Division problem of moment functionals.
The authors study moment generating functionals \(\sigma\), i.e., linear functions on the space \textbf{P} of all polynomials, satisfying the Hamburger condition \[ \Delta:=\text{det}\,\left( \sigma_{i+j}\right)_{i,j,=0}^n \not= 0,\;n\geq 0, \] with \(\sigma_n:=\langle \sigma, x^n\rangle,\;n\geq 0\). The paper gives necessary and sufficient conditions for a moment functional \(\tau\) to be also quasi-definite, where \(\tau\) is connected with \(\sigma\) by \[ D(x)\tau = A(x)\sigma, \] where \(A(x),\,D(x)\) are co-prime, non-zero polynomials. When \(\tau\) is quasi-definite, they exhibit a rather simple representation of the monic orthogonal polynomials with respect to \(\tau\) in terms of those orthogonal relative to \(\sigma\). As corollaries some previous results are recovered, due to \textit{S. Belmehdi} [IMACS Ann. Comput. Appl. Math. 9, 169--176 (1991; Zbl 0837.42013)] and \textit{A. Branquinho} and \textit{F. Marcellán} [Int. J. Math. Math. Sci. 19, No. 4 , 643--656 (1996; Zbl 0846.33008)]. The final section of the paper treats two examples: \[ x^3\tau=\sigma,\;\sigma\text{ the Laguerre moment functional on }(0,\infty), \] \[ (1+x)\tau=(1-x)\sigma,\;\sigma\text{ the Jacobi moment functional on }(-1,1). \]
scientific article; zbMATH DE number 2012116