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We study the open question on determination of jumps for functions raised by Shi and Hu in 2009. An affirmative answer is given for the case that spline-wavelet series are used to approximate the functions.

In this paper, we discuss the pointwise convergence of conjugate convolution operators with some applications to wavelets. Some criteria of convergence at (C, 1) continuous points, Lebesgue points and almost everywhere are established.

In this paper, several new results on the boundedness of parammetric Marcinkiewicz integrals on the weighted Hardy spaces and the weak weighted Hardy spaces are established.

In this paper, we will introduce multiresolution analysis with composite dilations and give a characterization of generator for multiresolution analysis with composite dilations.

Let ,2 an ≥ be two natural numbers. C. K. Chui and X. L. Shi proved that for any affine frame ( ) ( ) { } 2 ;, ,, jj b j k x a a x kb j k Z ψψ = − ∈ of ( ) 2 LR , and the family { } 12 , ,, jk n n j k Z ψ − ∈ is also a frame with the same bounds if n is relatively prime to a . In this paper we prove that n is relatively prime to a which is also necessary.

Let V be a star shaped region. In 2006, Colzani, Meaney and Prestini proved that if function f satisfies some condition, then the multiplier transform with the characteristic function of tV as the multiplier tends to f almost everywhere, when t goes to ∞. In this paper we use a Theorem established by K.K.Chen to show that if we change their multiplier, then the condition on f can be weakened.

In this paper, a new result on pointwise convergence of wavelets of generalized Shannon type is proved, which improves a theorem established by Zayed.

Suppose that η1, ..., ηn are measurable functions in L2(ℝ). We call the n-tuple (η1, ..., ηn) a Parseval super frame wavelet of length n if $$\left\{ {2^{\tfrac{k} {2}} \eta _1 \left( {2^k t - \ell } \right) \oplus \cdots \oplus 2^{\tfrac{k} {2}} \eta _n \left( {2^k t - \ell } \right):k,\ell \in \mathbb{Z}} \right\}$$ is a Parseval frame for L2(ℝ)⊕n. In high dimensional case, there exists a similar notion of Parseval super frame wavelet with some expansive dilation matrix. In this paper, we will study the Parseval super frame wavelets of length n, and will focus on the path-connectedness of the set of all s-elementary Parseval super frame wavelets in one-dimensional and high dimensional cases. We will prove the corresponding path-connectedness theorems.

The single 2 dilation wavelet multipliers in one-dimensional case and single A-dilation (where A is any expansive matrix with integer entries and |detA| = 2) wavelet multipliers in twodimensional case were completely characterized by Wutam Consortium (1998) and Li Z., et al. (2010). But there exist no results on multivariate wavelet multipliers corresponding to integer expansive dilation matrix with the absolute value of determinant not 2 in L 2(ℝ2). In this paper, we choose $2I_2 = \left( {\begin{array}{*{20}c} 2 & 0 \\ 0 & 2 \\ \end{array} } \right)$ as the dilation matrix and consider the 2I 2-dilation multivariate wavelet Φ = {ψ 1, ψ 2, ψ 3}(which is called a dyadic bivariate wavelet) multipliers. Here we call a measurable function family f = {f 1, f 2, f 3} a dyadic bivariate wavelet multiplier if $\Psi _1 = \left\{ {\mathcal{F}^{ - 1} \left( {f_1 \widehat{\psi _1 }} \right),\mathcal{F}^{ - 1} \left( {f_2 \widehat{\psi _2 }} \right),\mathcal{F}^{ - 1} \left( {f_3 \widehat{\psi _3 }} \right)} \right\}$ is a dyadic bivariate wavelet for any dyadic bivariate wavelet Φ = {ψ 1, ψ 2, ψ 3}, where $\hat f$ and F −1 denote the Fourier transform and the inverse transform of function f respectively. We study dyadic bivariate wavelet multipliers, and give some conditions for dyadic bivariate wavelet multipliers. We also give concrete forms of linear phases of dyadic MRA bivariate wavelets.

In this paper, the authors consider the behaviors of a class of parametric Marcinkiewicz integrals μ Ω , μ Ω, *, and μ Ω, on BMO(ℝ n ) and Campanato spaces with complex parameter ρ and the kernel Ω in Llog+ L(S n−1). Here μ Ω, *, and μ Ω, are parametric Marcinkiewicz functions corresponding to the Littlewood-Paley g λ *-function and the Lusin area function S, respectively. Under certain weak regularity condition on Ω, the authors prove that if f belongs to BMO(ℝ n ) or to a certain Campanato space, then [μ Ω, *, (f)]2, [μ Ω, (f)]2 and [μ Ω (f)]2 are either infinite everywhere or finite almost everywhere, and in the latter case, some kind of boundedness are also established.

Recently, Shi Xianliang and Hu Lan published the method of concentration factors for determination of jumps of functions via MCM conjugate wavelets. Usually, it is difficult to calculate the Hilbert transform of general window functions. The aim of this paper is to discuss determination of jumps for functions based on derivative Gabor series. The results will simplify the calculation of jump values.

In this paper we discuss determination of jumps for non-periodic function based on Malvar- Cofiman-Meyer (MCM) conjugate wavelets. We prove the equality of Lukacs type. Furthermore we establish several criteria on concentration factors for functions that satisfy weak-smoothness condition of Dini type.

The goal of this paper is to establish a set of necessary conditions for Gabor frames. These conditions are also sufficient for tight frames.

If ψ \psi generates an affine frame ψ j , k ( x ) = 2 j / 2 ψ ( 2 j x − k ) , j , k ∈ Z {\psi _{j,k}}(x) = {2^{j/2}}\psi ({2^j}x - k),j,k \in \mathbb {Z} , of L 2 ( R ) {L^2}(\mathbb {R}) , we prove that { n − 1 / 2 ψ j , k / n } \{ {n^{ - 1/2}}{\psi _{j,k/n}}\} is also an affine frame of L 2 ( R ) {L^2}(\mathbb {R}) with the same frame bounds for any positive odd integer n. This establishes the result stated as the title of this paper. A counterexample of this statement for n = 2 n = 2 is also given.

Some local Poincaré and Sobolev inequalities involving weights in product spaces are established.

We use the theory of operational research to describe a model to quantify the manufacturing process flexibility’s effect on supply chain risk. From the result we can see flexibility can mitigating risk and a little flexibility can achieve nearly all the benefits of total flexibility. Also by comparing the result we can find that the effects on different enterprises are the same.

The main goal of this paper is to establish a set of necessary conditions for affine frames. These conditions are also sufficient for tight frames.