Now we claim that the function j is an example of oscillatory integral. In mathematics, in the field of harmonic analysis, an oscillatory integral operator is an integral operator of the form. Oscillatory integrals and extremal problems in harmonic. Oscillatory integrals have played a central role in harmonic analysis since its very beginning. In this representation the oscillatory integral is the fourier transform of the function given by the inside integral. In this article, a theory of generalized oscillatory integrals ois is developed whose phase functions as well as amplitudes may be generalized functions of colombeau type. Modi ed fourier series use highly oscillatory integrals to obtain a function approximation scheme that converges faster than the standard fourier series 51. Weighted estimates for rough oscillatory singular integrals. The authors focus on key results in each area, highlighting their importance and the organic unity of the subject. Besides the obvious fact that the fourier transform is itself an example of an oscillatory integral, the early works on asymptotic analysis by bessel, airy, and. One often considers sx,y to be realvalued and smooth, and ax,y smooth and compactly supported. The reader should have some background knowledge in analysis distributions and fourier transformations and differential geometry. The integrand is highly oscillatory along the real axis if. As a consequence, oscillatory integrals have deep connections with problems in fourier analysis, but also partial di erential equations and special functions.
Journal of functional analysis 73, 179194 1987 harmonic analysis on nilpotent groups and singular integrals i. Of course, in some cases undergraduate calculus allows one to compute such integrals exactly, after some e. Stein dipartimento di matematica, politecnico di torino, 10129 torino, italy, and department of mathematics, princeton university, fine hall, princeton, new jersey 08544 communicated by l. The fourier transform of a function is an important example of an such an integral. Fourier and fourierstieltjes transforms and other transforms of fourier type citation watson, david k. Oscillatory integral theory has its origins at the very heart of harmonic analysis. Oscillatory integrals of the second kind can be usually written in the form t. Oscillatory integrals and fourier transforms of surface carried measures article pdf available in transactions of the american mathematical society 3041. Levin methods for highly oscillatory integrals with. Oscillatory integrals make rigorous many arguments that, on a naive level, appear to use divergent integrals. Oscillatory integrals, lagrange immersions and unfolding. A levin method for logarithmically singular oscillatory. Decay of multilinear oscillatory integrals oscillatory integrals have played a central role in harmonic analysis since its very beginning. Oscillatory integrals real analysis pdes fourier analysis o.
Fast computation of singular oscillatory fourier transforms kang, hongchao and shao, xinping, abstract and applied analysis, 2014. The canonical relation associated to the generalized radon transform is the twisted conormal bundle of the incidence relation. Rational approximation, oscillatory cauchy integrals and. Oscillatory integral operators with loworder degeneracies greenleaf, allan and seeger, andreas, duke mathematical journal, 2002. These solutions were put together by the participants of the. Pdf fourier analysis an introduction princeton lectures. Besides the obvious fact that the fourier transform is itself.
Sogge notes that the book evolved out of his 1991 ucla lecture notes, and this indicates the level of preparation expected from the reader. Asymptotic expansions of oscillatory integrals with complex phase. Weighted estimates for singular integrals via fourier transform estimates. This book contains an exposition of some of the main developments of the last twenty years in the following areas of harmonic analysis. Asymptotic expansions of oscillatory integrals with. Rational approximation, oscillatory cauchy integrals and fourier transforms. The fourier transform is a fundamental tool in microlocal analysis and its application to the theory of pdes and. Some estimates may differ a constant multiple from the real situation because the author is familiar with the fourier coefficients f. Research statement lechao xiao 1 oscillatory integrals. Browse other questions tagged integration numericalmethods fourieranalysis or ask your own question. Numerical approximation of highly oscillatory integrals. The book is completed with applications to the cauchy problem for strictly hyperbolic equations and caustics in oscillatory integrals. Rational approximation, oscillatory cauchy integrals and fourier transforms thomas trogdon1 courant institute of mathematical sciences new york university 251 mercer st new york, ny 10012, usa march 5, 2014 abstract we develop the convergence theory for a wellknown method for the interpolation of functions on the real axis with rational functions. Oscillatory integrals a basic problem which comes up whenever performing a computation in harmonic analysis is how to quickly and e.
Today, fourier analysis is, among other things, perhaps the single most important mathematical tool used in what we call signal. The purpose of this paper is to obtain estimates for the decay at infinity of certain oscillatory integrals related to the fourier transform of surface carried measures. We show that fourier integral operators with slowscale phase function and regular amplitude map g. Harmonic analysis on nilpotent groups and singular. Modified fourier series use highly oscillatory integrals to obtain a function approximation scheme that converges faster than the standard. To appear in beijing lectures on harmonic analysis, annals of math. Siam journal on numerical analysis siam society for. Lecture notes on oscillatory integrals a large part of.
To avoid singularity, the technique of singularity separation is applied and then the singular ode occurring in classic levin methods is converted into two kinds of nonsingular odes. Asymptotic expansions of oscillatory integrals 227 proof. We look at a spike, a step function, and a rampand smoother functions too. This is motivated by the need of a general framework for partial differential operators with nonsmooth. The branch of mathematics we will consider is called fourier analysis, after the french mathematician jean baptiste joseph fourier1 17681830, whose treatise on heat. In mathematical analysis an oscillatory integral is a type of distribution. Thanks for contributing an answer to mathematics stack exchange. It is possible to represent approximate solution operators for many differential equations as oscillatory integrals. Numericalapproximationofhighly oscillatoryintegrals. Pdf oscillatory integrals and fourier transforms of. Pdf fourier analysis, stein and shakarchi chapter 3.
In this paper, new levin methods are presented for calculating oscillatory integrals with algebraic andor logarithmic singularities. The analysis here can be seen as an extension of that of weber 1980 and weideman 1995 in a. Oscillatory integrals and extremal problems in harmonic analysis by. The introduction of fouriers famous transform, the original and perhaps best example of an oscillatory integral, led to the consideration of more general oscillatory integrals. Realvariable methods, orthogonality, and oscillatory integrals, princeton mathematical series, vol. Fourier integrals and classical analysis is an excellent book on a beautiful subject seeing a lot of highlevel activity.
Based on this, generalized fourier integral operators fios acting on colombeau algebras are defined. Lately, they have emerged as powerful analytic tools in various problems, ranging from pdes to geometry and number theory. It can be derived in a rigorous fashion but here we will follow the timehonored approach of considering nonperiodic functions as functions with a period t. Fourier integral fourier transformation of an oscillatory function with fft.
J is simply the fourier transform of the spherical measure d. Wave packets resonances euler equations as geometric pdes vortex sheet. The solutions of one can be obtained explicitly, while those of the other can. Scalar oscillatory integrals, commonly referred to as oscillatory integrals of the rst kind, concern.
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