International Journal of Wavelets, Multiresolution and Information Processing

The main purpose of this paper is to survey some of the work on spherical approximation done by the BNU group under the direction of Professor Sun.

The equiconvergent operators of Cesàro means, and their interesting applications are described. The Jackson inequality for spherical polynomials and some moduli of smoothness on the sphere are investigated. The equivalence between moduli of smoothness and K-functionals is also discussed. We also describe several weighted polynomial inequalities on the sphere, including the Remez-type and the Nikolskii-type inequalities, the Marcinkiewicz–Zygmund inequality, the Bernstein-type and the Schur-type inequalities. Positive cubature formulas on the sphere, and their relation to the Marcinkiewicz–Zygmund inequality are also discussed. A survey on recent results on asymptotic orders of the n-widths of Sobolev's classes on the sphere is also given.

Let q > 0, α ≥ 0, f ∈ C[0, 1], and be the q-Bernstein–Stancu polynomials. In the case α = 0, reduces to the well-known q-Bernstein polynomials introduced by Phillips in 1997. In this paper, we study the rate of convergence of the sequence . Both a theorem on convergence and a Voronovskaya-type theorem on the rate of convergence are proved.

This article considers regularized least square regression on the sphere. It develops a theoretical analysis of the generalization performances of regularized least square regression algorithm with spherical polynomial kernels. The explicit bounds are derived for the excess risk error. The learning rates depend on the eigenvalues of spherical polynomial integral operators and on the dimension of spherical polynomial spaces.

In this paper, we consider the relative n-widths of two kinds of periodic convolution classes, and , whose convolution kernels K and G are NCVD-kernel and B-kernel. The asymptotic estimations of and are obtained for p = 1 and ∞, 1 ≤ q ≤ ∞. We also defined a new concept of the average relative widths and obtained the exact results of the average relative widths of the classes of some smooth functions in L₂(R^d).

In this paper, we study two problems: one is of the worst case cubature error over the Sobolev classes on the sphere and another is of the approximation by hyperinterpolation operators on the sphere in the Sobolev space setting. We obtain lower estimates for the worst case cubature error over the Sobolev classes, which are optimal in the sense of order. We also obtain optimal estimates for the approximation by hyperinterpolation operators on the sphere in the Sobolev space setting.

We research the simultaneous approximation problem of the quasi-Hermite interpolation that based on the zeros of Chebyshev polynomials under weighted L_p-norm where our estimation is sharp.

The study of regularized learning algorithms is a very important issue and functional data analysis extends classical methods. We establish the learning rates of the least square regularized regression algorithm in reproducing kernel Hilbert space for functional data. With the iteration method, we obtain fast learning rate for functional data. Our result is a natural extension for least square regularized regression algorithm when the dimension of input data is finite.

This paper is devoted to exploring asymptotic behavior of the Szász–Mirakian-type operators including the Szász–Mirakian operators S_n(f;x) and its two kinds of integral modifications , . For these operators, we obtain the complete asymptotic expansions in the form of

In this paper, an estimate of Jacobi polynomials with complex indices is proved. It improves a corresponding result of Ref. 2.

Feature extraction is one of the most challenging problems in pattern recognition fields and has attracted great attention recently. In this paper, we propose a novel feature extraction algorithm named tensor locality sensitive discriminant analysis which accepts tensors as inputs. The algorithm preserves the key structure of data by using the labeled samples and has high performance as well as low time complexity. Experiments on the three standard databases show that the proposed method has better performance and achieves high accuracy.