Some Research On Wavelet Approximations

This paper studies the approximations of the partial sum f_m of wavelet series, concludiong one-dimensional wavelet series and high-dimensional wavelet series, to f, and the exact estimation of the approximation rate. This paper is devided into three chapters.Chapter 1 studies the convergence and approximation rate of one-dimensional wavelet series. First, we give several common wavelets. Second, we establish two exact estimations of the approximation rate of the partial sum f_m to f when the scaling funciton satisfies(contants).At last, we give several estimations of the remainder of the wavelet expansion when the wavelet function satisfieswhere the series converges.Chapter 2 is devoted to the study of the convergence of high-dimensional wavelet series. By introducing the concept of Quasi-positive δ Sequence, we discuss its properties, construct a uniformly approximation sequence, and then obtain the pointwise convergence of this sequence. At last, we prove that when the scaling function satisfies(constants),the reproducing kernel sequence {q_m}_m∈Z is a Quasi-positive δ Sequence, and then get a theorem of uniform convergence. This theorem generalizes a theorem due to G. G. Walter's in paper 'Pointwise Convergence of Wavelet Expansions'.Chapter 3 discusses the convergence of Shannon wavelet series. We prove the convergence and give the estimations of the rate of approximation when the function f is of totally variation, or of bounded variation, or of ∧— bounded variation on some interval [a, b], or f is a monotonic-like function,...