scholarly journals Toeplitz operators and Wiener-Hopf factorisation: an introduction

2017 ◽  
Vol 4 (1) ◽  
pp. 130-145 ◽  
Author(s):  
M. Cristina Câmara
Keyword(s):  
Integral Equations ◽  
Half Plane ◽  
Singular Integral ◽  
Hardy Spaces ◽  
Toeplitz Operators ◽  
Sufficient Conditions ◽  
Singular Integral Equations ◽  
Necessary And Sufficient Conditions ◽  
Upper Half Plane ◽  
Necessary And Sufficient

Abstract Wiener-Hopf factorisation plays an important role in the theory of Toeplitz operators. We consider here Toeplitz operators in the Hardy spaces Hp of the upper half-plane and we review how their Fredholm properties can be studied in terms of a Wiener-Hopf factorisation of their symbols, obtaining necessary and sufficient conditions for the operator to be Fredholm or invertible, as well as formulae for their inverses or one-sided inverses when these exist. The results are applied to a class of singular integral equations in L−1(ℝ)

Solution of a Linear Integral Equation of Third Kind

2002 ◽  
Vol 9 (1) ◽  
pp. 179-196
Author(s):  
D. Shulaia
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Singular Integral ◽  
Sufficient Conditions ◽  
Singular Integral Equations ◽  
Fredholm Theory ◽  
Linear Integral Equation ◽  
Hölder Functions ◽  
Linear Integral ◽  
Necessary And Sufficient

Abstract The aim of this paper is to study, in the class of Hölder functions, a nonhomogeneous linear integral equation with coefficient cos 𝑥. Necessary and sufficient conditions for the solvability of this equation are given under some assumptions on its kernel. The solution is constructed analytically, using the Fredholm theory and the theory of singular integral equations.

scholarly journals Some properties of Glisson distances in the upper half-plane

2021 ◽  
Vol 2131 (3) ◽  
pp. 032039
Author(s):  
M Ovchintsev
Keyword(s):  
Unit Ball ◽  
Unit Disk ◽  
Half Plane ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Ordinate Axis ◽  
Upper Half Plane ◽  
Necessary And Sufficient

Abstract The author compares the Gleason distance with the distance of Euclid in the unit disk in the upper half plane. The concept of “the Gleason distance” was formulated in the work of H.S. Bear [1] The Gleason distance is defined as follows (see [1]): d = sup |f(z2)-f(z1)|, f(Z)εB1(K) where B 1 (K) is the unit ball in the space of bounded analytic in K functions. The author of the article proves that in the circle K the distances of Gleason and Euclid are equal only when the points are opposite. He found necessary and sufficient conditions, when the distances are equal for the two given points which are symmetrical about the ordinate axis.

scholarly journals Hyponormality of Toeplitz operators with non-harmonic symbols on the Bergman spaces

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Sumin Kim ◽  
Jongrak Lee
Keyword(s):  
Toeplitz Operator ◽  
Bergman Space ◽  
Toeplitz Operators ◽  
Bergman Spaces ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient

AbstractIn this paper, we present some necessary and sufficient conditions for the hyponormality of Toeplitz operator $T_{\varphi }$ T φ on the Bergman space $A^{2}(\mathbb{D})$ A 2 ( D ) with non-harmonic symbols under certain assumptions.

Partial slip contact analysis for a monoclinic half plane

2020 ◽  
pp. 108128652096283
Author(s):  
İ Çömez ◽  
Y Alinia ◽  
MA Güler ◽  
S El-Borgi
Keyword(s):  
Integral Equations ◽  
Half Plane ◽  
Singular Integral ◽  
Integral Transform ◽  
Mixed Boundary ◽  
Normal Load ◽  
Fibrous Composite ◽  
Singular Integral Equations ◽  
Contact Stresses ◽  
Partial Slip

In this paper, the nonlinear partial slip contact problem between a monoclinic half plane and a rigid punch of an arbitrary profile subjected to a normal load is considered. Applying Fourier integral transform and the appropriate boundary conditions, the mixed-boundary value problem is reduced to a set of two coupled singular integral equations, with the unknowns being the contact stresses under the punch in addition to the stick zone size. The Gauss–Chebyshev discretization method is used to convert the singular integral equations into a set of nonlinear algebraic equations, which are solved with a suitable iterative algorithm to yield the lengths of the stick zone in addition to the contact pressures. Following a validation section, an extensive parametric study is performed to illustrate the effects of material anisotropy on the contact stresses and length of the stick zone for typical monoclinic fibrous composite materials.

Interaction of Three Interfacial Cracks between an Orthotropic Half-Plane Bonded to a Dissimilar Orthotropic Layer with Punch

2017 ◽  
Vol 72 (11) ◽  
pp. 1021-1029
Author(s):  
P.K. Mishra ◽  
P. Singh ◽  
S. Das
Keyword(s):  
Integral Equations ◽  
Half Plane ◽  
Singular Integral ◽  
Jacobi Polynomials ◽  
Singular Integral Equations ◽  
Orthotropic Layer ◽  
Central Crack ◽  
Interfacial Cracks ◽  
Crack Shielding

AbstractThis article deals with the interactions between a central crack and a pair of outer cracks situated at the interface of an orthotropic elastic half-plane bonded to a dissimilar orthotropic layer with a punch. The problem is reduced to the solution of three simultaneous singular integral equations that are finally solved using Jacobi polynomials. The phenomena of crack shielding and crack amplification have been depicted through graphs for different particular cases.

scholarly journals A convergence theorem for singular integral equations

1981 ◽  
Vol 22 (4) ◽  
pp. 539-552 ◽  
Author(s):  
David Elliott
Keyword(s):  
Integral Equation ◽  
Singular Integral Equation ◽  
Integral Equations ◽  
Singular Integral ◽  
Sufficient Conditions ◽  
Singular Integral Equations ◽  
Convergence Theory ◽  
Cauchy Kernel ◽  
Algebraic Equations ◽  
Linear Algebraic Equations

AbstractThe principal result of this paper states sufficient conditions for the convergence of the solutions of certain linear algebraic equations to the solution of a (linear) singular integral equation with Cauchy kernel. The motivation for this study has been the need to provide a convergence theory for a collocation method applied to the singular integral equation taken over the arc (−1, 1). However, much of the analysis will be applicable both to other approximation methods and to singular integral equations taken over other arcs or contours. An estimate for the rate of convergence is also given.

SINGULAR INTEGRAL EQUATION METHOD FOR MULTIPLE FLAT PUNCH PROBLEM FOR AN ELASTIC HALF-PLANE

2009 ◽  
Vol 06 (04) ◽  
pp. 605-614
Author(s):  
Y. Z. CHEN ◽  
Z. X. WANG ◽  
X. Y. LIN
Keyword(s):  
Stress Distribution ◽  
Integral Equations ◽  
Half Plane ◽  
Singular Integral ◽  
Singular Integral Equations ◽  
Integral Equation Method ◽  
Angle Distribution ◽  
Flat Punch ◽  
Singular Stress ◽  
Punch Problem

When a flat punch is indented on elastic half-plane, the singular stress distribution at the vicinity of the punch corners is studied. The angle distribution for the stress components is also achieved in an explicit form. From obtained singular stress distribution, the punch singular stress factor is defined. The multiple punch problem can be considered as a superposition of many single punch problems. Taking the stress distribution under the punch base as the unknown function and the deformation under punch as the right-hand term, a set of the singular integral equations for the multiple punch problem can be achieved. After the singular integral equations are solved, the stress distributions under punches can be obtained. In addition, the exerting locations of the resultant forces under punches can also be determined. Two numerical examples with the calculated results are presented.

scholarly journals On the approximate solution of nonlinear singular integral equations with positive index

1996 ◽  
Vol 19 (2) ◽  
pp. 389-396 ◽  
Author(s):  
S. M. Amer
Keyword(s):  
Approximate Solution ◽  
Integral Equations ◽  
Collocation Method ◽  
Jordan Curve ◽  
Singular Integral ◽  
Sufficient Conditions ◽  
Singular Integral Equations ◽  
Holder Space ◽  
Smooth Jordan Curve ◽  
Positive Index

This paper is devoted to investigating a class of nonlinear singular integral equations with a positive index on a simple closed smooth Jordan curve by the collocation method. Sufficient conditions are given for the convergence of this method in Holder space.

Dynamic Response of a Rigid Footing Bonded to an Elastic Half Space

1972 ◽  
Vol 39 (2) ◽  
pp. 527-534 ◽  
Author(s):  
J. E. Luco ◽  
R. A. Westmann
Keyword(s):  
Dynamic Response ◽  
Integral Equations ◽  
Half Plane ◽  
Singular Integral ◽  
Singular Integral Equations ◽  
Elastic Half Space ◽  
Vertical Shear ◽  
Fredholm Integral Equations ◽  
Interface Conditions ◽  
Rigid Footing

The problem of determining the response of a rigid strip footing bonded to an elastic half plane is considered. The footing is subjected to vertical, shear, and moment forces with harmonic time-dependence; the bond to the half plane is complete. Using the theory of singular integral equations the problem is reduced to the numerical solution of two Fredholm integral equations. The results presented permit the evaluation of approximate footing models where assumptions are made about the interface conditions.

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