On the order of indeterminate moment problems

TY - JOUR

T1 - On the order of indeterminate moment problems

AU - Berg, Christian

AU - Szwarc, Ryszard

PY - 2014/1/15

Y1 - 2014/1/15

N2 - 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)=(∑n=0∞|Pn(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 -1P n -1(z), we show that the function P is of order α with 0 < α < 1, if and only if the sequence (P n(z)) is absolutely summable to any power greater than 2α. Furthermore, the order α 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 n.

AB - 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)=(∑n=0∞|Pn(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 -1P n -1(z), we show that the function P is of order α with 0 < α < 1, if and only if the sequence (P n(z)) is absolutely summable to any power greater than 2α. Furthermore, the order α 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 n.

KW - Faculty of Science

KW - matematik, momentproblemer

U2 - 10.1016/j.aim.2013.09.020

DO - 10.1016/j.aim.2013.09.020

M3 - Journal article

SN - 0001-8708

VL - 250

SP - 105

EP - 143

JO - Advances in Mathematics

JF - Advances in Mathematics

ER -