Approximation Properties of Vallée Poussin Means for Special Series of Ultraspherical Jacobi Polynomials

2021 ◽  
pp. 107-120
Author(s):  
M. G. Magomed-Kasumov
Keyword(s):  
Jacobi Polynomials ◽  
Approximation Properties ◽  
Special Series

Forthcoming Special Series

2008 ◽  
Vol 37 (2) ◽  
pp. 283-283
Keyword(s):  
Special Series

Modified algorithm for fast bandwidth selection for kernel estimates of multidimensional probability densities

2020 ◽  
pp. 9-13
Author(s):  
A. V. Lapko ◽  
V. A. Lapko
Keyword(s):  
Probability Density ◽  
Optimal Parameter ◽  
Random Variables ◽  
Kernel Functions ◽  
Independent Random Variables ◽  
Functional Dependencies ◽  
Approximation Properties ◽  
Probability Density Estimation ◽  
Multidimensional Probability ◽  
Selection Of

An original technique has been justified for the fast bandwidths selection of kernel functions in a nonparametric estimate of the multidimensional probability density of the Rosenblatt–Parzen type. The proposed method makes it possible to significantly increase the computational efficiency of the optimization procedure for kernel probability density estimates in the conditions of large-volume statistical data in comparison with traditional approaches. The basis of the proposed approach is the analysis of the optimal parameter formula for the bandwidths of a multidimensional kernel probability density estimate. Dependencies between the nonlinear functional on the probability density and its derivatives up to the second order inclusive of the antikurtosis coefficients of random variables are found. The bandwidths for each random variable are represented as the product of an undefined parameter and their mean square deviation. The influence of the error in restoring the established functional dependencies on the approximation properties of the kernel probability density estimation is determined. The obtained results are implemented as a method of synthesis and analysis of a fast bandwidths selection of the kernel estimation of the two-dimensional probability density of independent random variables. This method uses data on the quantitative characteristics of a family of lognormal distribution laws.

Approximation properties of repeated de la Vallée-Poussin means for piecewise smooth functions

2017 ◽  
Vol 60 (3) ◽  
pp. 695-713
Author(s):  
I. I. Sharapudinov ◽  
T. I. Sharapudinov ◽  
M. G. Magomed-Kasumov
Keyword(s):  
Piecewise Smooth ◽  
Approximation Properties ◽  
Smooth Functions ◽  
Piecewise Smooth Functions
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