The single 2 dilation orthogonal 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 high dimensional case were completely characterized by the Wutam Consortium (1998) and Z. Y. Li, et al. (2010). But there exist no more results on orthogonal multivariate wavelet matrix multipliers corresponding integer expansive dilation matrix with the absolute value of determinant not 2 in $L^2(\mathbb{R}^2)$. In this paper, we choose $$2I_2=\left(\begin{array}{cc}2 & 0\\0 & 2\end{array}\right)$$ as the dilation matrix and consider the $2I_2$-dilation orthogonal multivariate wavelet$\Psi=\{\psi_1,\psi_2,\psi_3\}$, (which is called a dyadic bivariate wavelet) multipliers. We call the $3\times 3$ matrix-valued function $A(s)=[f_{i,j}(s)]_{3\times 3}$, where $f_{i,j}$ are measurable functions, a dyadic bivariate matrix Fourier wavelet multiplier if the inverse Fourier transform of $A(s)(\widehat{\psi_{1}}(s),\widehat{\psi_{2}}(s),\widehat{\psi_{3}}(s))^{\top}=(\widehat{g_1}(s),\widehat{g_2}(s),\widehat{g_3}(s))^{\top}$ is a dyadic bivariate wavelet whenever $(\psi_{1},\psi_{2},\psi_{3})$ is any dyadic bivariate wavelet. We give some conditions for dyadic matrix bivariate wavelet multipliers. The results extended that of Z. Y. Li and X. L. Shi (2011). As an application, we construct some useful dyadic bivariate wavelets by using dyadic Fourier matrix wavelet multipliers and use them to image denoising.

Let $P(z)$ be a polynomial of degree $n$ and for any complex number $\alpha$, let $D_{\alpha}P(z)=nP(z)+(\alpha-z)P'(z)$ denote the polar derivative of the polynomial $P(z)$ with respect to $\alpha$. In this paper, we obtain inequalities for the polar derivative of a polynomial having all zeros inside a circle. Our results shall generalize and sharpen some well-known results of Turan, Govil, Dewan et al. and others.

For the commutators of multilinear Calderón-Zygmund singular integral operators with BMO functions, the weak type weighted norm inequalities with respect to $A_{\vec{p}}$ weights are obtained.

In this short note, we show the behavior in Orlicz spaces of best approximations by algebraic polynomials pairs on union of neighborhoods, when the measure of them tends to zero.

With the aid of Mullin-Rota's substitution rule, we show that the Sheffer-type differential operators together with the delta operators $\Delta$ and $D$ could be used to construct a pair of expansion formulas that imply a wide variety of summation formulas in the discrete analysis and combinatorics. A convergence theorem is established for a fruitful source formula that implies more than 20 noted classical formulas and identities as consequences. Numerous new formulas are also presented as illustrative examples. Finally, it is shown that a kind of lifting process can be used to produce certain chains of $(\infty^m)$ degree formulas for $m\geq 3$ with $m\equiv 1$ (mod 2) and $m\equiv 1$ (mod 3), respectively.

In this paper we introduce generalized cyclic $C$-contractions through $p$ number of subsets of a probabilistic metric space and establish two fixed point results for such contractions. In our first theorem we use the Hadzic type $t$-norm. In our next theorem we use Lukasiewicz $t$-norm. Our results generalize the results of Choudhury and Bhandari [11]. A control function [3] has been utilized in our second theorem. The results are illustrated with some examples.

In this paper, we show that new modified double cosine trigonometric sums introduced in [1] are inappropriate, the class of double sequences $J_d$ introduced there is unusable for such sums and consequently the results obtained in it are completely incorrect. We here introduce appropriate modified double cosine trigonometric sums making the class $J_d$ usable considering a particular double cosine trigonometric series.

In this paper some approximation formulae for a class of convolution type double singular integral operators depending on three parameters of the type $$( T_{\lambda }f) ( x,y)=\int_{a}^{b}\int_{a}^{b}f(t,s) K_{\lambda}(t-x,s-y) dsdt,  \quad   x,y\in (a,b), \quad \lambda \in \Lambda \subset[ 0,\infty ), $$ are given. Here $f$ belongs to the function space $L_{1}( \langle a,b\rangle ^{2}),$ where $\langle a,b\rangle $ is an arbitrary interval in $\mathbb{R}$. In this paper three theorems are proved, one for existence of the operator $( T_{\lambda }f)( x,y) $ and the others for its Fatou-type pointwise convergence to $f(x_{0},y_{0}),$ as $(x,y,\lambda )$ tends to $(x_{0},y_{0},\lambda_{0}).$ In contrast to previous works, the kernel functions $K_{\lambda}(u,v)$ don't have to be $2\pi$-periodic, positive, even and radial. Our results improve and extend some of the previous results of [1,6,8,10,11,13] in three dimensional frame and especially the very recent paper [15].