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On quadrature methods for highly oscillatory integrals and their implementation
 BIT Numerical Mathematics
, 2004
"... On quadrature methods for highly oscillatory integrals and their implementations by ..."
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Cited by 45 (8 self)
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On quadrature methods for highly oscillatory integrals and their implementations by
Preconditioned GMRES for oscillatory integrals
, 2008
"... None of the existing methods for computing the oscillatory integral ∫ b a f(x)e iωg(x) dx achieve all of the following properties: high asymptotic order, stability, avoiding the computation of the path of steepest descent and insensitivity to oscillations in f. We present a new method that satisfie ..."
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Cited by 4 (2 self)
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None of the existing methods for computing the oscillatory integral ∫ b a f(x)e iωg(x) dx achieve all of the following properties: high asymptotic order, stability, avoiding the computation of the path of steepest descent and insensitivity to oscillations in f. We present a new method
oscillatory integral transforms
, 2011
"... The classical theory of Gaussian quadrature assumes a positive weight function. We will show that in some cases Gaussian rules can be constructed with respect to an oscillatory weight, yielding methods with complex quadrature nodes and positive weights. These rules are well suited for highly oscilla ..."
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oscillatory integrals because they attain optimal asymptotic order. We show that for the Fourier oscillator this approach yields the numerical method of steepest descent, a method with optimal asymptotic order that has previously been proposed for this class of integrals. However, the approach readily extends
On Oscillatory Integrals with Two Smooth Phases
"... Abstract: Oscillatory integrals appear in many problems, including the 50year old unsolved problem in the Anderson model, where oscillatory integrals with two phases arise when one formulates the meansquare distance in terms of the Heisenberg position and momentum operators. The asymptotic behavio ..."
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Abstract: Oscillatory integrals appear in many problems, including the 50year old unsolved problem in the Anderson model, where oscillatory integrals with two phases arise when one formulates the meansquare distance in terms of the Heisenberg position and momentum operators. The asymptotic
Decay estimates for weighted oscillatory integrals
 in R 2 , Indiana
"... ABSTRACT. In this paper, we study decay estimates for a twodimensional scalar oscillatory integral with degenerate realanalytic phase and amplitude. Integrals such as these form a model for certain higherdimensional degenerate oscillatory integrals, for which it is known that many of the twodim ..."
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Cited by 6 (0 self)
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ABSTRACT. In this paper, we study decay estimates for a twodimensional scalar oscillatory integral with degenerate realanalytic phase and amplitude. Integrals such as these form a model for certain higherdimensional degenerate oscillatory integrals, for which it is known that many of the two
Complex Gaussian quadrature of oscillatory integrals
, 2008
"... We construct and analyze Gausstype quadrature rules with complexvalued nodes and weights to approximate oscillatory integrals with stationary points of high order. The method is based on substituting the original interval of integration by a set of contours in the complex plane, corresponding to t ..."
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Cited by 19 (8 self)
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We construct and analyze Gausstype quadrature rules with complexvalued nodes and weights to approximate oscillatory integrals with stationary points of high order. The method is based on substituting the original interval of integration by a set of contours in the complex plane, corresponding
Asymptotic Estimates for Oscillatory Integral Operators
, 1997
"... Asymptotic Estimates for Oscillatory Integral Operators Andrew Comech Asymptotics of the estimates on oscillatory integral operators with degenerate phase functions are considered. The main result is the estimate for the operator associated to the canonical relation which is a twosided Whitney fold ..."
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Cited by 2 (0 self)
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Asymptotic Estimates for Oscillatory Integral Operators Andrew Comech Asymptotics of the estimates on oscillatory integral operators with degenerate phase functions are considered. The main result is the estimate for the operator associated to the canonical relation which is a twosided Whitney
Oscillatory Integral Operators In Scattering Theory
 COMM. PARTIAL DIFFERENTIAL EQUATIONS
, 1997
"... We consider a particular Fourier integral operator with folding canonical relations, which arises in scattering theory: the Radon Transform of Melrose and Taylor. We obtain the regularity properties of this operator when the obstacle admits tangent planes with contact of precise order k (Theorem 1.1 ..."
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Cited by 6 (3 self)
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.1 and its Corollary). For these purposes, we derive asymptotic estimates for oscillatory integral operators in R n with folding canonical relations (Theorem 2.2). Asymptotics correspond to vanishing principal curvature of a fold of one of the projections from the canonical relation, and to small support
Homogeneous estimates for oscillatory integrals
 Acta. Math. Comenian. (N.S
"... Abstract. Let u(x, t) be the solution to the free timedependent Schrödinger equation at the point (x, t) in spacetime Rn+1 with initial data f. We characterize the size of u in terms Lp(Lq)estimates with power weights. Our bounds are given by norms of f in homogeneous Sobolev spaces Ḣs (Rn). O ..."
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Cited by 2 (0 self)
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Abstract. Let u(x, t) be the solution to the free timedependent Schrödinger equation at the point (x, t) in spacetime Rn+1 with initial data f. We characterize the size of u in terms Lp(Lq)estimates with power weights. Our bounds are given by norms of f in homogeneous Sobolev spaces Ḣs (Rn). Our methods include use of spherical harmonics, uniformity properties of Bessel functions and interpolation of vector valued weighted Lebesgue spaces. 1.
Results 1  10
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860