Let K⊂ R^d be a set with positive and finite Lebesgue measure. Let Λ=M( Z^2d) be a lattice in R^2... more Let K⊂ R^d be a set with positive and finite Lebesgue measure. Let Λ=M( Z^2d) be a lattice in R^2d with density dens(Λ)=1. It is well-known that if M is a diagonal block matrix with diagonal matrices A and B, then G(|K|^-1/2χ_K, Λ) is an orthonormal basis for L^2( R^d) if and only if K tiles both by A( Z^d) and B^-t( Z^d). However, there has not been any intensive study when M is not a diagonal matrix. We investigate this problem for a large class of important cases of M. In particular, if M is any lower block triangular matrix with diagonal matrices A and B, we prove that if G(|K|^-1/2χ_K, Λ) is an orthonormal basis, then K can be written as a finite union of fundamental domains of A( Z^d) and at the same time, as a finite union of fundamental domains of B^-t( Z^d). If A^tB is an integer matrix, then there is only one common fundamental domain, which means K tiles by a lattice and is spectral. However, surprisingly, we will also illustrate by an example that a union of more than on...
Given a domain Ω⊂ R^d with positive and finite Lebesgue measure and a discrete set Λ⊂ R^d, we say... more Given a domain Ω⊂ R^d with positive and finite Lebesgue measure and a discrete set Λ⊂ R^d, we say that (Ω, Λ) is a frame spectral pair if the set of exponential functions ℰ(Λ):={e^2π i λ· x: λ∈Λ} is a frame for L^2(Ω). Special cases of frames include Riesz bases and orthogonal bases.In the finite setting Z_N^d, d, N≥ 1, a frame spectral pair can be defined in a similar way. We show how to construct and obtain a new frame spectral pair in R^d by "adding" frame spectral pairs in R^d and Z_N^d. Our construction unifies the well-known examples of exponential frames for the union of cubes with equal volumes. In this paper, we will also obtain a connection between frame spectral pairs and the Whittaker-Shannon interpolation formula when the frame is an orthogonal basis.
We establish wavelet characterizations of homogeneous Besov spaces on stratified Lie groups, both... more We establish wavelet characterizations of homogeneous Besov spaces on stratified Lie groups, both in terms of continuous and discrete wavelet systems. We first introduce a notion of homogeneous Besov spaceḂ s p,q in terms of a Littlewood-Paley-type decomposition, in analogy to the well-known characterization of the Euclidean case. Such decompositions can be defined via the spectral measure of a suitably chosen sub-Laplacian. We prove that the scale of Besov spaces is independent of the precise choice of Littlewood-Paley decomposition. In particular, different sub-Laplacians yield the same Besov spaces. We then turn to wavelet characterizations, first via continuous wavelet transforms (which can be viewed as continuous-scale Littlewood-Paley decompositions), then via discretely indexed systems. We prove the existence of wavelet frames and associated atomic decomposition formulas for all homogeneous Besov spacesḂ s p,q , with 1 ≤ p, q < ∞ and s ∈ R.
Function spaces are central topic in analysis. Often those spaces and related analysis involves s... more Function spaces are central topic in analysis. Often those spaces and related analysis involves symmetries in form of an action of a Lie group. Coorbit theory as introduced by Feichtinger and Gröchenig and then later extended in gives a unified method to construct Banach spaces of functions based on representations of Lie groups. In this article we identify the homogeneous Besov spaces on stratified Lie groups introduced in [13] as coorbit spaces in the sense of and use this to derive atomic decompositions for the Besov spaces.
We characterize the Besov spaces associated to the Gelfand pairs on the Heisenberg group. The cha... more We characterize the Besov spaces associated to the Gelfand pairs on the Heisenberg group. The characterization is given in terms of bandlimited wavelet coefficients where the bandlimitedness is introduced using spherical Fourier transform. To obtain these results we develop an approach to the characterization of Besov spaces in abstract Hilbert spaces through compactly supported admissible functions.
Let M be a smooth compact oriented Riemannian manifold, and let Δ_ M be the Laplace-Beltrami oper... more Let M be a smooth compact oriented Riemannian manifold, and let Δ_ M be the Laplace-Beltrami operator on M. Say 0 ≠ f ∈S(^+), and that f(0) = 0. For t > 0, let K_t(x,y) denote the kernel of f(t^2 Δ_ M). We show that K_t is well-localized near the diagonal, in the sense that it satisfies estimates akin to those satisfied by the kernel of the convolution operator f(t^2Δ) on ^n. We define continuous S-wavelets on M, in such a manner that K_t(x,y) satisfies this definition, because of its localization near the diagonal. Continuous S-wavelets on M are analogous to continuous wavelets on ^n in S(^n). In particular, we are able to characterize the Hölder continuous functions on M by the size of their continuous S-wavelet transforms, for Hölder exponents strictly between 0 and 1. If M is the torus ^2 or the sphere S^2, and f(s)=se^-s (the "Mexican hat" situation), we obtain two explicit approximate formulas for K_t, one to be used when t is large, and one to be used when t is s...
In this article wavelets (admissible vectors) on the Heisenberg group H are studied from the poin... more In this article wavelets (admissible vectors) on the Heisenberg group H are studied from the point of view of Calderón’s formula. We shall define Calderón admissible vectors in Definition 2.1. Further in Theorem 2.2 we show that for the class of Schwartz functions the Calderón admissibility condition is equivalent to the usual admissibility property which will be introduced in this work. Furthermore motivated by a well-known example on the real line, the Mexican-Hat wavelet, we demonstrate the existence and construction of an analogous wavelet on the Heisenberg Lie group with 2 vanishing moments, which together with all of its derivatives has “Gaussian” decay. The precise proof can be found in Theorem 3.2. Keywords.
We recall Mexican needlets from [5]. We derive an estimate for certain types of Legendre series, ... more We recall Mexican needlets from [5]. We derive an estimate for certain types of Legendre series, which we apply to the statistical properties of Mexican needlets. More precisely, we shall show that, under isotropy and Gaussianity assumptions, the Mexican needlet coefficients of a random field on the sphere are asymptotically uncorrelated, as the frequency parameter goes to infinity. This property is important in the analysis of cosmic microwave background radiation. Keywords. Spherical harmonics, spherical Laplacian operator, Mexican needlets,
We show that one can characterize the Besov spaces on a smooth compact oriented Riemannian manifo... more We show that one can characterize the Besov spaces on a smooth compact oriented Riemannian manifold, for the full range of indices, through a knowledge of the size of frame coefficients, using the smooth, nearly tight frames we have constructed in [8].
In this work we introduce a class of discrete groups containing subgroups of abstract translation... more In this work we introduce a class of discrete groups containing subgroups of abstract translations and dilations, respectively. A variety of wavelet systems can appear as π(Γ)ψ, where π is a unitary representation of a wavelet group and Γ is the abstract pseudo-lattice Γ. We prove a condition in order that a Parseval frame π(Γ)ψ can be dilated to an orthonormal basis of the form τ (Γ)Ψ where τ is a super-representation of π. For a subclass of groups that includes the case where the translation subgroup is Heisenberg, we show that this condition always holds, and we cite familiar examples as applications.
Let M be a smooth compact oriented Riemannian manifold, and let ∆ be the Laplace-Beltrami operato... more Let M be a smooth compact oriented Riemannian manifold, and let ∆ be the Laplace-Beltrami operator on M. Say 0 ̸ = f ∈ S(R +), and that f(0) = 0. For t> 0, let Kt(x, y) denote the kernel of f(t 2 ∆). Suppose f satisfies Daubechies ’ criterion, and b> 0. For each j, write M as a disjoint union of measurable sets Ej,k with diameter at most ba j, and comparable to ba j if ba j is sufficiently small. Take xj,k ∈ Ej,k. We then show that the functions φj,k(x) = [µ(Ej,k)] 1/2 Kaj (xj,k, x) form a frame for (I − P)L 2 (M), for b sufficiently small (here P is the projection onto the constant functions). Moreover, we show that the ratio of the frame bounds approaches 1 nearly quadratically as the dilation parameter approaches 1, so that the frame quickly becomes nearly tight (for b sufficiently small). Moreover, based upon how well-localized a function F ∈ L 2 is in space and in frequency, we can describe which terms in the summation F ∼ SF = P j P k 〈F, φj,k〉φj,k are so small that they...
Given a simply connected nilpotent Lie group having unitary irreducible representations that are ... more Given a simply connected nilpotent Lie group having unitary irreducible representations that are square-integrable modulo the center (SI/Z), we use operator-valued periodization to give a range-function type characterization of shift-invariant spaces of function on the group. We then give characterizations of frame and Riesz families for shift-invariant spaces. keywords: Nilpotent Lie groups; square-integrable; range function; periodization operator; shift-invariant subspaces; frame and Reisz bases;
Given a simply connected nilpotent Lie group having unitary irreducible representations that are ... more Given a simply connected nilpotent Lie group having unitary irreducible representations that are square-integrable modulo the center (SI/Z), we develop a notion of periodization on the group Fourier transform side, and use this notion to give a characterization of shift-invariant spaces in $L^2(N)$ in terms of range functions. We apply these results to study the structure of frame and Reisz families for shift-invariant spaces. We illustrate these results for the Heisenberg group as well as for other groups with SI/Z representations.
Let $g(x)=\chi_B(x)$ be the indicator function of a bounded convex set in $\Bbb R^d$, $d\geq 2$, ... more Let $g(x)=\chi_B(x)$ be the indicator function of a bounded convex set in $\Bbb R^d$, $d\geq 2$, with a smooth boundary and everywhere non-vanishing Gaussian curvature. Using a combinatorial appraoch we prove that if $d \neq 1 \mod 4$, then there does not exist $S \subset {\Bbb R}^{2d}$ such that ${ \{g(x-a)e^{2 \pi i x \cdot b} \}}_{(a,b) \in S}$ is an orthonormal basis for $L^2({\Bbb R}^d)$.
We show that the spin wavelets on the sphere S 2 , which were constructed by the first author and... more We show that the spin wavelets on the sphere S 2 , which were constructed by the first author and Marinucci in [3], can be chosen so as to form a nearly tight frame. These spin wavelets can be applied (see [1] and [2]) to the study of the polarization of cosmic microwave background radiation. For certain of these frames, there is a positive C such that each frame element at scale a j is supported in a geodesic ball of radius Ca j .
In this paper we study how zeros of the Fourier transform of a function $f: \mathbb{Z}_p^d \to \m... more In this paper we study how zeros of the Fourier transform of a function $f: \mathbb{Z}_p^d \to \mathbb{C}$ are related to the structure of the function itself. In particular, we introduce a notion of bandwidth of such functions and discuss its connection with the decomposition of this function into wavelets. Connections of these concepts with the tomography principle and the Nyquist-Shannon sampling theorem are explored. We examine a variety of cases such as when the Fourier transform of the characteristic function of a set $E$ vanishes on specific sets of points, affine subspaces, and algebraic curves. In each of these cases, we prove properties such as equidistribution of $E$ across various surfaces and bounds on the size of $E$. We also establish a finite field Heisenberg uncertainty principle for sets that relates their bandwidth dimension and spatial dimension.
We study Gabor orthonormal windows in $L^2({\Bbb Z}_p^d)$ for translation and modulation sets $A$... more We study Gabor orthonormal windows in $L^2({\Bbb Z}_p^d)$ for translation and modulation sets $A$ and $B$, respectively, where $p$ is prime and $d\geq 2$. We prove that for a set $E\subset \Bbb Z_p^d$, the indicator function $1_E$ is a Gabor window if and only if $E$ tiles and is spectral. Moreover, we prove that for any function $g:\Bbb Z_p^d\to \Bbb C$ with support $E$, if the size of $E$ coincides with the size of the modulation set $B$ or if $g$ is positive, then $g$ is a unimodular function, i.e., $|g|=c1_E$, for some constant $c>0$, and $E$ tiles and is spectral. We also prove the existence of a Gabor window $g$ with full support where neither $|g|$ nor $|\hat g|$ is an indicator function and $|B|<<p^d$. We conclude the paper with an example and open questions.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this p... more The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
In this paper, we introduce a class of nonsmooth nonconvex optimization problems, and we propose ... more In this paper, we introduce a class of nonsmooth nonconvex optimization problems, and we propose to use a local iterative minimization-majorization (MM) algorithm to find an optimal solution for the optimization problem. The cost functions in our optimization problems are an extension of convex functions with MC separable penalty, which were previously introduced by Ivan Selesnick. These functions are not convex; therefore, convex optimization methods cannot be applied here to prove the existence of optimal minimum point for these functions. For our purpose, we use convex analysis tools to first construct a class of convex majorizers, which approximate the value of non-convex cost function locally, then use the MM algorithm to prove the existence of local minimum. The convergence of the algorithm is guaranteed when the iterative points $x^{(k)}$ are obtained in a ball centred at $x^{(k-1)}$ with small radius. We prove that the algorithm converges to a stationary point (local minimum...
Let K⊂ R^d be a set with positive and finite Lebesgue measure. Let Λ=M( Z^2d) be a lattice in R^2... more Let K⊂ R^d be a set with positive and finite Lebesgue measure. Let Λ=M( Z^2d) be a lattice in R^2d with density dens(Λ)=1. It is well-known that if M is a diagonal block matrix with diagonal matrices A and B, then G(|K|^-1/2χ_K, Λ) is an orthonormal basis for L^2( R^d) if and only if K tiles both by A( Z^d) and B^-t( Z^d). However, there has not been any intensive study when M is not a diagonal matrix. We investigate this problem for a large class of important cases of M. In particular, if M is any lower block triangular matrix with diagonal matrices A and B, we prove that if G(|K|^-1/2χ_K, Λ) is an orthonormal basis, then K can be written as a finite union of fundamental domains of A( Z^d) and at the same time, as a finite union of fundamental domains of B^-t( Z^d). If A^tB is an integer matrix, then there is only one common fundamental domain, which means K tiles by a lattice and is spectral. However, surprisingly, we will also illustrate by an example that a union of more than on...
Given a domain Ω⊂ R^d with positive and finite Lebesgue measure and a discrete set Λ⊂ R^d, we say... more Given a domain Ω⊂ R^d with positive and finite Lebesgue measure and a discrete set Λ⊂ R^d, we say that (Ω, Λ) is a frame spectral pair if the set of exponential functions ℰ(Λ):={e^2π i λ· x: λ∈Λ} is a frame for L^2(Ω). Special cases of frames include Riesz bases and orthogonal bases.In the finite setting Z_N^d, d, N≥ 1, a frame spectral pair can be defined in a similar way. We show how to construct and obtain a new frame spectral pair in R^d by "adding" frame spectral pairs in R^d and Z_N^d. Our construction unifies the well-known examples of exponential frames for the union of cubes with equal volumes. In this paper, we will also obtain a connection between frame spectral pairs and the Whittaker-Shannon interpolation formula when the frame is an orthogonal basis.
We establish wavelet characterizations of homogeneous Besov spaces on stratified Lie groups, both... more We establish wavelet characterizations of homogeneous Besov spaces on stratified Lie groups, both in terms of continuous and discrete wavelet systems. We first introduce a notion of homogeneous Besov spaceḂ s p,q in terms of a Littlewood-Paley-type decomposition, in analogy to the well-known characterization of the Euclidean case. Such decompositions can be defined via the spectral measure of a suitably chosen sub-Laplacian. We prove that the scale of Besov spaces is independent of the precise choice of Littlewood-Paley decomposition. In particular, different sub-Laplacians yield the same Besov spaces. We then turn to wavelet characterizations, first via continuous wavelet transforms (which can be viewed as continuous-scale Littlewood-Paley decompositions), then via discretely indexed systems. We prove the existence of wavelet frames and associated atomic decomposition formulas for all homogeneous Besov spacesḂ s p,q , with 1 ≤ p, q < ∞ and s ∈ R.
Function spaces are central topic in analysis. Often those spaces and related analysis involves s... more Function spaces are central topic in analysis. Often those spaces and related analysis involves symmetries in form of an action of a Lie group. Coorbit theory as introduced by Feichtinger and Gröchenig and then later extended in gives a unified method to construct Banach spaces of functions based on representations of Lie groups. In this article we identify the homogeneous Besov spaces on stratified Lie groups introduced in [13] as coorbit spaces in the sense of and use this to derive atomic decompositions for the Besov spaces.
We characterize the Besov spaces associated to the Gelfand pairs on the Heisenberg group. The cha... more We characterize the Besov spaces associated to the Gelfand pairs on the Heisenberg group. The characterization is given in terms of bandlimited wavelet coefficients where the bandlimitedness is introduced using spherical Fourier transform. To obtain these results we develop an approach to the characterization of Besov spaces in abstract Hilbert spaces through compactly supported admissible functions.
Let M be a smooth compact oriented Riemannian manifold, and let Δ_ M be the Laplace-Beltrami oper... more Let M be a smooth compact oriented Riemannian manifold, and let Δ_ M be the Laplace-Beltrami operator on M. Say 0 ≠ f ∈S(^+), and that f(0) = 0. For t > 0, let K_t(x,y) denote the kernel of f(t^2 Δ_ M). We show that K_t is well-localized near the diagonal, in the sense that it satisfies estimates akin to those satisfied by the kernel of the convolution operator f(t^2Δ) on ^n. We define continuous S-wavelets on M, in such a manner that K_t(x,y) satisfies this definition, because of its localization near the diagonal. Continuous S-wavelets on M are analogous to continuous wavelets on ^n in S(^n). In particular, we are able to characterize the Hölder continuous functions on M by the size of their continuous S-wavelet transforms, for Hölder exponents strictly between 0 and 1. If M is the torus ^2 or the sphere S^2, and f(s)=se^-s (the "Mexican hat" situation), we obtain two explicit approximate formulas for K_t, one to be used when t is large, and one to be used when t is s...
In this article wavelets (admissible vectors) on the Heisenberg group H are studied from the poin... more In this article wavelets (admissible vectors) on the Heisenberg group H are studied from the point of view of Calderón’s formula. We shall define Calderón admissible vectors in Definition 2.1. Further in Theorem 2.2 we show that for the class of Schwartz functions the Calderón admissibility condition is equivalent to the usual admissibility property which will be introduced in this work. Furthermore motivated by a well-known example on the real line, the Mexican-Hat wavelet, we demonstrate the existence and construction of an analogous wavelet on the Heisenberg Lie group with 2 vanishing moments, which together with all of its derivatives has “Gaussian” decay. The precise proof can be found in Theorem 3.2. Keywords.
We recall Mexican needlets from [5]. We derive an estimate for certain types of Legendre series, ... more We recall Mexican needlets from [5]. We derive an estimate for certain types of Legendre series, which we apply to the statistical properties of Mexican needlets. More precisely, we shall show that, under isotropy and Gaussianity assumptions, the Mexican needlet coefficients of a random field on the sphere are asymptotically uncorrelated, as the frequency parameter goes to infinity. This property is important in the analysis of cosmic microwave background radiation. Keywords. Spherical harmonics, spherical Laplacian operator, Mexican needlets,
We show that one can characterize the Besov spaces on a smooth compact oriented Riemannian manifo... more We show that one can characterize the Besov spaces on a smooth compact oriented Riemannian manifold, for the full range of indices, through a knowledge of the size of frame coefficients, using the smooth, nearly tight frames we have constructed in [8].
In this work we introduce a class of discrete groups containing subgroups of abstract translation... more In this work we introduce a class of discrete groups containing subgroups of abstract translations and dilations, respectively. A variety of wavelet systems can appear as π(Γ)ψ, where π is a unitary representation of a wavelet group and Γ is the abstract pseudo-lattice Γ. We prove a condition in order that a Parseval frame π(Γ)ψ can be dilated to an orthonormal basis of the form τ (Γ)Ψ where τ is a super-representation of π. For a subclass of groups that includes the case where the translation subgroup is Heisenberg, we show that this condition always holds, and we cite familiar examples as applications.
Let M be a smooth compact oriented Riemannian manifold, and let ∆ be the Laplace-Beltrami operato... more Let M be a smooth compact oriented Riemannian manifold, and let ∆ be the Laplace-Beltrami operator on M. Say 0 ̸ = f ∈ S(R +), and that f(0) = 0. For t> 0, let Kt(x, y) denote the kernel of f(t 2 ∆). Suppose f satisfies Daubechies ’ criterion, and b> 0. For each j, write M as a disjoint union of measurable sets Ej,k with diameter at most ba j, and comparable to ba j if ba j is sufficiently small. Take xj,k ∈ Ej,k. We then show that the functions φj,k(x) = [µ(Ej,k)] 1/2 Kaj (xj,k, x) form a frame for (I − P)L 2 (M), for b sufficiently small (here P is the projection onto the constant functions). Moreover, we show that the ratio of the frame bounds approaches 1 nearly quadratically as the dilation parameter approaches 1, so that the frame quickly becomes nearly tight (for b sufficiently small). Moreover, based upon how well-localized a function F ∈ L 2 is in space and in frequency, we can describe which terms in the summation F ∼ SF = P j P k 〈F, φj,k〉φj,k are so small that they...
Given a simply connected nilpotent Lie group having unitary irreducible representations that are ... more Given a simply connected nilpotent Lie group having unitary irreducible representations that are square-integrable modulo the center (SI/Z), we use operator-valued periodization to give a range-function type characterization of shift-invariant spaces of function on the group. We then give characterizations of frame and Riesz families for shift-invariant spaces. keywords: Nilpotent Lie groups; square-integrable; range function; periodization operator; shift-invariant subspaces; frame and Reisz bases;
Given a simply connected nilpotent Lie group having unitary irreducible representations that are ... more Given a simply connected nilpotent Lie group having unitary irreducible representations that are square-integrable modulo the center (SI/Z), we develop a notion of periodization on the group Fourier transform side, and use this notion to give a characterization of shift-invariant spaces in $L^2(N)$ in terms of range functions. We apply these results to study the structure of frame and Reisz families for shift-invariant spaces. We illustrate these results for the Heisenberg group as well as for other groups with SI/Z representations.
Let $g(x)=\chi_B(x)$ be the indicator function of a bounded convex set in $\Bbb R^d$, $d\geq 2$, ... more Let $g(x)=\chi_B(x)$ be the indicator function of a bounded convex set in $\Bbb R^d$, $d\geq 2$, with a smooth boundary and everywhere non-vanishing Gaussian curvature. Using a combinatorial appraoch we prove that if $d \neq 1 \mod 4$, then there does not exist $S \subset {\Bbb R}^{2d}$ such that ${ \{g(x-a)e^{2 \pi i x \cdot b} \}}_{(a,b) \in S}$ is an orthonormal basis for $L^2({\Bbb R}^d)$.
We show that the spin wavelets on the sphere S 2 , which were constructed by the first author and... more We show that the spin wavelets on the sphere S 2 , which were constructed by the first author and Marinucci in [3], can be chosen so as to form a nearly tight frame. These spin wavelets can be applied (see [1] and [2]) to the study of the polarization of cosmic microwave background radiation. For certain of these frames, there is a positive C such that each frame element at scale a j is supported in a geodesic ball of radius Ca j .
In this paper we study how zeros of the Fourier transform of a function $f: \mathbb{Z}_p^d \to \m... more In this paper we study how zeros of the Fourier transform of a function $f: \mathbb{Z}_p^d \to \mathbb{C}$ are related to the structure of the function itself. In particular, we introduce a notion of bandwidth of such functions and discuss its connection with the decomposition of this function into wavelets. Connections of these concepts with the tomography principle and the Nyquist-Shannon sampling theorem are explored. We examine a variety of cases such as when the Fourier transform of the characteristic function of a set $E$ vanishes on specific sets of points, affine subspaces, and algebraic curves. In each of these cases, we prove properties such as equidistribution of $E$ across various surfaces and bounds on the size of $E$. We also establish a finite field Heisenberg uncertainty principle for sets that relates their bandwidth dimension and spatial dimension.
We study Gabor orthonormal windows in $L^2({\Bbb Z}_p^d)$ for translation and modulation sets $A$... more We study Gabor orthonormal windows in $L^2({\Bbb Z}_p^d)$ for translation and modulation sets $A$ and $B$, respectively, where $p$ is prime and $d\geq 2$. We prove that for a set $E\subset \Bbb Z_p^d$, the indicator function $1_E$ is a Gabor window if and only if $E$ tiles and is spectral. Moreover, we prove that for any function $g:\Bbb Z_p^d\to \Bbb C$ with support $E$, if the size of $E$ coincides with the size of the modulation set $B$ or if $g$ is positive, then $g$ is a unimodular function, i.e., $|g|=c1_E$, for some constant $c>0$, and $E$ tiles and is spectral. We also prove the existence of a Gabor window $g$ with full support where neither $|g|$ nor $|\hat g|$ is an indicator function and $|B|<<p^d$. We conclude the paper with an example and open questions.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this p... more The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
In this paper, we introduce a class of nonsmooth nonconvex optimization problems, and we propose ... more In this paper, we introduce a class of nonsmooth nonconvex optimization problems, and we propose to use a local iterative minimization-majorization (MM) algorithm to find an optimal solution for the optimization problem. The cost functions in our optimization problems are an extension of convex functions with MC separable penalty, which were previously introduced by Ivan Selesnick. These functions are not convex; therefore, convex optimization methods cannot be applied here to prove the existence of optimal minimum point for these functions. For our purpose, we use convex analysis tools to first construct a class of convex majorizers, which approximate the value of non-convex cost function locally, then use the MM algorithm to prove the existence of local minimum. The convergence of the algorithm is guaranteed when the iterative points $x^{(k)}$ are obtained in a ball centred at $x^{(k-1)}$ with small radius. We prove that the algorithm converges to a stationary point (local minimum...
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