Let$K_{a,b}(x)=\int_{0}^{\infty}{\psi({\xi})\xi^{-b}e^{ix\xi\pm \xi^{a}}d\xi}\quad a,b>0$ where $\psi\in C^{\infty}$, equals to $0$ when $\xi<\frac{1}{2}$, and equals to $1$ when $\xi>1$.
oscillatory integrals in one dimension
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real-analysis
harmonic-analysis
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0@ Eric Angle:right,it's essentially what you have written down. – 2012-09-08