1,337,378 research outputs found
Approximation Theory for Matrices
We review the theory of optimal polynomial and rational Chebyshev
approximations, and Zolotarev's formula for the sign function over the range
(\epsilon \leq |z| \leq1). We explain how rational approximations can be
applied to large sparse matrices efficiently by making use of partial fraction
expansions and multi-shift Krylov space solvers.Comment: 10 pages, 7 figure
Approximation Theory XV: San Antonio 2016
These proceedings are based on papers presented at the international conference Approximation Theory XV, which was held May 22\u201325, 2016 in San Antonio, Texas. The conference was the fifteenth in a series of meetings in Approximation Theory held at various locations in the United States, and was attended by 146 participants. The book contains longer survey papers by some of the invited speakers covering topics such as compressive sensing, isogeometric analysis, and scaling limits of polynomials and entire functions of exponential type.
The book also includes papers on a variety of current topics in Approximation Theory drawn from areas such as advances in kernel approximation with applications, approximation theory and algebraic geometry, multivariate splines for applications, practical function approximation, approximation of PDEs, wavelets and framelets with applications, approximation theory in signal processing, compressive sensing, rational interpolation, spline approximation in isogeometric analysis, approximation of fractional differential equations, numerical integration formulas, and trigonometric polynomial approximation
Algorithms and error bounds for multivariate piecewise constant approximation
We review the surprisingly rich theory of approximation of functions of many vari- ables by piecewise constants. This covers for example the Sobolev-Poincar´e inequalities, parts of the theory of nonlinear approximation, Haar wavelets and tree approximation, as well as recent results about approximation orders achievable on anisotropic partitions
Spin-fluctuation theory beyond Gaussian approximation
A characteristic feature of the Gaussian approximation in the
functional-integral approach to the spin-fluctuation theory is the jump phase
transition to the paramagnetic state. We eliminate the jump and obtain a
continuous second-order phase transition by taking into account high-order
terms in the expansion of the free energy in powers of the fluctuating exchange
field. The third-order term of the free energy renormalizes the mean field, and
fourth-order term, responsible for the interaction of the fluctuations,
renormalizes the spin susceptibility. The extended theory is applied to the
calculation of magnetic properties of Fe-Ni Invar.Comment: 20 pages, 4 figure
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