On moment-discretization and least-squares solutions of linear integral equations of the first kind
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In this paper, a simple commutativity result between the operations of moment-discretization and least-squares solutions of linear integral equations of the first kind is established.About:
This article is published in Journal of Mathematical Analysis and Applications.The article was published on 1976-02-01 and is currently open access. It has received 52 citations till now. The article focuses on the topics: Integral equation & Numerical integration.read more
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Book ChapterDOI
Linear Inverse and III-Posed Problems
TL;DR: In this article, the authors considered linear inverse problems that have the following general structure: the first step is the definition of the direct problem, which must be linear, and then the solution of the original direct problem defines a linear mapping L from the space X of all functions characterizing the properties of the physical sample (such as the density function in the case of a vibrating string or the refraction index in a semi-transparent object, etc.) into the space Y of all corresponding measurable quantities, such as sequences of eigenvalues, scattering amplitudes, and so
Journal ArticleDOI
Algorithms for the regularization of ill-conditioned least squares problems
TL;DR: Two regularization methods for ill-conditioned least squares problems are studied from the point of view of numerical efficiency and it is shown that if they are transformed into a certain standard form, very efficient algorithms can be used for their solution.
Book ChapterDOI
The Gamma Function
Willi Freeden,Martin Gutting +1 more
TL;DR: The Gamma function as discussed by the authors is a generalized factorial function that can be used to estimate the probability distribution of a probability distribution, and it has been used in many applications, e.g., as part of probability distributions.
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Operator-theoretic and computational approaches to Ill-posed problems with applications to antenna theory
TL;DR: A general framework for regularization and approximation methods for ill-posed problems is developed in this paper, where three levels in the resolution processes are distinguished and emphasized: philosophy of resolution, regularization-approximation schema, and regularization algorithms.
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Convolution, Average Sampling, and a Calderon Resolution of the Identity for Shift-Invariant Spaces
TL;DR: In this article, the convolution/deconvolution problem, the uniformly sampled convolution and the reconstruction problem, and the sample convolution followed by sampling on irregular grid were studied.
References
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Book
Generalized inverses: theory and applications
Adi Ben-Israel,T. N. E. Greville +1 more
TL;DR: In this paper, the Moore of the Moore-Penrose Inverse is described as a generalized inverse of a linear operator between Hilbert spaces, and a spectral theory for rectangular matrices is proposed.
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Generalized Inverse of Matrices and Its Applications
TL;DR: In this article, the generalized inverse of matrices and its applications are discussed and discussed in terms of generalized inverse of matrix and its application in the context of generalization of matrix matrices.
Journal ArticleDOI
On the approximate minimization of functionals
TL;DR: In this article, the authors consider the problem of finding the minimum of a given functional f(u) over a set B by approximately minimizing a sequence of functionals over a "discretized" set Bn; theorems are given proving the convergence of the approximating points un in Bn to the desired point u in B.
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A Numerical Method for Solving Fredholm Integral Equations of the First Kind Using Singular Values
TL;DR: In this article, the integral equation in question is approximated by simple numerical quadrature formulas plus collocation, and each row of the resulting matrix equation for the unknown function values is weighted by the reciprocal of the standard deviation of the known function.