Vol. 31, No. 1, 1969

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Boundary behavior of random valued heat polynomial expansions

Robert Hughes

Vol. 31 (1969), No. 1, 61–72

This paper is concerned with random series of the form n=0Xn(ω)anvn(x,t) where the Xn’s are random variables, the an’s are real numbers, and the vn’s are heat polynomials as introduced by P. C. Rosenbloom and D. V. Widder. The sequences {an} are assumed to satisfy limsupn→∞|an|2∕n(2n∕e) = 1 which implies n=0anvn(x,t) has |t| < 1 as its strip of convergence, i.e., it converges to a C2-solution of the heat equation in |t| < 1 and does not converge everywhere in any larger open strip. Associated with each sequence {an} is its classification number from [0,1] which indicates how rapidly an tends to zero. Assumptions are placed on the random variables which imply that for almost every ω the series n=0Xn(ω)anvn(x,t) has |t| < 1 as its strip of convergence.

The main results of the paper are two theorems. The first states that if {an} has its classification number in [0,12), then for almost every ω the lines t = 1 and t = 1 form the natural boundary for n=0Xn(ω)anvn(x,t). The second is concerned with sequences having their classification numbers in (1/2.1]. The conclusion implies that for almost every ω no interval of either of the lines t = 1 or t = 1 is part of the natural boundary for n=0Xn(ω)anvn(x,t).

Mathematical Subject Classification
Primary: 42.16
Secondary: 35.00
Received: 23 July 1968
Published: 1 October 1969
Robert Hughes