Fourier integral operator pdf

Pdf a maximal fourier integral operator and an application. There are different approaches for defining a fractional derivative. More recent tools include singular integral operators pseudodifferential opera tors to treat elliptic equations, and also parametrices to study the heat equation. For arbitrary c,fairly hopeless,but can begin to see some structure by looking at fios arising in applications with least. N is of type kif it is the smallest integer such that 0. Evolution equations and generalized fourier integral operators. A maximal fourier integral operator and an application article pdf available in journal of pseudodifferential operators and applications 44. Fourierintegraloperator approximation of solutions to. The adjoint of this fourier integral operator then allows to form seismic images from seismic data. There exist meaningful applications of fio in di erent frameworks, in particular to the study of hyperbolic type equations, and the related literature is quite large. Geometry of pseudodifferential algebra bundles and fourier integral operators mathai, varghese and melrose, richard b. Knapp, basic real analysis, digital second edition east. Fourier integral operators by duistermaath ormander xiaolong han department of mathematics.

Looking at the bilinear pseudodi erential operator written in the form 2, it is only a matter of introducing an appropriate oscillatory factor to create a bilinear fourier integral operator. More specifically, an integral and a derivative operators based on the connection between a derivative operator and other operators such as fourier transform, laplace transform, nfold integral operator, finite difference and so on can be extended to upper dimensions. Here we will learn about fourier transform with examples lets start with what is fourier transform really is. First and foremost, the integrals in question as in. Integral operator an overview sciencedirect topics. The operator generated by the integral in 2, or simply the operator 2, is called a linear integral operator, and the function is called its kernel cf. This volume is a useful introduction to the subject of fourier integral operators. Fourier integral operators by duistermaathormander csun. If ais a properly supported operator in lm x, fourier transforms of distributions 71 3. Can there be a general theory of fourier integral operators. Moreover, the solution operator to typical cauchy problems that appear in exploration seismology can be approximated by the composition of global fourier integral operators with complex phases. Obrien as we will see in the next section, the fourier transform is developed from the fourier integral, so it shares many properties of the former. Fourier integral operators by duistermaath ormander. Define the oscillatory integral as the generalized integral.

The plancherel identity suggests that the fourier transform is a onetoone norm preserving map of the hilbert space l21. Hormander university of nijmegen, holland, and university of lund, sweden 1 preface the purpose of this paper is to give applications of the operator theory developed in the first part acta math. An integral operator with a generalized kernel that is a rapidlyoscillating function or the integral of such a function. On lp continuity of singular fourier integral operators 2455 we will say that at a point p2 the map m. More than twenty years ago i gave a course on fourier integral op erators at the. Just as the discrete fourier transform is the digital analogue of the continuous fourier transform, one can also introduce discrete fourier integral operators. Representation of fourier integral operators using shearlets. Because it constitutes a link between real and complex analysis because it is a model case for the general theory of singular integral operators.

The generating integral and the canonical maslov operator in the wkb method. Pseudodifferential operators are obtained when x y and the canonical transformation is the identity. Chapter 1 the fourier transform university of minnesota. In mathematical analysis, fourier integral operators have become an important tool in the theory of partial differential equations. Operators of this type arose when investigating the asymptotic expansions of rapidlyoscillating solutions to partial differential equations see, and in studying the singularities of the fundamental solutions of hyperbolic equations see. A main keyword in the theory of singular integrals and in analysis in. An integral transform is a particular kind of mathematical operator there are numerous useful integral transforms. Because of its relationship with summability for fourier integrals in lpnorms. This volume is a useful introduction to the subject of fourier integral operators and is based on the authors classic set of notes. An approximation ansatz for the solution operator, u z. Given the canonical transformation and principal symbol of the operator, a preprocessing step. Familiarity with analysis distributions and fourier transformation and differential geometry is useful. That sawtooth ramp rr is the integral of the square wave. An integral transform is any transform t of the following form.

The class of fourier integral operators contains differential operators as well as classical integral operators as special cases a fourier integral operator is given by. On the basis of the stationary phase method for oscillatory integrals with complex phase function, the authors prove the coincidence of fourier integral operators and maslovs canonical operator. We investigate the propagation of singularities for this ansatz and prove microlocal convergence. The aim of this paper is to present certain global regularity proper. Additionally, this book is designed for a onesemester introductory course on fourier integral operators aimed at a broad audience. The delta functions in ud give the derivative of the square wave. Integral operators with degenerate canonical relations 5 then the distribution kernel of r is. There is no reason for us at this time to consider the variables x and y. Fourier integral operator encyclopedia of mathematics. Pdf lpbounds for fourier integral operators on the torus.

Dedicato a ferruccio colombini in occasione del suo sessantesimo compleanno. In this framework, the forward modeling operator is a fourier integral operator which maps singularities of the subsurface into singularities of the wave. The notion of fourier integral operator fio arises in the theory of partial differ ential equations in two contexts. Approximation of fourier integral operators by gabor multipliers 3 operators of the form 3 m af m. Dooley abstract let t be a fourier integral operator on r n of order n. Fourier integral operators, generalized radon transform, separated repre. Moreover, the solution operator to typical cauchy problems. This is a wide and general theory, and thus we provide here only a short and comprehensive but rigorous description. Paper computing fourier integral operators with caustics. Fast computation of fourier integral operators mit math. Seeger, sogge, and stein showed among other things that t maps the hardy space h1 to l1.