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Article
Filter design based on the fractional Fourier transform associated with new convolutions and correlations
We introduce new convolutions and correlations associated with the Fractional Fourier Transform (FrFT) which present a significant simplicity in both the time and FrFT domains. This allows for several conseque...
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Article
New sampling theorem and multiplicative filtering in the FRFT domain
Having in consideration a fractional convolution associated with the fractional Fourier transform (FRFT), we propose a novel reconstruction formula for bandlimited signals in the FRFT domain without using the ...
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Chapter
Hyers-Ulam and Hyers-Ulam-Rassias Stability for a Class of Integro-Differential Equations
We analyse different kinds of stabilities for a class of very general nonlinear integro-differential equations of Volterra type within appropriate metric spaces. Sufficient conditions are obtained in view to g...
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Article
New Convolutions for Quadratic-Phase Fourier Integral Operators and their Applications
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 convolu...
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Chapter
On a Class of Integral Equations Involving Kernels of Cosine and Sine Type
We consider a class of integral equations characterized by kernels of cosine and sine type and study their solvability. Moreover, we analyse the integral operator T, which is generating those equations, by identi...
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Article
Two New Convolutions for the Fractional Fourier Transform
In this paper we introduce two novel convolutions for the fractional Fourier transforms, and prove natural algebraic properties of the corresponding multiplications such as commutativity, associativity and dis...
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Article
Solutions of Tikhonov Functional Equations and Applications to Multiplication Operators on Szegö Spaces
We consider a natural representation of solutions for Tikhonov functional equations. This will be done by applying the theory of reproducing kernels to the approximate solutions of general bounded linear opera...
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Chapter and Conference Paper
Reproducing Kernels and Discretization
We give a short survey of a general discretization method based on the theory of reproducing kernels. We believe our method will become the next generation method for solving analytical problems by computers. ...
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Article
A Fundamental Theorem on Initial Value Problems by Using the Theory of Reproducing Kernels
We introduce a new method for solving general initial value problems by using the theory of reproducing kernels. The results are depending on the specific structure of each problem. Here, we give the general p...
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Chapter
How to Catch Smoothing Properties and Analyticity of Functions by Computers?
We would like to propose a new method in view to catch smoothing properties and analyticity of functions by computers. Of course, in the strict sense, such goal is impossible. However, we would like to propose...
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Chapter
Initial Value Problems in Linear Integral Operator Equations
For some general linear integral operator equations, we investigate consequent initial value problems by using the theory of reproducing kernels. A new method is proposed which—in particular—generates a new fi...
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Chapter
Aveiro Discretization Method in Mathematics: A New Discretization Principle
We found a very general discretization method for solving wide classes of mathematical problems by applying the theory of reproducing kernels. An illustration of the generality of the method is here performed ...
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Article
Invertibility criteria for Wiener–Hopf plus Hankel operators with different almost periodic Fourier symbol matrices
Based on the different kinds of auxiliary operators and corresponding operator relations, we will present conditions which characterize the invertibility of matrix Wiener–Hopf plus Hankel operators having diff...
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Article
Optimal and Approximate Solutions of Singular Integral Equations by Means of Reproducing Kernels
A reproducing kernel method is proposed to obtain the optimal and approximate solutions of Carleman singular integral equations. Therefore, we will be mostly interested in singular integral equations with a Ca...
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Article
Fractional Functions and their Representations
For arbitrary non-identically zero functions f, we will introduce some natural fractional functions f 1 having f as denominators and we shall consider their representations f 1 by ...
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Article
General Inhomogeneous Discrete Linear Partial Differential Equations with Constant Coefficients on the Whole Spaces
In this paper we shall introduce new constructions of approximate solutions of general linear partial differential equations with constant coefficients on the whole spaces, and establish fundamental estimates ...
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Article
On the Solvability of Singular Integral Equations with Reflection on the Unit Circle
The solvability of a class of singular integral equations with reflection in weighted Lebesgue spaces is analyzed, and the corresponding solutions are obtained. The main techniques are based on the considerati...
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Article
Exterior Wedge Diffraction Problems with Dirichlet, Neumann and Impedance Boundary Conditions
Classes of problems of wave diffraction by a plane angular screen occupying an infinite 270 degrees wedge sector are studied in a Bessel potential spaces framework. The problems are subjected to different poss...
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Chapter
Invertibility of Singular Integral Operators with Flip Through Explicit Operator Relations
The integral equations which are characterized by singular integral operators with shift appear frequently in a large variety of applied problems (we refer to [KaSa01, KrLi94] for a general background on these...
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Chapter
Fredholm Characterization of Wiener–Hopf–Hankel Integral Operators with Piecewise Almost Periodic Symbols
This chapter is concerned with the Fredholm property of matrix Wiener–Hopf–Hankel operators (cf. [BoCa08], [BoCa], and [LMT92]) of the form
(7.1)
