Research Article A Family of Heat Functions as Solutions of Indeterminate Moment Problems

For a real-valued, measurable function f defined on [0,∞), its nth moment is defined as sn( f ) = ∫∞ 0 x n f (x)dx, n ∈N = {0,1, . . .}. Let (sn)n≥0 be a sequence of real numbers. 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). If the solution is unique, the moment problem is calledM-determinate. Otherwise, the moment problem is said to beM-indeterminate. When we replace N with Z we can formulate the same problem (the so-called strong Stieltjes moment problem). In [1–3], Stieltjes was the first to give examples ofM-indeterminatemoment problems. He showed that the log-normal distribution with density on (0,∞) given as