廣義Stieltjes-Post反演公式及漸近積分

<正> 在作者之一的文章1]中,曾定義過一種含有参數的正規變換函數類。對於以這類中的函數為核所構成的積分變換,即存在有一種廣義的Stieltjes-Post-Widder反演公式。在本文的第一節中,我們將對正規變換函數定義中的第二條件予以减弱,也就是把核函數的範圍加以放寬,而仍保持廣義反演公式的有效.在本文的第二節中,主要是改善先前一篇短文2]中的結果,我們將在較廣泛的條件下,重新建立某一漸近積分定理.

CONCERNING A GENERALIZED STIELTJES-POST INVERSION FORMULA AND AN ASYMPTOTIC INTEGRAL

It is known that there is a type of generalized Stieltjes-Post-Widder inversion formula for integral transforms of the form where λ. is a parameter and the kernel, Ψ, is a regular transform function satisfying 3 conditions, i.e., for a fixed t>0, log Ψ,(u, v, t)=F(u, v, t), F_u, F_(uu) are real continuous functions (u≥0, v>0) such thati) F_u(u,v,t)=0 has a solution u=φ(v,t) with,ii) F_(uu)(u,v.t)<0 for large v and all u>0,iii) F_(uu)(u,v,t)～F_(uu)(θ(t),v,t)→—∞ as u→θ(t), v→∞.In this note, it is shown that ii) can be replaced by a weaker condition, viz.ii)' F_u(u,v,t)≤0 for φ(v,t)0 such that for large v, F_u(u, v, t)|~(-1)≤u/{(1 + δ) log u}, (u≥U). Accordingly, the class K of regular transform functions may be re-defined by i), ii)', iii), and the general inversion formula is still valid: where λ→∞ through such a sequence (λ_n) .that {φ(λ_n,t)}together with lim φ(λ_n, t) = θ(t)>0 belongs to the Lebesgue set for g(n).This is done by suitable modification of Lemma 1 and its proof of 1]. Also, an example has been constructed, satisfying i), ii)', iii) but not ii). Clearly, the ordinary Post-Widder formula is a particular case with Ψ,(u, λ, t)= e~(-λu/t)u~λ. The second part of this note contains the following result:Let F(u. v), F_u(u. v), F_uu(u, v) be continuous in the domain D (— ∞N, F_u(u,v)≥0 (—∞1>q>0 and U, V such that |u F_u(u,v)|>log|u|+ plog log|u|—|F_(uu)(ξ,v)|~q (|u|>U,v>V)ThenThis is an extension of a theorem of a previous paper. It is obvious that the hypothesis iii) can be dropped if we are concerned with the asymptotic behavior of a definite integral instead of an infinite integral.