Abstract
We construct a family of functions satisfying the heat equation and show how they can be used to generate solutions to indeterminate moment problems. The following cases are considered: log-normal, generalized Stieltjes-Wigert, andq-Laguerre.
Highlights
For a real-valued, measurable function f defined on [0,∞), its nth moment is defined as sn( f ) = ∞ 0 xn f (x)dx, n ∈ N = {0, 1, . . . }.Letn≥0 be a sequence of real numbers.If f is a real-valued, measurable function defined on [0, ∞) with moment sequencen≥0, we say that f is a solution to the Stieltjes moment problem (related ton≥0)
We construct a family of functions satisfying the heat equation and show how they can be used to generate solutions to indeterminate moment problems
In [1,2,3], Stieltjes was the first to give examples of M-indeterminate moment problems. He showed that the log-normal distribution with density on (0,∞) given as dσ (x) = 2πσ2 −1/2x−1 exp log x 2 2σ 2
Summary
If f is a real-valued, measurable function defined on [0, ∞) with moment sequence (sn)n≥0, we say that f is a solution to the Stieltjes moment problem (related to (sn)n≥0). In [1,2,3], Stieltjes was the first to give examples of M-indeterminate moment problems He showed that the log-normal distribution with density on (0,∞) given as dσ (x) = 2πσ2 −1/2x−1 exp. If h is a 1-periodic, real-valued, measurable function, the last equality is equivalent to θ x, 2−1σ−2 h(x)dx = 0, where θ is the so-called theta function given by (1.6). The following 1-periodic, continuous function satisfies the condition (1.6): hy,t,α(x) = θα(y − x, t) − θα y, 2−1σ−2 + t , y ∈ [0, 1), t > 0. We show a nonperiodic, continuous function h fulfilling the condition (1.5)
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