Functions whose Fourier coefficients are moments of a probability distribution. (English. Russian original) Zbl 0783.42017

Mirzakhmedov, M. A.; Tursunov, G. T.

Theory Probab. Math. Stat. 43, 101-107 (1991); translation from Teor. Veroyatn. Mat. Stat., Kiev 43, 90-97 (1990).

Let \(A(t)\), \(t\in [0,1)\), be a probability distribution and consider the moment sequence of \(A(t)\) defined by \(a_ i:=\int^ 1_ 0 t^ i dA(t)\), \(i=0,1,2,\dots\;\). The authors characterize those functions \(f\) that admit a representation \(f(x)=\sum^ \infty_{i=0} a_ i \psi_ i(x)\), where \(\{\psi_ i(x):\;i\geq 0\}\) is the orthonormal system of either the Laguerre functions \((x\in \mathbb{R}^ +_ 1)\) or the Hermite functions \((x\in\mathbb{R}_ 1)\). They also prove an analogous result in the setting of the two-dimensional Hermite functions orthonormal on the whole plane \(\mathbb{R}_ 2\).

Reviewer: F.Móricz (Szeged)

MSC:

42C15	General harmonic expansions, frames
33C45	Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
62E20	Asymptotic distribution theory in statistics

Keywords:

Fourier coefficients; probability distribution; moment sequence; representation; Laguerre functions; Hermite functions