共查询到20条相似文献,搜索用时 640 毫秒
1.
《Advances in Mathematics》1985,55(3):242-315
The uniform asympotic behavior of the scattering amplitude near the forward peak, in the case of classical scattering of waves by a convex obstacle, is derived. A microlocal model is obtained for the scattering operator. This is achieved by use of a parametrix for diffractive boundary problems and by a new study of a class of Fourier integral operators, those with folding canonical relations. A crucial ingredient consists of putting a Fourier integral operator with folding canonical relation into a normal form. The analysis also gives the asymptotic behavior of the normal derivative of the scattered wave on a neighbourhood of the shadow boundary, thus providing a corrected version of the Kirchhoff approximation. 相似文献
2.
In this paper we develop a new approach to the theory of Fourier integral operators. It allows us to represent the Schwartz kernel of a Fourier integral operator by one oscillatory integral with a complex phase function. We consider Fourier integral operators associated with canonical transformations, having in mind applications to hyperbolic equations. As a by-product we obtain yet another formula for the Maslov index. © 1994 John Wiley & Sons, Inc. 相似文献
3.
Ivana Alexandrova 《Proceedings of the American Mathematical Society》2006,134(8):2295-2302
We study the semi-classical behavior of the spectral function of the Schrödinger operator with short range potential. We prove that the spectral function is a semi-classical Fourier integral operator quantizing the forward and backward Hamiltonian flow relations of the system. Under a certain geometric condition we explicitly compute the phase in an oscillatory integral representation of the spectral function.
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V. M. Bruk 《Russian Mathematics (Iz VUZ)》2013,57(2):13-24
We define families of maximal and minimal relations generated by integral equations with Nevanlinna operator measure and non-selfadjoint operator measure. We prove that if a restriction of a maximal relation is continuously invertible, then the inverse operator is integral. We study the case when the convergence of non-selfadjoint operator measures implies the convergence of the corresponding integral operators inverse to restrictions of maximal relations, and establish a sufficient condition for the validity of this implication. The obtained results are applicable to the study of differential equations with singular potentials. 相似文献
6.
A Rota-Baxter operator of weight λ is an abstraction of both the integral operator (when λ=0) and the summation operator (when λ=1). We similarly define a differential operator of weight λ that includes both the differential operator (when λ=0) and the difference operator (when λ=1). We further consider an algebraic structure with both a differential operator of weight λ and a Rota-Baxter operator of weight λ that are related in the same way that the differential operator and the integral operator are related by the First Fundamental Theorem of Calculus. We construct free objects in the corresponding categories. In the commutative case, the free objects are given in terms of generalized shuffles, called mixable shuffles. In the noncommutative case, the free objects are given in terms of angularly decorated rooted forests. As a byproduct, we obtain structures of a differential algebra on decorated and undecorated planar rooted forests. 相似文献
7.
We obtain new convolutions for quadratic-phase Fourier integral operators (which include, as subcases, e.g., the fractional Fourier transform and the linear canonical transform). The structure of these convolutions is based on properties of the mentioned integral operators and takes profit of weight-functions associated with some amplitude and Gaussian functions. Therefore, the fundamental properties of that quadratic-phase Fourier integral operators are also studied (including a Riemann–Lebesgue type lemma, invertibility results, a Plancherel type theorem and a Parseval type identity). As applications, we obtain new Young type inequalities, the asymptotic behaviour of some oscillatory integrals, and the solvability of convolution integral equations. 相似文献
8.
Shanzhen LU 《Frontiers of Mathematics in China》2011,6(5):821-833
In this paper, the author introduces some late results and puts forward a few problems on commutators of many important operators
in harmonic analysis, included the Bochner-Riesz operator below the critical index, the strongly singular integral operator,
the pseudo-differential operator, a class of convolution operators with oscillatory kernel, the Marcinkiewicz integral operator,
and the fractional integral operator with rough kernel. 相似文献
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In this article we investigate averaging properties of fully nonlinear PDEs in bounded domains with oscillatory Neumann boundary data. The oscillation is periodic and is present both in the operator and in the Neumann data. Our main result states that, when the domain does not have flat boundary parts and when the homogenized operator is rotation invariant, the solutions uniformly converge to the homogenized solution solving a Neumann boundary problem. Furthermore we show that the homogenized Neumann data is continuous with respect to the normal direction of the boundary. Our result is the nonlinear version of the classical result in [3] for divergence-form operators with co-normal boundary data. The main ingredients in our analysis are the estimate on the oscillation on the solutions in half-spaces (Theorem 3.1), and the estimate on the mode of convergence of the solutions as the normal of the half-space varies over irrational directions (Theorem 4.1). 相似文献
11.
V. M. Bruk 《Russian Mathematics (Iz VUZ)》2012,56(10):1-14
We define families of maximal and minimal linear relations generated by an integral equation with Nevanlinna operator measure and prove their holomorphic property. We also prove that if a restriction of a maximal relation is continuously invertible, then the operator inverse to this restriction is integral. We apply the obtained results for proving the constancy of deficiency indices of some integral and differential equations. 相似文献
12.
We analyze the singular spectrum of selfadjoint operators which arise from pasting a finite number of boundary relations with a standard interface condition. A model example for this situation is a Schrödinger operator on a star-shaped graph with continuity and Kirchhoff conditions at the interior vertex. We compute the multiplicity of the singular spectrum in terms of the spectral measures of the Weyl functions associated with the single (independently considered) boundary relations. This result is a generalization and refinement of a Theorem of I.S.Kac. 相似文献
14.
考虑如下的振荡积分算子:T_(m,k,n)f(x):=∫_(R~n)e~(i(x_1~2+…+x_n~2))~m(y_1~2+…+y_n~2)~kf(y)dy,其中函数f为定义在R~n上的Schwartz函数,并且满足m,k0.本文给出算子T_(m,k,n).从L~p(R~n)(1≤p∞)到L~q(R~n)有界的一个充分必要条件.此外,我们还证明了算子T_(m,k,n)把L~1(R~n)映到C_0(R~n). 相似文献
15.
Products of Toeplitz Operators on the Bergman Space 总被引:1,自引:0,他引:1
Issam Louhichi Elizabeth Strouse Lova Zakariasy 《Integral Equations and Operator Theory》2006,54(4):525-539
In 1962 Brown and Halmos gave simple conditions for the product of two Toeplitz operators on Hardy space to be equal to a
Toeplitz operator. Recently, Ahern and Cucković showed that a similar result holds for Toeplitz operators with bounded harmonic
symbols on Bergman space. For general symbols, the situation is much more complicated. We give necessary and sufficient conditions
for the product to be a Toeplitz operator (Theorem 6.1), an explicit formula for the symbol of the product in certain cases
(Theorem 6.4), and then show that almost anything can happen (Theorem 6.7). 相似文献
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Jan Janas 《Integral Equations and Operator Theory》1996,26(2):188-201
We establish an operator version of the Newman — Shapiro Isometry Theorem for operators satisfying generalized canonical commutation relations. An application to operator inequalities is also given. 相似文献
18.
Khaled Mehrez 《Integral Transforms and Special Functions》2017,28(8):616-628
In this paper our aim is to establish the Paley–Wiener Theorem for the Weinstein Transform. Furthermore, some applications are presents. In particular some properties for the generalized translation operator associated with the Weinstein operator are proved, an integral representation and a series representation for a function in the Paley–Wiener classes are investigated. 相似文献
19.
We consider an integral equation on half-line with Chebyshev polynomial nonlinearity, arising in dynamic theory of universe and p-adic string theory.We prove existence of the positive and monotonically increasing continuous solution in class of essentially bounded functions on half-line. We also found two sided estimates for obtained solution, as well as the limit of solution at infinity (Theorem 2.1). We prove uniqueness of a solution in the certain class of functions (Theorem 2.2). We generalize the results for more general integral equation with “double” nonlinearity (Theorem 2.3). At the end we give some examples of functions, describing nonlinearity. Using suggested constructive solution method, we present some results of numerical calculations, having direct application in cosmology. 相似文献
20.
Both oscillatory integral operators and level set operators appear naturally in the study of properties of degenerate Fourier
integral operators (such as generalized Randon transforms). The properties of oscillatory integral operators have a longer
history and are better understood. On the other hand, level set operators, while sharing many common characteristics with
oscillatory integral operators, are easier to handle.
We study L2-estimates on level set operators in dimension two and compare them with what is known about oscillatory integral operators.
The cases include operators with non-degenerate phase functions and the level set version of Melrose-Taylor transform (as
an example of a degenerate phase function). The estimates are formulated in terms of the Newton polyhedra and type conditions. 相似文献