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Register a new userA stationary set method for estimating oscillatory integrals
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We propose a new method of estimating oscillatory integrals, which we call a stationary set method. We use it to obtain the sharp convergence exponents of Tarrys problems in dimension two for every degree $kge 2$. As a consequence, we obtain sharp Fourier extension estimates for a family of monomial surfaces.
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In this paper, we prove an $L^2-L^2-L^2$ decay estimate for a trilinear oscillatory integral of convolution type in $mathbb{R}^d,$ which recovers the earlier result of Li (2013) when $d=1.$ We discuss the sharpness of our result in the $d=2$ case. Our main hypothesis has close connections to the property of simple nondegeneracy studied by Christ, Li, Tao and Thiele (2005).
The purpose of this paper is to establish some one-sided estimates for oscillatory singular integrals. The boundedness of certain oscillatory singular integral on weighted Hardy spaces $H^{1}_{+}(w)$ is proved. It is here also show that the $H^{1}_{+}(w)$ theory of oscillatory singular integrals above cannot be extended to the case of $H^{q}_{+}(w)$ when $0<q<1$ and $win A_{p}^{+}$, a wider weight class than the classical Muckenhoupt class. Furthermore, a criterion on the weighted $L^{p}$-boundednesss of the oscillatory singular integral is given.
This paper is devoted to $L^2$ estimates for trilinear oscillatory integrals of convolution type on $mathbb{R}^2$. The phases in the oscillatory factors include smooth functions and polynomials. We shall establish sharp $L^2$ decay estimates of trilinear oscillatory integrals with smooth phases, and then give $L^2$ uniform estimates for these integrals with polynomial phases.
The notion of integral Bailey pairs is introduced. Using the single variable elliptic beta integral, we construct an infinite binary tree of identities for elliptic hypergeometric integrals. Two particular sequences of identities are explicitly described.
We introduce a new method for evaluating the oscillatory integrals which describe natural interference patterns. As an illustrative example of contemporary interest, we consider astrophysical plasma lensing of coherent sources like pulsars and fast radio bursts in radioastronomy. Plasma lenses are known to occur near the source, in the interstellar medium, as well as in the solar wind and the earths ionosphere. Such lensing is strongest at long wavelengths hence it is generally important to go beyond geometric optics and into the full wave optics regime. Our computational method is a spinoff of new techniques two of us, and our collaborators, have developed for defining and performing Lorentzian path integrals. Cauchys theorem allows one to transform a computationally fragile and expensive, highly oscillatory integral into an exactly equivalent sum of absolutely and rapidly convergent integrals which can be evaluated in polynomial time. We require only that it is possible to analytically continue the lensing phase, expressed in the integrated coordinates, into the complex domain. We give a first-principles derivation of the Fresnel-Kirchhoff integral, starting from Feynmans path integral for a massless particle in a refractive medium. We then demonstrate the effectiveness of our method by computing the interference patterns of Thoms caustic catastrophes, both in their normal forms and within a variety of more realistic, local lens models, over all wavelengths. Our numerical method, implemented in a freely downloadable code, provides a fast, accurate tool for modeling interference patterns in radioastronomy and other fields of physics.
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