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On stationary phase integrals

Published online by Cambridge University Press:  18 May 2009

M. N. Huxley
M. N. Huxley
Affiliation:
School of Mathematics, University of Wales College of Cardiff, 23 Senghennydd Road, Cardiff, CF2 4YH
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Let f(x) and g(x) be real functions defined on the interval [a, b], with f(x) at least twice continuously differentiable, f′(x) monotone increasing, and f(x) of bounded variation. We consider the exponential integral

where e(t) denotes exp 2πit. The purpose of this note is to prove sharp forms of the well-known estimates:

A: If f′(x) is nonzero on [a, b], then I has order of magnitude

The constant of proportionality depends on the function g(x).

B: If f′(x) changes sign at x = c with a < c < b, then

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 36 , Issue 3 , September 1994 , pp. 355 - 362
Copyright
Copyright © Glasgow Mathematical Journal Trust 1994

References

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