dual frame

In this paper a class of Gabor frames with time shift parameter $a>0$, frequency shift parameter $b>0$ and bounded compactly supported generator function $g$ such that $supp\ g\subseteq\left[\left(k+2\right)a-\frac{2}{b},ka+\frac{1}{b}\right]$ or $supp\ g\subseteq\left[\left(k+1\right)a-\frac{1}{b},ka+\frac{1}{b}\right]$, where $k$ is an integer number is introduced. In particular, a sufficient condition on a function $g\in C_c^+\left( \mathbb{R}\right) $ with $supp\ g\subseteq\left[\left(k+2\right)a-\frac{2}{b},ka+\frac{1}{b}\right]$ and positive decreasing derivative $g^\prime$ on $\left(ka-\frac{1}{b},\left( k+2\right)a \right)$, that make $\left\{E_{mb}T_{na}g\right\}_{m,n\in\mathbb{Z}}$ into a Gabor frame, is given.

We study the concept of frame in tensor product of  $n$-Hilbert spaces as tensor product of  $n$-Hilbert spaces is again an  $n$-Hilbert space. We generalize some of the known results about bases to frames in this new Hilbert space. A relationship between frame and bounded linear operator in tensor product of  $n$-Hilbert spaces is studied. Finally,;the dual frame in tensor product of  $n$-Hilbert spaces is discussed.

The theory of algebraic frames for a Hilbert space $H$ is a generalization of the theory of frames and generalized frames. The paper applies the theory of unbounded operators to define the dual of algebraic frames with densely defined unbounded analysis operators. It is shown that every algebraic frame has an algebraic dual frame, and if an algebraic frame has a nonzero redundancy, then it is not Riesz-type. An example of an algebraic frame with finite redundancy is constructed which is not a Riesz-type algebraic frame. Finally, for a lower bounded analytic frame, the discreteness of its indexing measure space and the uniqueness of its algebraic dual are studied and shown to be interrelated.

The duals of Gabor frames have an essential role in reconstruction of signals. In this paper we find a necessary and sufficient  condition for two Gabor systems $left(chi_{left[c_1,d_1right)},a,bright)$ and $left(chi_{left[c_2,d_2right)},a,bright)$ to form dual frames for $L_2left(mathbb{R}right)$, where $a$ and $b$ are positive numbers and $c_1,c_2,d_1$ and $d_2$ are real numbers such that $c_1<d_1$ and $c_2<d_2$.

We introduced the continuous and discrete $p$-adic shearlet systems. We restrict ourselves to a brief description of the $p$-adic theory and shearlets in real case. Using the group $G_p$ consist of all $p$-adic numbers that all of its elements have a square root, we defined the continuous $p$-adic shearlet system associated with $L^2left(Q_p^{2}right)$. The discrete $p$-adic shearlet frames for $L^2left(Q_p^{2}right)$ is discussed. Also we prove that the frame operator $S$ associated with the group $G_p$ of all with the shearlet frame $SHleft( psi; Lambdaright)$ is a Fourier multiplier with a function in terms of $widehat{psi}$. For a measurable subset $H subset Q_p^{2}$, we considered a subspace $L^2left(Hright)^{vee}$ of $L^2left(Q_p^{2}right)$. Finally we give a necessary condition for two functions in $L^2left(Q_p^{2}right)$ to generate a p-adic dual shearlet tight frame via admissibility.

Hilbert frames theory have been extended to frames in Hilbert C∗ -modules. The paper introduces equivalent ∗-frames and presents ordinary duals of a constructed ∗-frame by an adjointable and invertible operator. Also, some necessary and sufficient conditions are studied such that ∗-frames and ordinary duals or operator duals of another ∗-frames are equivalent under these conditions. We obtain a ∗-frame by an orthogonal projection and a given ∗-frame, characterize its duals, and give a bilateral condition for commutating frame operator of a primary ∗-frame and an orthogonal projection. At the end of paper, pre-frame operator of a dual frame is computed by pre-frame operator of a general ∗ -frame and an orthogonal projection.

Generalized frames are an extension of frames in Hilbert spaces and Hilbert C-modules. In this paper, the concept\Similar" for modular g-frames is introduced and all of operator duals (ordinary duals) of similar g-frames with respect to each other are characterized. Also, an operator dual of a given g-frame is studied where g-frame is constructed by a primary gframe and an orthogo- nal projection. Moreover, a g-frame is obtained by two the g-frames and its operator duals are investigated. Finally, the dilation of g- frames is studied.

In this paper, g-dual function-valued frames in L2(0;1) are in- troduced. We can achieve more reconstruction formulas to ob- tain signals in L2(0;1) by applying g-dual function-valued frames in L2(0;1).

Let G be a locally compact abelian group with a uniform lattice sub-
group. In this paper, we verify extension of shift-invariant systems in L2(G) to tight frames. We show that any shift-invariant Bessel sequence with an at most countable number of generators in L2(G) can be extended to a tight frame for its closed linear span by adding another shift-invariant system with at most the same number of generators in L2(G). Also, we yield an extension of the given Bessel sequence to a pair of dual frame sequences.