Transform Methods for Obtaining Asymptotic Expansions of Definite Integrals

With the condition $\int_R {| {dh(t)} |} < \infty $, asymptotic approximations are obtained to the integral $\int_R {f(t)dh(\lambda t)} $ over the real line $R$ as $\lambda \to \infty $, (a) by approximating $\hat h(x)\int_R {e^{ixt} dh(t)} $ in a neighborhood of $x = 0$ and (b) by using a basis...

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Bibliographic Details
Published inSIAM journal on mathematical analysis Vol. 3; no. 1; pp. 20 - 30
Main Author Stenger, Frank
Format Journal Article
LanguageEnglish
Published Philadelphia Society for Industrial and Applied Mathematics 01.02.1972
Subjects
Approximation
Fourier transforms
Integrals
Neighborhoods
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Summary:With the condition $\int_R {| {dh(t)} |} < \infty $, asymptotic approximations are obtained to the integral $\int_R {f(t)dh(\lambda t)} $ over the real line $R$ as $\lambda \to \infty $, (a) by approximating $\hat h(x)\int_R {e^{ixt} dh(t)} $ in a neighborhood of $x = 0$ and (b) by using a basis $\{ {\psi _k (t)} \}_{k = 1}^n $, where in contrast to the usual case $\psi _k (t)$ need not be equal to $t^{k - 1} $.
ISSN:0036-1410
1095-7154
DOI:10.1137/0503003