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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

REFERENCES

1.Bombieri, E. and Iwaniec, H., Some mean value theorems for exponential sums, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 13 (1986), 473–486.Google Scholar
2.Corput, J. G. van der, Zur Methode der stationären Phase I, Compositio Math. 1 (1935), 15–38; II, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 3 (1936), 328–372.Google Scholar
3.Graham, S. W. and Kolesnik, G., Van der Corput's method of exponential sums, L. M. S. Lecture notes 126 (Cambridge University Press, 1991).CrossRefGoogle Scholar
4.Heath-Brown, D. R., The Pjatecki-Sapiro prime number theorem, J. Number Theory 16 (1983), 242–266.CrossRefGoogle Scholar
5.Hörmander, L., The existence of wave operators in scattering theory, Math. Zeitschrift 446 (1976), 69–91.CrossRefGoogle Scholar
6., H. and Jeffreys, B. S., Methods of mathematical physics (Cambridge University Press, 1956).Google Scholar
7.Karatsuba, A. A., Approximation of exponential sums by shorter ones, Proc. Indian Acad. Sci. (Math. Sci.) 97 (1987), 167–178.CrossRefGoogle Scholar
8.Murray, J. D., Asymptotic analysis (Oxford University Press, 1974).Google Scholar
9.Phillips, E., The zeta function of Riemann; further developments of van der Corput's method, Quarterly J. Maths. (Oxford) 4 (1933), 209–225.CrossRefGoogle Scholar
10.Titchmarsh, E. C., The theory of the Riemann zeta function, (Oxford University Press 1951, 1987).Google Scholar