A new approach to estimating approximation parameters is developed. In this approach, the distance of the approximating function from a given finite set of points is estimated by a vector criterion the components of w...
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A new approach to estimating approximation parameters is developed. In this approach, the distance of the approximating function from a given finite set of points is estimated by a vector criterion the components of which are the absolute values of residuals at all points. Using this criterion, the remoteness preference relation is defined, and the nondominated function with respect to this relation is considered to be the best approximating function. approximation for several preference relations is studied, including the Pareto relation and the relation generated by the information about the equal importance of the criteria. Computational issues are considered and the relationship between the introduced approximating functions and the classical ones (obtained by the methods of least squares, least modulus, and the least maximum absolute value of deviation) are considered.
The paper investigates the approximate properties of biharmonic Poisson integrals for the upper half-plane on the classes of Hölder functions in the uniform metric. The exact values for the upper bounds of the de...
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We use the methods of Fourier-Jacobi harmonic analysis to study problems of the approximation of functions in weighted function spaces on the half-axis . We define function spaces of Nikol'skii-Besov type and desc...
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We use the methods of Fourier-Jacobi harmonic analysis to study problems of the approximation of functions in weighted function spaces on the half-axis . We define function spaces of Nikol'skii-Besov type and describe them in terms of best approximations. We use as the approximation tool a class of functions with the bounded spectrum, that is a class of functions for which their Fourier-Jacobi transform are functions with compact support.
We use the methods of Fourier-Jacobi harmonic analysis to study problems of the approximation of functions in weighted function spaces on the half-axis . We prove analogues of Jackson's direct theorem for the modu...
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We use the methods of Fourier-Jacobi harmonic analysis to study problems of the approximation of functions in weighted function spaces on the half-axis . We prove analogues of Jackson's direct theorem for the moduli of smoothness of all orders constructed on the basis of Jacobi generalized translations. As a tool for approximation, we use functions with bounded spectrum.
Some problems in the theory of approximation of complex-valued functions on locally compact Vilenkin groups in the metric of L-p, 1 <= p <= infinity, by functions with bounded spectrum, are investigated. A descr...
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Some problems in the theory of approximation of complex-valued functions on locally compact Vilenkin groups in the metric of L-p, 1 <= p <= infinity, by functions with bounded spectrum, are investigated. A description of certain function spaces in terms of the best approximations are obtained and some imbedding theorems are proved.
Some problems in the theory of approximation of complex-valued functions on the group Q(p) in the metric of L-p, 1 <= rho <= infinity by functions with bounded spectrum, are investigated. A description of certai...
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Some problems in the theory of approximation of complex-valued functions on the group Q(p) in the metric of L-p, 1 <= rho <= infinity by functions with bounded spectrum, are investigated. A description of certain function spaces in terms of the best approximations are obtained and some imbedding theorems are proved.
We consider some questions about the approximation of functions on the infinite-dimensional torus by trigonometric polynomials. Our main results are analogues of the direct and inverse theorems in the classical theory...
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We consider some questions about the approximation of functions on the infinite-dimensional torus by trigonometric polynomials. Our main results are analogues of the direct and inverse theorems in the classical theory of approximation of periodic functions and a description of the Lipschitz spaces on the infinite-dimensional torus in terms of the best approximation.
In this paper, we prove direct and inverse theorems on the approximation of functions by Fourier-Laplace sums in the spaces S ((p,q))(sigma (m-1)), m a parts per thousand yen 3, in terms of best approximations and mod...
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In this paper, we prove direct and inverse theorems on the approximation of functions by Fourier-Laplace sums in the spaces S ((p,q))(sigma (m-1)), m a parts per thousand yen 3, in terms of best approximations and moduli of continuity and consider the constructive characteristics of function classes defined by the moduli of continuity of their elements. The given statements generalize the results of the author's work carried out in 2007.
approximation of functions on the infinite-dimensional torus by trigonometric polynomials is treated. The main results of the paper provide analogs of the Jackson theorem about estimates of the best approximation in t...
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approximation of functions on the infinite-dimensional torus by trigonometric polynomials is treated. The main results of the paper provide analogs of the Jackson theorem about estimates of the best approximation in terms of the modulus of continuity of a function.
One-hidden-layer feedforward neural networks are described as functions having many real-valued parameters. approximation properties of neural networks are established (universal approximation property), and the appro...
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One-hidden-layer feedforward neural networks are described as functions having many real-valued parameters. approximation properties of neural networks are established (universal approximation property), and the approximation error is related to the number of parameters in the network. The essentially optimal order of approximation error bounds was already derived in 1996. We focused on the numerical experiment that indicates the neural networks whose parameters contain stochastic perturbations gain better performance than ordinary neural networks and explored the approximation property of neural networks with stochastic perturbations. In this paper, we derived the quantitative order of variance of stochastic perturbations to achieve the essentially optimal approximation order and verified the justifiability of our theory by numerical experiments.
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