arXiv subjects
Publications and source records attributed to Xingde Dai.
Let $ψ$ be a Parceval wavelet in $L^2 (\R)$ with the space of negative dilates $V(ψ)$. The intersection of the dilates $V(ψ)$ is the zero space. In other words, we have \begin{align*} \bigcap_{n\in\Z} D^n \overline{\textrm{span}}\{D^{\textrm{-}m} T^\ell ψ\mid m\geq 0, m,\ell\in\Z\}=\{0\}. \end{align*}
We introduce a new example of unital commutative $n$-dimensional group algebra $\mathbb{R}_n$ for $n \geq 2$. The algebra $\mathbb{R}_n$ and the complex numbers $\mathbb{C}$ are astonishingly alike. The zero divisor set of the algebra has Lebesgue $μ_n$-measure zero. The formula for the Haar measure is established. Also, the analytic function theory in $\mathbb{R}_n,$ for $n=2k$ that similar to the classical theory in $\mathbb{C}$ is introduced. This includes the Cauchy-Riemann equations, mean-value theorem and Louisville theorem.
Two scaling functions $φ_A$ and $φ_B$ for Parseval frame wavelets are algebraically isomorphic, $φ_A \simeq φ_B$, if they have matching solutions to their (reduced) isomorphic systems of equations. Let $A$ and $B$ be $d\times d$ and $s\times s$ \thematrix matrices with $d, s\geq 1$ respectively and let $φ_A$ be a scaling function associated with matrix $A$ and generated by a finite solution. There always exists a scaling function $φ_B$ associated with matrix $B$ such that \begin{equation*} φ_B \simeq φ_A. \end{equation*} An example shows that the assumption on the finiteness of the solutions can not be removed.
Let $d\geq 1$ be a natural number and $A_0$ be a $d\times d$ expansive integral matrix with determinant $\pm 2.$ Then $A_0$ is integrally similar to an integral matrix $A$ with certain additional properties. A finite solution to the system of equations associated with the matrix $A$ will result in an iterated sequence $\{Ψ^k χ_{[0,1)^d}\}$ that converges to a function $φ_A$ in $L^2(\R^d)$-norm. With this (scaling) function $φ_A,$ we will construct the Parseval's wavelet function $ψ$ with compact support associated with matrix $A_0.$
We establish system of equations for single function normalized tight frame wavelets with compact supports associated with $2\times 2$ expansive integral matrices in $L^2(\R^2)$.
Let $a$, $b$ be two fixed non-zero constants. A measurable set $E\subset \mathbb{R}$ is called a Weyl-Heisenberg frame set for $(a, b)$ if the function $g=χ_{E}$ generates a Weyl-Heisenberg frame for $L^2(\mathbb{R})$ under modulates by $b$ and translates by $a$, i.e., $\{e^{imbt}g(t-na\}_{m,n\in\mathbb{Z}}$ is a frame for $L^2(\mathbb{R})$. It is an open question on how to characterize all frame sets for a given pair $(a,b)$ in general. In the case that $a=2π$ and $b=1$, a result due to Casazza and Kalton shows that the condition that the set $F=\bigcup_{j=1}^{k}([0,2π)+2n_{j}π)$ (where $\{n_{1}<n_{2}<...<n_{k}\}$ are integers) is a Weyl-Heisenberg frame set for $(2π,1)$ is equivalent to the condition that the polynomial $f(z)=\sum_{j=1}^{k}z^{n_{j}}$ does not have any unit roots in the complex plane. In this paper, we show that this result can be generalized to a class of more general measurable sets (called basic support sets) and to set theoretical functions and continuous functions defined on such sets.
Let $a$, $b$ be two fixed positive constants. A function $g\in L^2({\mathbb R})$ is called a \textit{mother Weyl-Heisenberg frame wavelet} for $(a,b)$ if $g$ generates a frame for $L^2({\mathbb R})$ under modulates by $b$ and translates by $a$, i.e., $\{e^{imbt}g(t-na\}_{m,n\in\mathbb{Z}}$ is a frame for $L^2(\mathbb{R})$. In this paper, we establish a connection between mother Weyl-Heisenberg frame wavelets of certain special forms and certain strongly positive definite quadratic forms of infinite dimension. Some examples of application in matrix algebra are provided.