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"Written by three leaders in the field of neural based algorithms, Neural Based Orthogonal Data Fitting proposes several neural networks, all endowed with a complete theory which not only explains their behavior, but also compares them with the existing neural and traditional algorithms. The algorithms are studied from different points of view, including: as a differential geometry problem, as a dynamic problem, as a stochastic problem, and as a numerical problem. All algorithms have also been analyzed on real time problems (large dimensional data matrices) and have shown accurate solutions. Where most books on the subject are dedicated to PCA (principal component analysis) and consider MCA (minor component analysis) as simply a consequence, this is the fist book to start from the MCA problem and arrive at important conclusions about the PCA problem."--

Neural networks (Computer science) --- Numerical analysis. --- Orthogonalization methods.

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L-functions --- Orthogonalization methods --- Symplectic and contact topology --- Number theory --- Numerical analysis

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The very classical orthogonal polynomials named after Hermite, Laguerre and Jacobi, satisfy many common properties. For instance, they satisfy a second-order differential equation with polynomial coefficients and they can be expressed in terms of a hypergeometric function. Replacing the differential equation by a second-order difference equation results in (discrete) orthogonal polynomial solutions with similar properties. Generalizations of these difference equations, in terms of Hahn's q-difference operator, lead to both continuous and discrete orthogonal polynomials with similar properties. For instance, they can be expressed in terms of (basic) hypergeometric functions. Based on Favard's theorem, the authors first classify all families of orthogonal polynomials satisfying a second-order differential or difference equation with polynomial coefficients. Together with the concept of duality this leads to the families of hypergeometric orthogonal polynomials belonging to the Askey scheme. For each family they list the most important properties and they indicate the (limit) relations. Furthermore the authors classify all q-orthogonal polynomials satisfying a second-order q-difference equation based on Hahn's q-operator. Together with the concept of duality this leads to the families of basic hypergeometric orthogonal polynomials which can be arranged in a q-analogue of the Askey scheme. Again, for each family they list the most important properties, the (limit) relations between the various families and the limit relations (for q --> 1) to the classical hypergeometric orthogonal polynomials belonging to the Askey scheme. These (basic) hypergeometric orthogonal polynomials have several applications in various areas of mathematics and (quantum) physics such as approximation theory, asymptotics, birth and death processes, probability and statistics, coding theory and combinatorics.

Orthogonal polynomials --- Orthogonalization methods --- Polynômes orthogonaux --- Orthogonalisation, Méthodes d' --- Orthogonal polynomials. --- Orthogonalization methods. --- Hypergeometrische orthogonale Polynome --- Hypergeometrische orthogonale Polynome. --- Polynômes orthogonaux --- Orthogonalisation, Méthodes d' --- EPUB-LIV-FT LIVMATHE LIVSTATI SPRINGER-B

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The very classical orthogonal polynomials named after Hermite, Laguerre and Jacobi, satisfy many common properties. For instance, they satisfy a second-order differential equation with polynomial coefficients and they can be expressed in terms of a hypergeometric function. Replacing the differential equation by a second-order difference equation results in (discrete) orthogonal polynomial solutions with similar properties. Generalizations of these difference equations, in terms of Hahn's q-difference operator, lead to both continuous and discrete orthogonal polynomials with similar properties. For instance, they can be expressed in terms of (basic) hypergeometric functions. Based on Favard's theorem, the authors first classify all families of orthogonal polynomials satisfying a second-order differential or difference equation with polynomial coefficients. Together with the concept of duality this leads to the families of hypergeometric orthogonal polynomials belonging to the Askey scheme. For each family they list the most important properties and they indicate the (limit) relations. Furthermore the authors classify all q-orthogonal polynomials satisfying a second-order q-difference equation based on Hahn's q-operator. Together with the concept of duality this leads to the families of basic hypergeometric orthogonal polynomials which can be arranged in a q-analogue of the Askey scheme. Again, for each family they list the most important properties, the (limit) relations between the various families and the limit relations (for q --> 1) to the classical hypergeometric orthogonal polynomials belonging to the Askey scheme. These (basic) hypergeometric orthogonal polynomials have several applications in various areas of mathematics and (quantum) physics such as approximation theory, asymptotics, birth and death processes, probability and statistics, coding theory and combinatorics.

Mathematical analysis --- Mathematics --- analyse (wiskunde) --- functies (wiskunde) --- wiskunde --- Orthogonal polynomials --- Orthogonalization methods --- Polynômes orthogonaux --- Orthogonalisation, Méthodes d' --- EPUB-LIV-FT LIVMATHE LIVSTATI SPRINGER-B

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Annotation

Wireless communication systems. --- Multiplexing. --- Orthogonalization methods. --- Functions, Orthogonal family of --- Functions, Orthogonal set of --- Orthogonal family of functions --- Orthogonal set of functions --- Orthogonal sets --- Orthogonal systems --- Orthogonalization (Numerical analysis) --- Sets, Orthogonal --- Systems, Orthogonal --- Algebras, Linear --- Numerical analysis --- Telecommunication --- Communication systems, Wireless --- Wireless data communication systems --- Wireless information networks --- Wireless telecommunication systems --- Telecommunication systems