Given a piecewise smooth function,it is possible to construct a global expansion in some complete orthogonal basis,such as the Fourier basis.However,the local discontinuities of the function will destroy the convergen...Given a piecewise smooth function,it is possible to construct a global expansion in some complete orthogonal basis,such as the Fourier basis.However,the local discontinuities of the function will destroy the convergence of global approximations,even in regions for which the underlying function is analytic.The global expansions are contaminated by the presence of a local discontinuity,and the result is that the partial sums are oscillatory and feature non-uniform convergence.This characteristic behavior is called the Gibbs phenomenon.However,David Gottlieb and Chi-Wang Shu showed that these slowly and non-uniformly convergent global approximations retain within them high order information which can be recovered with suitable postprocessing.In this paper we review the history of the Gibbs phenomenon and the story of its resolution.展开更多
This note derives the relationship between the Pearson product-moment coefficient of correlation and the Spearman rank-based coefficient of correlation for the bivariate normal distribution. This new derivation shows ...This note derives the relationship between the Pearson product-moment coefficient of correlation and the Spearman rank-based coefficient of correlation for the bivariate normal distribution. This new derivation shows the relationship between the two correlation coefficients through an infinite cosine series. A computationally efficient algorithm is also provided to estimate the relationship between the Pearson product-moment coefficient of correlation and the Spearman rank-based coefficient of correlation. The algorithm can be implemented with relative ease using current modern mathematical or statistical software programming languages e.g. R, SAS, Mathematica, Fortran, et al. The algorithm is also available from the author of this article.展开更多
In this paper the asymptotic behavior of Gibbs function for a class of M-band wavelet expansions is given. In particular, the Daubechies’ wavelets are included in this class.
文摘Given a piecewise smooth function,it is possible to construct a global expansion in some complete orthogonal basis,such as the Fourier basis.However,the local discontinuities of the function will destroy the convergence of global approximations,even in regions for which the underlying function is analytic.The global expansions are contaminated by the presence of a local discontinuity,and the result is that the partial sums are oscillatory and feature non-uniform convergence.This characteristic behavior is called the Gibbs phenomenon.However,David Gottlieb and Chi-Wang Shu showed that these slowly and non-uniformly convergent global approximations retain within them high order information which can be recovered with suitable postprocessing.In this paper we review the history of the Gibbs phenomenon and the story of its resolution.
文摘This note derives the relationship between the Pearson product-moment coefficient of correlation and the Spearman rank-based coefficient of correlation for the bivariate normal distribution. This new derivation shows the relationship between the two correlation coefficients through an infinite cosine series. A computationally efficient algorithm is also provided to estimate the relationship between the Pearson product-moment coefficient of correlation and the Spearman rank-based coefficient of correlation. The algorithm can be implemented with relative ease using current modern mathematical or statistical software programming languages e.g. R, SAS, Mathematica, Fortran, et al. The algorithm is also available from the author of this article.
基金The authors are partially supported by the Chinese National Natural Science Foundation(19571972)the Key Project Foundation(69735020)the Zhejiang Provincial Science Foundation of China(196083)
文摘In this paper the asymptotic behavior of Gibbs function for a class of M-band wavelet expansions is given. In particular, the Daubechies’ wavelets are included in this class.
