown that any solution of "the polynomial moment problem", which asks to describe polynomials Q orthogonal to all powers of a given polynomial P on a segment, may be obtained as a sum of some "reducible" solutions related to different decompositions of P into a composition of two polynomials of lesser degrees. However, the methods of arXiv:0710.4085 do not permit to estimate the number of necessary reducible solutions or to describe them explicitly. In this paper we provide a description of the polynomial solutions of the functional equation P=P_1(W_1)=P_2(W_2)=...=P_r(W_r), and on this base describe solutions of the polynomial moment problem in an explicit form suitable for applications. With respect to the previous version a more general form of the generalized "secon Ritt theorem" is proved and the proof is considerably simplified. Besides, a missed case in Theorem 1.2 was added and the proof is corrected.

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