1968/02/01 by David Drasin · 2 citations
Mathematics · #Nonlinear Differential Equations Analysis
paper · pdf · doi:10.1090/s0002-9947-1968-0226017-4
openalex publication_date 1968/02/01 · openalex created_date 2025/10/10 · openalex updated_date 2025/11/06
More recently, Feller [3], [4] noted that (4) and ( 5) are in turn equivalent to (6) sUrxA+iy*for every fixed a, as x-*-qo.Note that ( 6) is formally a much stronger condition than (2) (set a -0) ; one consequence of our results is that (2) and ( 6) are in fact equivalent.For purposes of comparison, we record that T(A+1) is the bilateral Laplace transform of k evaluated at A :(7) r(A+l)=n e-Mk(t)dt = jSPJfc(A). J -COThe goal of the present paper is to derive ( 4) and ( 3) from ( 2), making minimal assumptions on k and/; this is the content of Theorem 1. Examples presented in 9 indicate that best results are to be obtained when neither k nor/are allowed to vary sign ; thus k and / are assumed nonnegative, and further restrictions will be listed in 1.A converse to Theorem 1 is the content of 8.In 10, we use Theorem 1 to derive a recent result due to Edrei and Fuchs [2].It was this result that provided the impetus for the present work.The final section is devoted to extending Theorem 1 to the situation that g(x), defined by (8) g(x) = j\x-y)f(y)dy satisfies (2) (because of (1.6), this is not covered by Theorem 1).This is an important special case, since Karamata's own characterization of functions of the form (4) and ( 3) is that (8) and ( 2) are satisfied with / nonnegative and continuous, and kt) = e~si (/sO) (the dependence on A and s is given in the statement of Theorem 6 in 11).Similar characterizations appear in Hardy-Rogosinski [5].