Natural exponential families of probability distributions and exponential-polynomial approximation (Q1316164)

A Dirichlet polynomial in a finite linear combination of the functions \(e^{\lambda_ k x}, e^{\lambda_ k x},\dots, x^{m_ k-1} e^{\lambda_ k x}\), \(k=1,2,3,\dots\), where \(\{\lambda_ k\}\) is a sequence of complex numbers and \(\{m_ k\}\) is a sequence of positive integers. The authors [Appl. Math. Comput. 53, No. 2/3, 277-298 (1992; Zbl 0763.41005)] have shown that for every probability measure on a real line with finite moments there exists a naturally associated sequence of polynomials which are, in fact, classical Appel polynomials. They derived an explicit formula for the remainder term in the expansion of any smooth function in series with respect to these polynomial. In the present paper they generalize these results to the case of expansions with respect to Dirichlet polynomials.