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On stationary phase integrals
Published online by Cambridge University Press: 18 May 2009
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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
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- Copyright © Glasgow Mathematical Journal Trust 1994
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