Wavelet and Linear Algebra

Abstract We regard the shearlet group as a semidirect product group and show that its standard representation is,typically, a quasiregu- lar representation. As a result we can characterize irreducible as well as square-integrable subrepresentations of the shearlet group.

Abstract Finite affine groups are given by groups of translations and di- lations on finite cyclic groups. For cyclic groups of prime order we develop a time-scale (wavelet) analysis and show that for a large class of non-zero window signals/vectors, the generated full cyclic wavelet system constitutes a frame whose canonical dual is a cyclic wavelet frame.

Abstract Consider an n × n matrix polynomial P(λ). A spectral norm distance from P(λ) to the set of n × n matrix polynomials that have a given scalar µ ∈ C as a multiple eigenvalue was introduced and obtained by Papathanasiou and Psarrakos. They computed lower and upper bounds for this distance, constructing an associated perturbation of P(λ). In this paper, we extend this result to the case of two given distinct complex numbers µ1 and µ2. First, we compute a lower bound for the spectral norm distance from P(λ) to the set of matrix polynomials that have µ1, µ2 as two eigenvalues. Then we construct an associated perturbation of P(λ) such that the perturbed matrix polynomial has two given scalars µ1 and µ2 in its spectrum. Finally, we derive an upper bound for the distance by the constructed perturbation of P(λ). Numerical examples are provided to illustrate the validity of the method.

Abstract 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).

Abstract For A ∈ M n, the Schur multiplier of A is defined as S A(X) = A ◦ X for all X ∈ M n and the spectral norm of S A can be state as ∥S A∥ = supX,0 ∥A ∥X ◦X ∥ ∥. The other norm on S A can be defined as ∥S A∥ω = supX,0 ω(ω S( AX (X ) )) = supX,0 ωω (A (X ◦X ) ), where ω(A) stands for the numerical radius of A. In this paper, we focus on the relation between the norm of Schur multiplier of product of matrices and the product of norm of those matrices. This relation is proved for Schur product and geometric product and some applications are given. Also we show that there is no such relation for operator product of matrices. Furthermore, for positive definite matrices A and B with ∥S A∥ω ⩽ 1 and ∥S B∥ω ⩽ 1, we show that A♯B = n(I − Z)1/2C(I + Z)1/2, for some contraction C and Hermitian contraction Z.

Abstract In this paper, we establish some new results in ultra Bessel sequences and ultra Bessel sequences of subspaces. Also, we investigate ultra Bessel sequences in direct sums of Hilbert spaces. Specially, we show that {( fi, gi)}∞ i=1 is a an ultra Bessel sequence for Hilbert space H ⊕ K if and only if { fi}∞ i=1 and {gi}∞ i=1 are ultra Bessel sequences for Hilbert spaces H and K, respectively.

Abstract In this paper, we study ε-generalized weak subdifferential for vector valued functions defined on a real ordered topological vector space X. We give various characterizations of ε-generalized weak subdifferential for this class of functions. It is well known that if the function f : X → R is subdifferentiable at x0 ∈ X, then f has a global minimizer at x0 if and only if 0 ∈ ∂ f(x0). We show that a similar result can be obtained for ε-generalized weak subdifferential. Finally, we investigate some relations between ε-directional derivative and ε-generalized weak subdifferential. In fact, in the classical subdifferential theory, it is well known that if the function f : X → R is subdifferentiable at x0 ∈ X and it has directional derivative at x0 in the direction u ∈ X, then the relation f ′(x0, u) ≥ ⟨u, x∗⟩, ∀ x∗ ∈ ∂ f(x0) is satisfied. We prove that a similar result can be obtained for ε- generalized weak subdifferential.