Convolution Type Operators Research Articles

Stochastic homogenization of convolution type operators

This paper deals with the homogenization problem for convolution type non-local operators in random statistically homogeneous ergodic media. Assuming that the convolution kernel has a finite second moment and satisfies certain symmetry and uniform ellipticity conditions, we prove the almost sure homogenization result and show that the limit operator is a second order elliptic differential operator with constant deterministic coefficients.

Homogenization of biased convolution type operators

The final publication is available at IOS Press through http://dx.doi.org/10.3233/ASY-191533

NON-STATIONARY PROBLEM OF THE PLANE OBLIQUE PRESSURE WAVE DIFFRACTION ON THIN SHELL IN THE SHAPE OF PARABOLIC CYLINDER

A non-stationary plane problem of the dynamics of thin elastic shell in the form of parabolic cylinder immersed in the fluid under the impact of the plane oblique pressure wave is considered. To solve this problem, a system of equations in the related formulation is constructed. Herewith, the hydroelasticity problems are reduced to the equations of the shell dynamics, the damping effect of fluid is taken into account by introducing an integral convolution type operator in the time domain which in the first approximation allows for accounting the capillary porosity of the shell material. The operator core is a surface transition function of the auxiliary problem of the plane acoustic pressure wave diffraction on a convex surface. The problem is solved approximately based on the thin layer hypothesis. The integral and differential equations of shell motion are solved numerically based on the difference discretization of differential operators and the representation of the integral operator by sum using the trapezium rule.

On Invertibility of Convolution Type Operators in Morrey Spaces

On Invertibility of Convolution Type Operators in Morrey Spaces

Theory of the linear sampling method for time-dependent fields

The linear sampling method seeks to localize the unknown source of an observed, time-dependent field. The unknown source could be, for example, a scatterer embedded within a medium, or an impulsive excitation such as an earthquake or explosion. The source of the observed field is localized by means of solving the so-called near-field equation and mapping the obtained solutions through an indicator functional over a test region assumed to contain the source. In its current formulation, however, the linear sampling method suffers from an ambiguous time parameter that strongly influences its ability to localize the unknown source. Our paper consists of two fundamental results central to the theoretical understanding of the linear sampling method and its numerical implementation. First, we prove the so-called blowup behavior of solutions to the near-field equation for a general source function that is separable in space and time. Second, we show that the linear sampling method can be formulated such that the ambiguous time parameter is irrelevant. We demonstrate that a dependence of the linear sampling method on the time parameter arises from an incorrect implementation of a convolution-type operator found in the near-field equation. When the operator is implemented correctly, the dependence on the time parameter vanishes. We provide detailed algorithms for efficient and proper implementations of the convolutional operator in both the time and frequency domains. The crucial result of the improved implementations is that they allow the linear sampling method to be completely automated, as one does not need to know the space-time dependence of the unknown source. We demonstrate the effectiveness of the improved time- and frequency-domain implementations using several numerical examples applied to imaging scatterers.

C⁎-algebra of nonlocal convolution type operators

The C⁎-subalgebra B of all bounded linear operators on the space L2(R), which is generated by all multiplication operators by piecewise slowly oscillating functions, by all convolution operators with piecewise slowly oscillating symbols and by the range of a unitary representation of the group of all affine mappings on R, is studied. A faithful representation of the quotient C⁎-algebra Bπ=B/K in a Hilbert space, where K is the ideal of compact operators on the space L2(R), is constructed by applying an appropriate spectral measure decompositions, a local-trajectory method and the Fredholm symbol calculus for the C⁎-algebra of convolution type operators without shifts. This gives a Fredholm symbol calculus for the C⁎-algebra B and a Fredholm criterion for the operators B∈B.

Diffraction by semi-infinite cone formed with electric and magnetic surfaces: analytical regularization and Wiener–Hopf techniques

The problem of axially symmetric E-polarized electromagnetic waves diffraction from the circular cone formed by the junction of the perfectly electro-conducting (PEC) semi-infinite truncated cone and the perfectly magneto-conducting (PMC) finite cone is studied rigorously by applying the Kontorovich–Lebedev integral transformation technique, the mode-matching method, and the analytical regularization procedure. The convolution-type operator and its inverse operator are used to conduct the regularization procedure and to reduce the problem to the infinite systems of linear algebraic equations of the second kind. The left side and the right side of the regularization procedures are considered. Transition from the cone to the PEC plane with the circular magnetic window is analysed. The exact solution of the problem is obtained for the static limit, and the low-frequency approximate is derived. The Wiener–Hopf technique is also applied to obtain the solution. Both solutions obtained using the analytical regularization method and by the Wiener–Hopf technique are compared, and the relation between them is derived. Within the framework of the Macdonald’s model, the solution is applied to analyse the far field scattered from the two partially absorbing conical scatterers; (1) the PEC semi-infinite cone with the “black” vertex; and (2) the semi-infinite PMC cone with the “black” truncated conical surface.

Algebra of convolution type operators with continuous data on Banach function spaces

We show that if the Hardy-Littlewood maximal operator is bounded on a reflexive Banach function space $X(\mathbb{R})$ and on its associate space $X'(\mathbb{R})$, then the space $X(\mathbb{R})$ has an unconditional wavelet basis. As a consequence of the existence of a Schauder basis in $X(\mathbb{R})$, we prove that the ideal of compact operators $\mathcal{K}(X(\mathbb{R}))$ on the space $X(\mathbb{R})$ is contained in the Banach algebra generated by all operators of multiplication $aI$ by functions $a\in C(\dot{\mathbb{R}})$, where $\dot{\mathbb{R}}=\mathbb{R}\cup\{\infty\}$, and by all Fourier convolution operators $W^0(b)$ with symbols $b\in C_X(\dot{\mathbb{R}})$, the Fourier multiplier analogue of $C(\dot{\mathbb{R}})$.

A weighted inequality for potential type operators

AbstractWe establish a weighted inequality for fractional maximal and convolution type operators, between weak Lebesgue spaces and Wiener amalgam type spaces on {\mathbb{R}} endowed with a measure which needs not to be doubling.

Systems with Fractional Variable-Order Difference Operator of Convolution Type and Its Stability

In the paper the Grunwald–Letnikov-type linear fractional variable order discrete-time systems are studied. The conditions for stability and instability are formulated. The regions of the systems stability are determined accordingly to locus of eigenvalues of a matrix associated to the considered system.DOI: http://dx.doi.org/10.5755/j01.eie.24.5.21846

Optimal Reinsurance via Dirac-Feynman Approach

In this paper, the Dirac-Feynman path calculation approach is applied to analyse finite time ruin probability of a surplus process exposed to reinsurance by capital injections. Several reinsurance optimization problems on optimum insurance and reinsurance premium with respect to retention level are investigated and numerically illustrated. The retention level is chosen to decrease the finite time ruin probability and to guarantee that reinsurance premium covers an average of overall capital injections. All computations are based on Dirac-Feynman path calculation approach applied to the convolution type operators perturbed by Injection operator (shift type operator). In addition, the effect of the Injection operator on ruin probability is analysed.

A characterization of BMO in terms of endpoint bounds for commutators of singular integrals

We provide a characterization of BMO in terms of endpoint boundedness of commutators of singular integrals. In particular, in one dimension, we show that ∥b∥BMO ≃ B, where B is the best constant in the endpoint L log L modular estimate for the commutator [H, b]. We provide a similar characterization of the space BMO in terms of endpoint boundedness of higher order commutators of the Hilbert transform. In higher dimension we give the corresponding characterization of BMO in terms of the first order commutators of the Riesz transforms. We also show that these characterizations can be given in terms of commutators of more general singular integral operators of convolution type.

Compactness of Some Operators of Convolution Type in Generalized Morrey Spaces

Sufficient conditions for the compactness in generalized Morrey spaces of the composition of a convolution operator and the operator of multiplication by an essentially bounded function are obtained. Very weak conditions on the function are also obtained under which the commutator of the operator of multiplication by such a function and a convolution operator is compact. The compactness of convolution operators in domains of cone type is investigated.

Regularity of maximal functions on Hardy–Sobolev spaces

We prove that maximal operators of convolution type associated to smooth kernels are bounded in the homogeneous Hardy–Sobolev spaces H ̇ 1 , p ( R d ) when p > d / ( d + 1 ) . This range of exponents is sharp. As a by-product of the proof, we obtain similar results for the local Hardy–Sobolev spaces h ̇ 1 , p ( R d ) in the same range of exponents.

A One–parameter Family of Bounded Solutions for a System of Nonlinear Integral Equations on the Whole Line

A system of nonlinear integral equations with a convolution type operator arising in the p–adic string theory for the scalar tachyons field is studied. The existence of a one–parameter family of monotone continuous and bounded solutions for this system is proved. The limits of the constructed solutions at ±∞ are calculated.

Welland's type inequalities for fractional operators of convolution with kernels satisfying a Hörmander type condition and its commutators

ABSTRACTLet μ be a non-negative Ahlfors n-dimensional measure on . In this context we shall consider convolution type operators , , where the kernels are supposed to satisfy certain size and regularity conditions. We prove Welland's type inequalities for the operator and its commutator , with , that include the case . As far as we know both estimates are new even in the case of the Lebesgue measure. We shall also give sufficient conditions on a pair of weights that guarantee the boundedness of between two different weighted Lebesgue spaces when the underlying measure is Ahlfors n-dimensional.

On the variation of maximal operators of convolution type II

In this paper we establish that several maximal operators of convolution type, associated to elliptic and parabolic equations, are variation-diminishing. Our study considers maximal operators on the Euclidean space \mathbb R^d , on the torus \mathbb T^d and on the sphere \mathbb S^d . The crucial regularity property that these maximal functions share is that they are subharmonic in the corresponding detachment sets.

Pointwise estimates for heat kernels of convolution-type operators

The paper deals with point-wise estimates for the heat kernel of a nonlocal convolution type operator with a kernel that decays at least exponentially at infinity. It is shown that the large time behaviour of the heat kernel depends essentially on whether $|x|\ll t$, or $|x|\sim t$, or $|x|\gg t$. We obtain sharp point-wise upper bounds for the heat kernel in all these regions. In the first region the nonlocal heat kernel behaves like the classical one, while in the other regions we observe an essential difference.

Extension methods for convolution type operators

The purpose of this work is to present a study of extension methods for convolution type operators defined bywhere Ω is defined by the union of intervals in different ways.The extension methods presented are useful to construct relations between operators and is well known that the relations between operators presented in this work plays an important role in the study of a large number of mathematical and physical problems in particular in the problems of wave diffraction and receive great importance in many other applications.

Compactness of Some Operators of Convolution Type in Generalized Morrey Spaces

Получены достаточные условия компактности в обобщенных пространствах Морри композиции оператора свертки и оператора умножения на существенно ограниченную функцию. Также получены весьма слабые условия на функцию, при которых коммутатор оператора умножения на эту функцию и оператора свертки является компактным. Исследована компактность операторов свертки в областях типа конуса. Библиография: 15 названий.