Oscillatory Singular Integrals Research Articles

A Collocation Numerical Method for Highly Oscillatory Algebraic Singular Volterra Integral Equations

The highly oscillatory algebraic singular Volterra integral equations cannot be solved directly. A collocation numerical method is proposed to overcome the difficulty created by the highly oscillatory algebraic singular kernel. This paper is composed primarily of two methods—the piecewise constant collocation method and the piecewise linear collocation method—in which uniformly distributed nodes serve as collocation points. For the efficient computation of highly oscillatory and algebraic singular integrals, the steepest descent method as well as the Gauss–Laguerre and generalized Gauss–Laguerre quadrature rules are employed. Consequently, the resulting linear system is solved for the unknown function approximated by the Lagrange interpolation polynomial. Detailed theoretical analysis is carried out and numerical experiments showing high accuracy are also presented to confirm our analysis.

A Class of Oscillatory Singular Integrals with Rough Kernels and Fewnomials Phases

A Class of Oscillatory Singular Integrals with Rough Kernels and Fewnomials Phases

The Weighted $L^p$-Boundedness for a Class of Multilinear Oscillatory Singular Integrals Related to Block Spaces

The Weighted $L^p$-Boundedness for a Class of Multilinear Oscillatory Singular Integrals Related to Block Spaces

On the boundedness of a family of oscillatory singular integrals

On the boundedness of a family of oscillatory singular integrals

Numerical analysis of the spectrum for the highly oscillatory integral equation with weak singularity

In this paper, we focus on the spectra numerical computation for the integral equation with the absolute oscillation and power-law or logarithmic singularity. Finite section method is applied to transform the integral equation to an algebraic eigenvalue problem. The entries of the coefficient matrix appearing in the bivariate highly oscillatory singular integrals can be represented explicitly in Gamma or the exponential integral functions. The decay rate of the entries is established to construct the truncation scheme. Then the infinite algebraic eigenvalue problem can be simplified to be the finite one. The corresponding error of the infinite and finite algebraic systems is also bounded. Finally, the numerical experiments are provided to illustrate the theoretical analysis.

On oscillatory singular integrals and their commutators with non-convolutional Hölder class kernels

On oscillatory singular integrals and their commutators with non-convolutional Hölder class kernels

The Hilbert transform along the parabola, the polynomial Carleson theorem and oscillatory singular integrals

We make progress on an interesting problem on the boundedness of maximal modulations of the Hilbert transform along the parabola. Namely, if we consider the multiplier arising from it and restrict it to lines, we prove uniform $$L^p$$ bounds for maximal modulations of the associated operators. Our methods consist of identifying where to use effectively the polynomial Carleson theorem, and where we can take advantage of the presence of oscillation to obtain decay through the $$TT^*$$ method.

Fast hybrid numerical-asymptotic boundary element methods for high frequency screen and aperture problems based on least-squares collocation

We present a hybrid numerical-asymptotic (HNA) boundary element method (BEM) for high frequency scattering by two-dimensional screens and apertures, whose computational cost to achieve any prescribed accuracy remains bounded with increasing frequency. Our method is a collocation implementation of the high order hp HNA approximation space of Hewett et al. (IMA J Numer Anal 35:1698–1728, 2015), where a Galerkin implementation was studied. An advantage of the current collocation scheme is that the one-dimensional highly oscillatory singular integrals appearing in the BEM matrix entries are significantly easier to evaluate than the two-dimensional integrals appearing in the Galerkin case, which leads to much faster computation times. Here we compute the required integrals at frequency-independent cost using the numerical method of steepest descent, which involves complex contour deformation. The change from Galerkin to collocation is nontrivial because naive collocation implementations based on square linear systems suffer from severe numerical instabilities associated with the numerical redundancy of the HNA basis, which produces highly ill-conditioned BEM matrices. In this paper we show how these instabilities can be removed by oversampling, and solving the resulting overdetermined collocation system in a weighted least-squares sense using a truncated singular value decomposition. On the basis of our numerical experiments, the amount of oversampling required to stabilise the method is modest (around 25% typically suffices), and independent of frequency. As an application of our method we present numerical results for high frequency scattering by prefractal approximations to the middle-third Cantor set.

A Mode-Matching Solution for TE-Backscattering from an Arbitrary 2D Rectangular Groove in a PEC

In this paper, the simple yet effective mode-matching technique is utilized to compute TE-backscattering from a 2D filled rectangular groove in an infinite perfect electric conductor (PEC). The tangential magnetic fields inside and outside of the groove are represented as the sums of infinite series of cosine harmonics (half-range Fourier cosine series). By applying the continuity of the tangential magnetic field, these modes are matched on the groove to obtain the series coefficients by solving a system of linear equations. For this purpose, some oscillatory logarithmic singular integrals involving Hankel and trigonometric functions are solved numerically, starting by removing the logarithmic singularity via integration by parts. In the following, the new well-behaved highly oscillatory integrals are computed using efficient methods, and several comparisons are made to demonstrate the validity and ability of the presented procedure.

Numerical evaluation of oscillatory-singular integrals

ABSTRACTIn this paper, we consider an asymptotic Filon-type method for computing the oscillatory Cauchy principal value integrals and end-point oscillatory-singular integrals. The method is constructed by using a Lagrange interpolating polynomial to approximate the non-oscillatory function so that it successfully avoids the singularity and solving the systems of equations. Numerical examples and theoretical results show the accuracy and efficiency of the proposed method.

Oscillatory singular integral operators with Hölder class kernels on Hardy spaces

Abstract A sharp logarithmic bound is established for the H 1 {H^{1}} -norm of oscillatory singular integrals with quadratic phases and Hölder class kernels. Prior results had relied on a C 1 {C^{1}} -assumption on the kernel.

On the approximate evaluation of oscillatory-singular integrals

AbstractIn this paper an efficient numerical scheme is proposed for the numerical computation of the Cauchy type oscillatory integral ∫-11coswxxf(x)dx; where f(x) is a well-behaved function without having any kind of singularity in the range of integration [−1; 1]. The scheme is devised with the help of quadrature rule meant for the approximate evaluation of Cauchy principal value of integrals of the type ∫-11f(x)dx; and a quasi exact quadrature meant for the numerical integration of Filon-type integrals. The error bounds are determined and the scheme numerically verified by some standard test integrals.

L p estimates of rough maximal functions along surfaces with applications

In this paper, we study the Lp mapping properties of certain class of maximal oscillatory singular integral operators. We prove a general theorem for a class of maximal functions along surfaces. As a consequence of such theorem, we establish the Lp boundedness of various maximal oscillatory singular integrals provided that their kernels belong to the natural space Llog L(Sn−1). Moreover, we highlight some additional results concerning operators with kernels in certain block spaces. The results in this paper substantially improve previously known results.

Norm inequalities for higher-order commutators of one-sided oscillatory singular integrals

In the present paper, we study the weighted norm inequalities for higher-order commutators formed by a class of one-sided oscillatory singular integrals and $BMO$ functions. We obtain that the boundedness of these commutators can be deduced by that of one-sided Calderon-Zygmund singular integral operators.

Logarithmic Bounds for Oscillatory Singular Integrals on Hardy Spaces

We establish a logarithmic bound for oscillatory singular integrals with quadratic phases on the Hardy spaceH1(Rn). The logarithmic rate of growth is the best possible.

Some one-sided estimates for oscillatory singular integrals

The purpose of this paper is to establish some one-sided estimates for oscillatory singular integrals. The boundedness of certain oscillatory singular integrals on weighted Hardy spaces H+1(w) is proved. It is here also shown 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 w∈Ap+, a wider weight class than the classical Muckenhoupt class. Furthermore, a criterion on the weighted Lp-boundedness of the oscillatory singular integral is given.

LpBounds for the Commutators of Oscillatory Singular Integrals with Rough Kernels

We establish theLpboundedness for some commutators of oscillatory singular integrals with the kernel condition which was introduced by Grafakos and Stefanov. Our theorems contain various conditions on the phase function.

Boundedness of oscillatory singular integral with rough kernels on Triebel-Lizorkin spaces

In this paper, we will prove the Triebel-Lizorkin boundedness for some oscillatory singular integrals with the kernel Ω(x) satisfying a condition introduced by Grafakos and Stefanov. Our theorems will be proved under various conditions on the phase function, radial and nonradial. Since the L p boundedness of these operators is not complete yet, the theorems extend many known results.

Weighted Estimates for Oscillatory Singular Integrals

We establish uniform bounds for oscillatory singular integrals as well as oscillatory singular integral operators. We allow the singular kernel to be given by a function in the Hardy space <svg style="vertical-align:-2.3205pt;width:61.875px;" id="M1" height="18.725" version="1.1" viewBox="0 0 61.875 18.725" width="61.875" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,15.775)"><path id="x1D43B" d="M865 650q-1 -4 -4 -14t-4 -14q-62 -5 -77 -19.5t-29 -82.5l-74 -394q-12 -61 -0.5 -77t75.5 -21l-6 -28h-273l8 28q64 5 82 21t29 76l36 198h-380l-37 -197q-11 -64 0.5 -78.5t79.5 -19.5l-6 -28h-268l6 28q60 6 75.5 21.5t26.5 76.5l75 394q13 66 2 81.5t-77 20.5l8 28&#xA;h263l-6 -28q-58 -5 -75.5 -21t-30.5 -81l-26 -153h377l29 153q12 67 2 81t-74 21l5 28h268z" /></g> <g transform="matrix(.012,-0,0,-.012,14.975,7.613)"><path id="x31" d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" /></g> <g transform="matrix(.017,-0,0,-.017,21.3,15.775)"><path id="x28" d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" /></g><g transform="matrix(.017,-0,0,-.017,27.182,15.775)"><path id="x1D412" d="M484 692v-218h-28q-25 83 -70 130q-54 56 -123 56q-51 0 -80.5 -27t-29.5 -72q0 -37 21 -59q23 -25 82 -55q14 -8 68 -36q67 -34 80 -41q50 -28 79.5 -74.5t29.5 -101.5q0 -96 -69 -154q-68 -59 -179 -59q-61 0 -130 24q-30 10 -41 10q-23 0 -30 -34h-29v248h29&#xA;q23 -97 60 -143q57 -72 144 -72q58 0 92 30.5t34 82.5q0 47 -57 88q-30 21 -122 65q-91 44 -131 91.5t-40 118.5q0 95 58.5 148t160.5 53q54 0 117 -23q27 -9 42 -9q14 0 20.5 6.5t11.5 26.5h30z" /></g> <g transform="matrix(.012,-0,0,-.012,36.638,7.613)"><path id="x1D45B" d="M495 86q-46 -47 -87 -72.5t-63 -25.5q-43 0 -16 107l49 210q7 34 8 50.5t-3 21t-13 4.5q-35 0 -109.5 -72.5t-115.5 -140.5q-21 -75 -38 -159q-50 -10 -76 -21l-6 8l84 340q8 35 -4 35q-17 0 -67 -46l-15 26q44 44 85.5 70.5t64.5 26.5q35 0 10 -103l-24 -98h2&#xA;q42 56 97 103.5t96 71.5q46 26 74 26q9 0 16 -2.5t14 -11.5t9.5 -24.5t-1 -44t-13.5 -68.5q-30 -117 -47 -200q-4 -19 -3.5 -25t6.5 -6q21 0 70 48z" /></g><g transform="matrix(.012,-0,0,-.012,42.611,7.613)"><path id="x2212" d="M535 230h-483v50h483v-50z" /></g><g transform="matrix(.012,-0,0,-.012,49.595,7.613)"><use xlink:href="#x31"/></g> <g transform="matrix(.017,-0,0,-.017,55.925,15.775)"><path id="x29" d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" /></g> </svg>, while such results were known previously only for kernels in <svg style="vertical-align:-0.0pt;width:9.8249998px;" id="M2" height="11.175" version="1.1" viewBox="0 0 9.8249998 11.175" width="9.8249998" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path id="x1D43F" d="M559 163q-23 -66 -68 -163h-474l6 26q62 4 79.5 19.5t28.5 75.5l78 409q7 35 8.5 49t-8 25t-24 13t-51.5 5l5 28h266l-6 -28q-65 -5 -79.5 -18t-25.5 -74l-76 -406q-10 -57 14 -75q12 -13 96 -13q93 0 126 29q41 40 76 109z" /></g> </svg> log <svg style="vertical-align:-2.3205pt;width:50.337502px;" id="M3" height="18.725" version="1.1" viewBox="0 0 50.337502 18.725" width="50.337502" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,15.775)"><use xlink:href="#x1D43F"/></g><g transform="matrix(.017,-0,0,-.017,9.769,15.775)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,15.65,15.775)"><use xlink:href="#x1D412"/></g> <g transform="matrix(.012,-0,0,-.012,25.1,7.613)"><use xlink:href="#x1D45B"/></g><g transform="matrix(.012,-0,0,-.012,31.073,7.613)"><use xlink:href="#x2212"/></g><g transform="matrix(.012,-0,0,-.012,38.058,7.613)"><use xlink:href="#x31"/></g> <g transform="matrix(.017,-0,0,-.017,44.387,15.775)"><use xlink:href="#x29"/></g> </svg>, a proper subspace of <svg style="vertical-align:-2.3205pt;width:61.875px;" id="M4" height="18.725" version="1.1" viewBox="0 0 61.875 18.725" width="61.875" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,15.775)"><use xlink:href="#x1D43B"/></g> <g transform="matrix(.012,-0,0,-.012,14.975,7.613)"><use xlink:href="#x31"/></g> <g transform="matrix(.017,-0,0,-.017,21.3,15.775)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,27.182,15.775)"><use xlink:href="#x1D412"/></g> <g transform="matrix(.012,-0,0,-.012,36.638,7.613)"><use xlink:href="#x1D45B"/></g><g transform="matrix(.012,-0,0,-.012,42.611,7.613)"><use xlink:href="#x2212"/></g><g transform="matrix(.012,-0,0,-.012,49.595,7.613)"><use xlink:href="#x31"/></g> <g transform="matrix(.017,-0,0,-.017,55.925,15.775)"><use xlink:href="#x29"/></g> </svg>. One of our results established a <svg style="vertical-align:-2.3205pt;width:113.95px;" id="M5" height="17.3375" version="1.1" viewBox="0 0 113.95 17.3375" width="113.95" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,14.388)"><use xlink:href="#x1D43F"/></g> <g transform="matrix(.012,-0,0,-.012,9.763,6.225)"><path id="x1D45D" d="M570 304q0 -108 -87 -199q-40 -42 -94.5 -74t-105.5 -43q-41 0 -65 11l-29 -141q-9 -45 -1.5 -58t45.5 -16l26 -2l-5 -29l-241 -10l4 26q51 10 67.5 24t26.5 60l113 520q-54 -20 -89 -41l-7 26q38 28 105 53l11 49q20 25 77 58l8 -7l-17 -77q39 14 102 14q82 0 119 -36&#xA;t37 -108zM482 289q0 114 -113 114q-26 0 -66 -7l-70 -327q12 -14 32 -25t39 -11q59 0 118.5 81.5t59.5 174.5z" /></g> <g transform="matrix(.017,-0,0,-.017,17.45,14.388)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,23.332,14.388)"><path id="x1D464" d="M689 332q0 -81 -40 -161.5t-103.5 -131.5t-127.5 -51q-32 0 -58.5 16t-40.5 46q-70 -62 -139 -62q-45 0 -77.5 30t-32.5 86q0 8 7 52q11 77 45 187q6 19 6 34q0 6 -7 6q-24 0 -78 -64l-20 23q32 48 73 77t78 29q32 0 32 -43q0 -30 -12 -73q-27 -95 -38 -152&#xA;q-8 -44 -8 -58q0 -77 68 -77q31 0 60 33.5t39 77.5l62 260l75 16l5 -6l-63 -246q-8 -29 -8 -58q0 -77 68 -77q66 0 116 75.5t50 194.5q0 43 -12 57q-11 12 -11 24q0 19 15 35.5t34 16.5q18 0 30.5 -34.5t12.5 -81.5z" /></g><g transform="matrix(.017,-0,0,-.017,35.435,14.388)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,48.864,14.388)"><path id="x2192" d="M901 255q-71 -62 -185 -187l-22 15l102 147h-727v50h727l-102 147l22 15q114 -125 185 -187z" /></g><g transform="matrix(.017,-0,0,-.017,72.628,14.388)"><use xlink:href="#x1D43F"/></g> <g transform="matrix(.012,-0,0,-.012,82.35,6.225)"><use xlink:href="#x1D45D"/></g> <g transform="matrix(.017,-0,0,-.017,90.025,14.388)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,95.907,14.388)"><use xlink:href="#x1D464"/></g><g transform="matrix(.017,-0,0,-.017,108.01,14.388)"><use xlink:href="#x29"/></g> </svg> bound for certain weights. At the same time, it provides a solution to an open problem in Lu (2005).

Boundedness of Oscillatory Integrals with Variable Calderón-Zygmund Kernel on Weighted Morrey Spaces

Oscillatory integral operators play a key role in harmonic analysis. In this paper, the authors investigate the boundedness of the oscillatory singular integrals with variable Calderón-Zygmund kernel on the weighted Morrey spacesLp,k(ω). Meanwhile, the corresponding results for the oscillatory singular integrals with standard Calderón-Zygmund kernel are established.