55 research outputs found
On the order of indeterminate moment problems
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 general types
of slow growth. To prove these results we introduce a concept of an order
function and its dual.
We also relate the order of P with the order of certain entire functions
defined in terms of the moments or the leading coefficient of P_nComment: 45 pages. To appear in Adv. Mat
Orthogonal polynomials of discrete variable and boundedness of Dirichlet kernel
For orthogonal polynomials defined by compact Jacobi matrix with exponential
decay of the coefficients, precise properties of orthogonality measure is
determined. This allows showing uniform boundedness of partial sums of
orthogonal expansions with respect to norm, which generalize
analogous results obtained for little -Legendre, little -Jacobi and
little -Laguerre polynomials, by the authors
A determinant characterization of moment sequences with finitely many mass-points
To a sequence (s_n)_{n\ge 0} of real numbers we associate the sequence of
Hankel matrices \mathcal H_n=(s_{i+j}),0\le i,j \le n. We prove that if the
corresponding sequence of Hankel determinants D_n=\det\mathcal H_n satisfy
D_n>0 for n<n_0 while D_n=0 for n\ge n_0, then all Hankel matrices are positive
semi-definite, and in particular (s_n) is the sequence of moments of a discrete
measure concentrated in n_0 points on the real line. We stress that the
conditions D_n\ge 0 for all n do not imply the positive semi-definiteness of
the Hankel matrices.Comment: 8 page
Closable Hankel operators and moment problems
In a paper from 2016 D. R. Yafaev considers Hankel operators associated with
Hamburger moment sequences q_n and claims that the corresponding Hankel form is
closable if and only if the moment sequence tends to 0. The claim is not
correct, since we prove closability for any indeterminate moment sequence but
also for certain determinate moment sequences corresponding to measures with
finite index of determinacy. It is also established that Yafaev's result holds
if the moments satisfy \root{2n}\of{q_{2n}}=o(n).Comment: 10 pages. The notation for the closure of an operator A is changed to
\overline{A} from \o A on pages 7,
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