The Approximation Of Multivariate Functions In Besov Classes From Noisy Information

Using finite function values to approximate the original function is one of the most basic and important problems in approximation theory. In this dissertation, we study the approximation of the function by noisy function values, i.e., noisy information, and consider two kinds of noisy information:one is that the function values is measured sampled values, i.e., the values of a linear functional and its integer translates acting on a undergoing function, and the other is a set of random samples that are controlled by an unknown probability. We give estimates of errors arising from using these two kinds of noise information to approximate functions in multivariate Besov class. The full text is divided into two parts.The first part is that, under the framework of Shannon sampling theory, we use two kinds of truncation to truncate Shannon sampling series with not exact sampled values to approximate functions from multivariate Besov class. The Shannon sampling theorem states that a bandlimited function can be exactly recovered from an infinite sequence of its samples if the bandlimit is no greater than half the sampling rate. Sev-eral types of errors such as aliasing error, truncated error, jitter error, and amplitude error appear when the Shannon sampling series is applied to approximate a function in real life. According to two different ways of the truncation, we will describe this part in two chapters. In chapter3, we truncate the Shannon sampling series with mea-sured sampled values by symmetric linear sampling, and obtain the uniform bounds of truncation errors for bandlimited functions with polynomial decay condition. We also derive the the uniform bounds of aliasing and truncation errors by intermediate approxiamtion method for not bandlimited functions with the same decay condition. The strong "bandlimited" condition is replaced by a weak "smooth" condition. We have two applications, the first one is that the measured sampled values are given by average of a function, and the second one is a general model applying linear functionals to cover aliasing error, truncated error, jitter error and amplitude error in one formula. In chapter4, we truncate the multidimensional Shannon sampling series via localized sampling but for functions without any decay condition, and obtain the uniform bounds of aliasing and truncation errors by intermediate approximation method for functions from anisotropic Besov class. The bounds are optimal up to a logarithmic factor. We also derive the corresponding results for the case that the multidimensional Shannon sampling series with the measured sampled values. We also apply the theorem to the examples as that in chapter3.The second part is to learn Besov smooth regression functions from random sam-ples under the framework of learning theory. Learning theory has various forms such as the theory of Neural Networks, Statistical Learning Theory and PAC(Probably Approx-imation Correct). Common characteristics of these forms are:the regression functions is unknown; samples are not given by function exactly, but with noise or some other uncertainties; the goal is to "learn" an unknown function from random samples. In chapter5, We investigate approximation errors with Lp norm,1≤p≤∞, which close the gap between the L2norm and the uniform norm, and obtain the decay order of approximation errors for inverse multiquadrics kernel and Gaussian kernel when the regression function is from Besov space. We also derive the corresponding result for the varying Gaussian kernel. The decay order of the approximation error for inverse multiquadrics kernel is improved.