Search results for "Moment problem"
showing 3 items of 3 documents
Local moment problem
2014
The work is devoted to the local moment problem, which consists in finding of non-decreasing functions on the real axis having given first 2n + 1, n ≥ 0, power moments on the whole axis and also 2m + 1 first power moments on a certain finite axis interval. Considering the local moment problem as a combination of the Hausdorff and Hamburger truncated moment problems we obtain the conditions of its solvability and describe the class of its solutions with minimal number of growth points if the problem is solvable. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)
Vanishing Abelian integrals on zero-dimensional cycles
2011
In this paper we study conditions for the vanishing of Abelian integrals on families of zero-dimensional cycles. That is, for any rational function $f(z)$, characterize all rational functions $g(z)$ and zero-sum integers $\{n_i\}$ such that the function $t\mapsto\sum n_ig(z_i(t))$ vanishes identically. Here $z_i(t)$ are continuously depending roots of $f(z)-t$. We introduce a notion of (un)balanced cycles. Our main result is an inductive solution of the problem of vanishing of Abelian integrals when $f,g$ are polynomials on a family of zero-dimensional cycles under the assumption that the family of cycles we consider is unbalanced as well as all the cycles encountered in the inductive proce…
On the order of indeterminate moment problems
2013
For an indeterminate moment problem we denote the orthonormal polynomials by P_n. We study the relation between the growth of the function P(z)=(\sum_{n=0}^\infty|P_n(z)|^2)^{1/2} and summability properties of the sequence (P_n(z)). Under certain assumptions on the recurrence coefficients from the three term recurrence relation zP_n(z)=b_nP_{n+1}(z)+a_nP_n(z)+b_{n-1}P_{n-1}(z), we show that the function P is of order \alpha with 0<\alpha<1, if and only if the sequence (P_n(z)) is absolutely summable to any power greater than 2\alpha. Furthermore, the order \alpha is equal to the exponent of convergence of the sequence (b_n). Similar results are obtained for logarithmic order and for more ge…