LAPLACE INTEGRAL ON RATIONAL NUMBERS

摘要

In 1934 D. V. Widder proved [5, p. 325] that a function cp: I -> R denned on an open interval of real numbers is of the form cp(t) = JR e'x dpi(x), where f.t is a positive Radon measure on R, if and only if q> is continuous and positive definite on I, which means that is continuous and satisfies the following condition: for each natural number n > 0 and each family of real numbers cx,..., c,,, ru.-.., rm such that rtejl, we have We prove in Section 2 of this paper a similar theorem for a function denned on an open interval of rational numbers.