ADVANCES IN COMPUTATIONAL MATHEMATICS, ISSN 1019-7168, 12/2019, Volume 45, Issue 5-6, pp. 2273 - 2286

Kolmogorov n-widths and Hankel singular values are two commonly used concepts in model reduction. Here, we show that for the special case of linear...

MATHEMATICS, APPLIED | REDUCED-BASIS APPROXIMATIONS | RATES | PARTIAL-DIFFERENTIAL-EQUATIONS | NONLINEAR-SYSTEMS | Kolmogorov n-width | Active subspaces | Model reduction | Hankel operator | Reduced basis method | MODEL ORDER REDUCTION | PRIORI CONVERGENCE THEORY | Hankel singular values

MATHEMATICS, APPLIED | REDUCED-BASIS APPROXIMATIONS | RATES | PARTIAL-DIFFERENTIAL-EQUATIONS | NONLINEAR-SYSTEMS | Kolmogorov n-width | Active subspaces | Model reduction | Hankel operator | Reduced basis method | MODEL ORDER REDUCTION | PRIORI CONVERGENCE THEORY | Hankel singular values

Journal Article

Foundations of Computational Mathematics, ISSN 1615-3375, 12/2013, Volume 13, Issue 6, pp. 965 - 1003

In this paper, we study linear trigonometric hyperbolic cross approximations, Kolmogorov n-widths d n (W,H γ ), and ε-dimensions n ε (W,H γ ) of periodic...

Economics general | Sobolev space | Linear and Multilinear Algebras, Matrix Theory | Mathematics | Kolmogorov n -widths | 41A25 | 41A63 | Numerical Analysis | Trigonometric hyperbolic cross space | 42A10 | High-dimensional approximation | Applications of Mathematics | Math Applications in Computer Science | Computer Science, general | Function classes with anisotropic smoothness | ε -dimensions | ε-dimensions | Kolmogorov n-widths | MATHEMATICS, APPLIED | GRIDS | SPACES | ELECTRONIC SCHRODINGER-EQUATION | SPARSE FINITE-ELEMENTS | epsilon-dimensions | INTERPOLATION | MATHEMATICS | ELLIPTIC PROBLEMS | VARIABLES | COMPUTER SCIENCE, THEORY & METHODS | BOUNDED MIXED DERIVATIVES

Economics general | Sobolev space | Linear and Multilinear Algebras, Matrix Theory | Mathematics | Kolmogorov n -widths | 41A25 | 41A63 | Numerical Analysis | Trigonometric hyperbolic cross space | 42A10 | High-dimensional approximation | Applications of Mathematics | Math Applications in Computer Science | Computer Science, general | Function classes with anisotropic smoothness | ε -dimensions | ε-dimensions | Kolmogorov n-widths | MATHEMATICS, APPLIED | GRIDS | SPACES | ELECTRONIC SCHRODINGER-EQUATION | SPARSE FINITE-ELEMENTS | epsilon-dimensions | INTERPOLATION | MATHEMATICS | ELLIPTIC PROBLEMS | VARIABLES | COMPUTER SCIENCE, THEORY & METHODS | BOUNDED MIXED DERIVATIVES

Journal Article

数学学报：英文版, ISSN 1439-8516, 2015, Volume 31, Issue 9, pp. 1475 - 1486

This paper considers the problem of n-widths of a Sobolev function class Ωr∞ determined by Pr（D） = DσПlj=1 （D2 - tj2I） in Orlicz spaces. The relationship...

线性算子 | Orlicz空间 | 泛函分析 | Sobolev函数类 | Kolmogorov宽度 | 极值问题 | 度理论 | 精确值 | 41A46 | 41A30 | Mathematics, general | Mathematics | Orlicz space | n -width | optimal linear operator | extremal subspace | n-width | MATHEMATICS | ELLIPTIC PROBLEMS | MATHEMATICS, APPLIED | APPROXIMATION | Studies | Mathematical analysis | Approximations | Functions (mathematics) | Functional analysis | Subspaces | Linear operators | Optimization | Extreme values

线性算子 | Orlicz空间 | 泛函分析 | Sobolev函数类 | Kolmogorov宽度 | 极值问题 | 度理论 | 精确值 | 41A46 | 41A30 | Mathematics, general | Mathematics | Orlicz space | n -width | optimal linear operator | extremal subspace | n-width | MATHEMATICS | ELLIPTIC PROBLEMS | MATHEMATICS, APPLIED | APPROXIMATION | Studies | Mathematical analysis | Approximations | Functions (mathematics) | Functional analysis | Subspaces | Linear operators | Optimization | Extreme values

Journal Article

Set-Valued and Variational Analysis, ISSN 1877-0533, 3/2016, Volume 24, Issue 1, pp. 83 - 99

The problem of computing the asymptotic order of the Kolmogorov n-width of the unit ball of the space of multivariate periodic functions induced by a...

Geometry | Kolmogorov n -widths | Non-degenerate differential operator | 41A63 | Analysis | Convex duality | 41A10 | Mathematics | Asymptotic order | 41A50 | Kolmogorov n-widths | MATHEMATICS, APPLIED | SPACES | HYPERBOLIC CROSSES | MIXED SMOOTHNESS

Geometry | Kolmogorov n -widths | Non-degenerate differential operator | 41A63 | Analysis | Convex duality | 41A10 | Mathematics | Asymptotic order | 41A50 | Kolmogorov n-widths | MATHEMATICS, APPLIED | SPACES | HYPERBOLIC CROSSES | MIXED SMOOTHNESS

Journal Article

Mathematics of Computation of the American Mathematical Society, ISSN 0025-5718, 01/2006, Volume 75, Issue 253, pp. 241 - 258

Let S_{\beta}:=\{z\in {\mathbb C}:|{\rm Im}z|<\beta\} be a strip in complex plane. \widetilde{H}_{\infty,\beta}^{r} denotes those 2\pi-periodic, real-valued...

Integers | Zero | Mathematical theorems | Approximation | Analytic functions | Rolles theorem | Information economics | Mathematical functions | Polynomials | Kolmogorov type comparison theorem | n-widths | Hardy-Sobolev class | N-widths | Hardy–Sobolev class | MATHEMATICS, APPLIED | PERIODIC-FUNCTIONS | OPTIMAL RECOVERY | OPERATORS | ANALYTIC-FUNCTIONS

Integers | Zero | Mathematical theorems | Approximation | Analytic functions | Rolles theorem | Information economics | Mathematical functions | Polynomials | Kolmogorov type comparison theorem | n-widths | Hardy-Sobolev class | N-widths | Hardy–Sobolev class | MATHEMATICS, APPLIED | PERIODIC-FUNCTIONS | OPTIMAL RECOVERY | OPERATORS | ANALYTIC-FUNCTIONS

Journal Article

Journal of Approximation Theory, ISSN 0021-9045, 2009, Volume 157, Issue 1, pp. 19 - 31

We consider relative widths characterizing the best approximation of a fixed set by its sections of given dimension. For Sobolev classes W p 1 on [ 0 , 1 ]...

Relative [formula omitted]-width of a function class | Approximation of Sobolev classes | Kolmogorov width | Width of a function class | Relative n-width of a function class | MATHEMATICS

Relative [formula omitted]-width of a function class | Approximation of Sobolev classes | Kolmogorov width | Width of a function class | Relative n-width of a function class | MATHEMATICS

Journal Article

Mathematical Notes, ISSN 0001-4346, 6/2011, Volume 89, Issue 5, pp. 645 - 651

We describe isometric embeddings of the Wiener spiral in complex Hilbert space and obtain the asymptotics of the Kolmogorov n-widths of specific embeddings. We...

complex Hilbert space | real Hilbert space | isometric embedding | Kolmogorov n-width | Mathematics, general | Mathematics | Wiener spiral | correlation function of a mapping | Hermitian kernel | MATHEMATICS

complex Hilbert space | real Hilbert space | isometric embedding | Kolmogorov n-width | Mathematics, general | Mathematics | Wiener spiral | correlation function of a mapping | Hermitian kernel | MATHEMATICS

Journal Article

Analysis in Theory and Applications, ISSN 1672-4070, 6/2007, Volume 23, Issue 2, pp. 180 - 187

In this paper, we give some optimal algorithms for diagonal operator T from space l p (1 ≦ p ≦ 2) to l 2 on n-widths in different computational setting.

41A46 | linear average n-width | Analysis | Kolmogorov n-width | 47A58 | Approximations and Expansions | linear stochastic n-width | Mathematics | linear n-width | 42A61 | Linear n-width | Linear stochastic n-width | Linear average n-width

41A46 | linear average n-width | Analysis | Kolmogorov n-width | 47A58 | Approximations and Expansions | linear stochastic n-width | Mathematics | linear n-width | 42A61 | Linear n-width | Linear stochastic n-width | Linear average n-width

Journal Article

Journal of Approximation Theory, ISSN 0021-9045, 2006, Volume 140, Issue 2, pp. 141 - 146

Consider the Hardy-type operator T : L p ( a , b ) → L p ( a , b ) ,- ∞ ⩽ a < b ⩽ ∞ , which is defined by ( Tf ) ( x ) = v ( x ) ∫ a x u ( t ) f ( t ) dt . It...

Weighted Hardy-type operators | Kolmogorov | Integral operators | Approximation | Weighted spaces | Gel’fand and Bernstein numbers | Gel'fand and Bernstein numbers | MATHEMATICS | weighted spaces | approximation | weighted hardy-type operators | integral operators | Geffand and Bernstein numbers

Weighted Hardy-type operators | Kolmogorov | Integral operators | Approximation | Weighted spaces | Gel’fand and Bernstein numbers | Gel'fand and Bernstein numbers | MATHEMATICS | weighted spaces | approximation | weighted hardy-type operators | integral operators | Geffand and Bernstein numbers

Journal Article

Computational Mathematics and Mathematical Physics, ISSN 0965-5425, 10/2017, Volume 57, Issue 10, pp. 1559 - 1576

Some problems in computational mathematics and mathematical physics lead to Fourier series expansions of functions (solutions) in terms of special functions,...

Computational Mathematics and Numerical Analysis | generalized modulus of continuity | Fourier–Jacobi sums | functions in two variables | estimates of best approximations | Mathematics | Kolmogorov N -width | Kolmogorov N-width | MATHEMATICS, APPLIED | Fourier-Jacobi sums | PHYSICS, MATHEMATICAL | Computer science | Functions (mathematics) | Bivariate analysis | Approximation method | Approximations | Computational mathematics | Fourier series | Physics | Sums

Computational Mathematics and Numerical Analysis | generalized modulus of continuity | Fourier–Jacobi sums | functions in two variables | estimates of best approximations | Mathematics | Kolmogorov N -width | Kolmogorov N-width | MATHEMATICS, APPLIED | Fourier-Jacobi sums | PHYSICS, MATHEMATICAL | Computer science | Functions (mathematics) | Bivariate analysis | Approximation method | Approximations | Computational mathematics | Fourier series | Physics | Sums

Journal Article

Annals of Functional Analysis, ISSN 2008-8752, 2015, Volume 6, Issue 4, pp. 114 - 133

Weakly singular Volterra integral equations of the different types are considered. The construction of accuracy-optimal numerical methods for one-dimensional...

Optimal approximation | Volterra integral equation | Collocation method | Babenko and kolmogorov n-widths | Weakly singular kernel | collocation method | MATHEMATICS | MATHEMATICS, APPLIED | weakly singular kernel | Babenko and Kolmogorov n-widths | optimal approximation

Optimal approximation | Volterra integral equation | Collocation method | Babenko and kolmogorov n-widths | Weakly singular kernel | collocation method | MATHEMATICS | MATHEMATICS, APPLIED | weakly singular kernel | Babenko and Kolmogorov n-widths | optimal approximation

Journal Article

Mathematical Notes, ISSN 0001-4346, 3/2002, Volume 71, Issue 3, pp. 477 - 485

For classes of $$2\pi $$ -periodic functions whose K-functionals are majorized by functions satisfying certain constraints, exact values of Kolmogorov,...

periodic Lebesgue p th power integrable function | K -functional | Mathematics, general | Kolmogorov n -width | Mathematics | 2\pi $$ -periodic measurable function | Bernstein n -width | Bernstein n-width | 2π-periodic measurable function | Periodic Lebesgue pth power integrable function | K-functional | Kolmogorov n-width | MATHEMATICS | periodic Lebesgue pth power integrable function | 2 pi-periodic measurable function

periodic Lebesgue p th power integrable function | K -functional | Mathematics, general | Kolmogorov n -width | Mathematics | 2\pi $$ -periodic measurable function | Bernstein n -width | Bernstein n-width | 2π-periodic measurable function | Periodic Lebesgue pth power integrable function | K-functional | Kolmogorov n-width | MATHEMATICS | periodic Lebesgue pth power integrable function | 2 pi-periodic measurable function

Journal Article

IEEE Transactions on Information Theory, ISSN 0018-9448, 05/1993, Volume 39, Issue 3, pp. 930 - 945

Approximation properties of a class of artificial neural networks are established. It is shown that feedforward networks with one layer of sigmoidal...

Fourier transforms | Neural networks | Linear approximation | Statistical distributions | Artificial neural networks | Approximation error | Feedforward neural networks | Feeds | Statistics | Information theory | REGRESSION | KOLMOGOROV N-WIDTHS | APPROXIMATION OF FUNCTIONS | FOURIER ANALYSIS | ARTIFICIAL NEURAL NETWORKS | MULTILAYER FEEDFORWARD NETWORKS | ENGINEERING, ELECTRICAL & ELECTRONIC | Numerical analysis | Research

Fourier transforms | Neural networks | Linear approximation | Statistical distributions | Artificial neural networks | Approximation error | Feedforward neural networks | Feeds | Statistics | Information theory | REGRESSION | KOLMOGOROV N-WIDTHS | APPROXIMATION OF FUNCTIONS | FOURIER ANALYSIS | ARTIFICIAL NEURAL NETWORKS | MULTILAYER FEEDFORWARD NETWORKS | ENGINEERING, ELECTRICAL & ELECTRONIC | Numerical analysis | Research

Journal Article

Complex Analysis and Operator Theory, ISSN 1661-8254, 05/2009, Volume 3, Issue 2, pp. 501 - 524

Let $$G \subset{\mathbb{C}}$$ be a bounded simply connected domain with boundary Γ and let $$E \subset G$$ be a regular compact set with connected complement....

Operator Theory | 30E10 | Analysis | extremal problems | Mathematics, general | Kolmogorov n -width | Mathematics | potential theory | 41A16 | Potential theory | Extremal problems | Kolmogorov n-width | MATHEMATICS | MATHEMATICS, APPLIED | ANALYTIC-FUNCTIONS | Universities and colleges

Operator Theory | 30E10 | Analysis | extremal problems | Mathematics, general | Kolmogorov n -width | Mathematics | potential theory | 41A16 | Potential theory | Extremal problems | Kolmogorov n-width | MATHEMATICS | MATHEMATICS, APPLIED | ANALYTIC-FUNCTIONS | Universities and colleges

Journal Article

Siberian Mathematical Journal, ISSN 0037-4466, 5/2005, Volume 46, Issue 3, pp. 535 - 539

We find an estimate for the nth minimal error of linear algorithms for some problem defined in a finite-dimensional space with values in an arbitrary normed...

Mathematics, general | Kolmogorov n -width | Mathematics | minimal error of linear algorithms | least norm polynomial | Minimal error of linear algorithms | Kolmogorov n-width | Least norm polynomial | MATHEMATICS

Mathematics, general | Kolmogorov n -width | Mathematics | minimal error of linear algorithms | least norm polynomial | Minimal error of linear algorithms | Kolmogorov n-width | Least norm polynomial | MATHEMATICS

Journal Article

Constructive Approximation, ISSN 0176-4276, 1/2008, Volume 27, Issue 1, pp. 99 - 120

Presenting a unified approach, we establish a Kolmogorov-type comparison theorem for classes of 2π-periodic functions defined by a special class of operators...

Numerical Analysis | Analysis | Mathematics | Comparison theorems of Kolmogorov type | Oscillation properties | n-widths | MATHEMATICS | PERIODIC-FUNCTIONS | APPROXIMATION | oscillation properties | n-Widths | VALUES | ANALYTIC-FUNCTIONS | comparison theorems of Kolmogorov type

Numerical Analysis | Analysis | Mathematics | Comparison theorems of Kolmogorov type | Oscillation properties | n-widths | MATHEMATICS | PERIODIC-FUNCTIONS | APPROXIMATION | oscillation properties | n-Widths | VALUES | ANALYTIC-FUNCTIONS | comparison theorems of Kolmogorov type

Journal Article

Mathematical Notes, ISSN 0001-4346, 7/2006, Volume 80, Issue 1, pp. 11 - 18

The sharp Jackson-type inequalities obtained by Taikov in the space L 2 and containing the best approximation and the modulus of continuity of first order are...

Gelfand n-widths | width of function classes | Kolmogorov | Mathematics, general | Mathematics | Bernstein | periodic function | Jackson-type inequalities | modulus of continuity of kth order | Modulus of continuity of kth order | Periodic function | Width of function classes | MATHEMATICS

Gelfand n-widths | width of function classes | Kolmogorov | Mathematics, general | Mathematics | Bernstein | periodic function | Jackson-type inequalities | modulus of continuity of kth order | Modulus of continuity of kth order | Periodic function | Width of function classes | MATHEMATICS

Journal Article

Mathematical Notes, ISSN 0001-4346, 11/2002, Volume 72, Issue 5, pp. 615 - 619

In the Hardy space H p,ρ (p≥1, 0<ρ≤ 1, H p,1≡ H p) we develop best linear approximation methods (previously studied by Taikov and Ainulloev) for the classes...

and informational n -width of classes of functions | linear | Kolmogorov | Mathematics, general | Mathematics | Gelfand | modulus of continuity | Hardy space | analytic functions on the unit disk | Analytic functions on the unit disk | Modulus of continuity | Informational n-width of classes of functions | Linear | MATHEMATICS | hardy space | and informational n-width of classes of functions

and informational n -width of classes of functions | linear | Kolmogorov | Mathematics, general | Mathematics | Gelfand | modulus of continuity | Hardy space | analytic functions on the unit disk | Analytic functions on the unit disk | Modulus of continuity | Informational n-width of classes of functions | Linear | MATHEMATICS | hardy space | and informational n-width of classes of functions

Journal Article

CHINESE ANNALS OF MATHEMATICS SERIES B, ISSN 0252-9599, 07/1991, Volume 12, Issue 3, pp. 272 - 281

Let B approximately p(r),1 = {f: g(r-1) is abs. cont. on I = [a, b], f is periodic with periodic with period H(= b - a), f(x1) = 0, parallel-to f(r)...

MATHEMATICS

MATHEMATICS

Journal Article

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Full Text
Grid approximation of singularly perturbed parabolic equations with moving boundary layers

Mathematical Modelling and Analysis, ISSN 1392-6292, 01/2008, Volume 13, Issue 3, pp. 421 - 442

A grid approximation of a boundary value problem is considered for a singularly perturbed parabolic reaction-diffusion equation in a domain with boundaries...

finite difference approximation | ϵ-uniform convergence | Kolmogorov's widths | boundary value problem | parabolic reaction-diffusion equation | perturbation parameter ϵ | moving boundary layer | Finite difference approximation | Boundary value problem | ε-uniform convergence | Moving boundary layer | Parabolic reaction-diffusion equation | Perturbation parameter ε | MATHEMATICS | N-WIDTHS | perturbation parameter epsilon | epsilon-uniform convergence | Kolmogorov’s widths | parabolic reaction– diffusion equation | perturbation parameter ε

finite difference approximation | ϵ-uniform convergence | Kolmogorov's widths | boundary value problem | parabolic reaction-diffusion equation | perturbation parameter ϵ | moving boundary layer | Finite difference approximation | Boundary value problem | ε-uniform convergence | Moving boundary layer | Parabolic reaction-diffusion equation | Perturbation parameter ε | MATHEMATICS | N-WIDTHS | perturbation parameter epsilon | epsilon-uniform convergence | Kolmogorov’s widths | parabolic reaction– diffusion equation | perturbation parameter ε

Journal Article

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