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We prove that the HRT (Heil, Ramanathan, and Topiwala) Conjecture is equivalent to the conjecture that co-central translates of square-integrable functions on the Heisenberg group are linearly independent.
Let
$\{M_{n}\}_{n=1}^{\infty }$
be a sequence of expanding matrices with
$M_{n}=\operatorname{diag}(p_{n},q_{n})$
, and let
$\{{\mathcal{D}}_{n}\}_{n=1}^{\infty }$
be a sequence of digit sets with
${\mathcal{D}}_{n}=\{(0,0)^{t},(a_{n},0)^{t},(0,b_{n})^{t},\pm (a_{n},b_{n})^{t}\}$
, where
$p_{n}$
,
$q_{n}$
,
$a_{n}$
and
$b_{n}$
are positive integers for all
$n\geqslant 1$
. If
$\sup _{n\geqslant 1}\{\frac{a_{n}}{p_{n}},\frac{b_{n}}{q_{n}}\}<\infty$
, then the infinite convolution
$\unicode[STIX]{x1D707}_{\{M_{n}\},\{{\mathcal{D}}_{n}\}}=\unicode[STIX]{x1D6FF}_{M_{1}^{-1}{\mathcal{D}}_{1}}\ast \unicode[STIX]{x1D6FF}_{(M_{1}M_{2})^{-1}{\mathcal{D}}_{2}}\ast \cdots \,$
is a Borel probability measure (Cantor–Dust–Moran measure). In this paper, we investigate whenever there exists a discrete set
$\unicode[STIX]{x1D6EC}$
such that
$\{e^{2\unicode[STIX]{x1D70B}i\langle \unicode[STIX]{x1D706},x\rangle }:\unicode[STIX]{x1D706}\in \unicode[STIX]{x1D6EC}\}$
is an orthonormal basis for
$L^{2}(\unicode[STIX]{x1D707}_{\{M_{n}\},\{{\mathcal{D}}_{n}\}})$
.
The main result of this note implies that any function from the product of several vector spaces to a vector space can be uniquely decomposed into the sum of mutually orthogonal functions that are odd in some of the arguments and even in the other arguments. Probabilistic notions and facts are employed to simplify statements and proofs.
We provide a general program for finding nice arrangements of points in real or complex projective space from transitive actions of finite groups. In many cases, these arrangements are optimal in the sense of maximizing the minimum distance. We introduce our program in terms of general Schurian association schemes before focusing on the special case of Gelfand pairs. Notably, our program unifies a variety of existing packings with heretofore disparate constructions. In addition, we leverage our program to construct the first known infinite family of equiangular lines with Heisenberg symmetry.
We give a characterization of all Parseval wavelet frames arising from a given frame multiresolution analysis. As a consequence, we obtain a description of all Parseval wavelet frames associated with a frame multiresolution analysis. These results are based on a version of Oblique Extension Principle with the assumption that the origin is a point of approximate continuity of the Fourier transform of the involved refinable functions. Our results are written for reducing subspaces.
Suppose that
$0<|\unicode[STIX]{x1D70C}|<1$
and
$m\geqslant 2$
is an integer. Let
$\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m}$
be the self-similar measure defined by
$\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m}(\cdot )=\frac{1}{m}\sum _{j=0}^{m-1}\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m}(\unicode[STIX]{x1D70C}^{-1}(\cdot )-j)$
. Assume that
$\unicode[STIX]{x1D70C}=\pm (q/p)^{1/r}$
for some
$p,q,r\in \mathbb{N}^{+}$
with
$(p,q)=1$
and
$(p,m)=1$
. We prove that if
$(q,m)=1$
, then there are at most
$m$
mutually orthogonal exponential functions in
$L^{2}(\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m})$
and
$m$
is the best possible. If
$(q,m)>1$
, then there are any number of orthogonal exponential functions in
$L^{2}(\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m})$
.
We show the existence of a measurable selector in Carpenter’s Theorem due to Kadison. This solves a problem posed by Jasper and the first author in an earlier work. As an application we obtain a characterization of all possible spectral functions of shift-invariant subspaces of
$L^{2}(\mathbb{R}^{d})$
and Carpenter’s Theorem for type
$\text{I}_{\infty }$
von Neumann algebras.
We investigate the Gibbs–Wilbraham phenomenon for generalized sampling series, and related interpolation series arising from cardinal functions. We prove the existence of the overshoot characteristic of the phenomenon for certain cardinal functions, and characterize the existence of an overshoot for sampling series.
In this paper, we prove some reverse discrete inequalities with weights of Muckenhoupt and Gehring types and use them to prove some higher summability theorems on a higher weighted space
$l_{w}^{p}({\open N})$
form summability on the weighted space
$l_{w}^{q}({\open N})$
when p>q. The proofs are obtained by employing new discrete weighted Hardy's type inequalities and their converses for non-increasing sequences, which, for completeness, we prove in our special setting. To the best of the authors' knowledge, these higher summability results have not been considered before. Some numerical results will be given for illustration.
(1) For any finite metric space
$M$
the Lipschitz-free space on
$M$
contains a large well-complemented subspace that is close to
$\ell _{1}^{n}$
.
(2) Lipschitz-free spaces on large classes of recursively defined sequences of graphs are not uniformly isomorphic to
$\ell _{1}^{n}$
of the corresponding dimensions. These classes contain well-known families of diamond graphs and Laakso graphs.
Interesting features of our approach are: (a) We consider averages over groups of cycle-preserving bijections of edge sets of graphs that are not necessarily graph automorphisms. (b) In the case of such recursive families of graphs as Laakso graphs, we use the well-known approach of Grünbaum (1960) and Rudin (1962) for estimating projection constants in the case where invariant projections are not unique.
We consider three special and significant cases of the following problem. Let
$D\subset \mathbb{R}^{d}$
be a (possibly unbounded) set of finite Lebesgue measure. Let
$E(\mathbb{Z}^{d})=\{e^{2\unicode[STIX]{x1D70B}ix\cdot n}\}\text{}_{n\in \mathbb{Z}^{d}}$
be the standard exponential basis on the unit cube of
$\mathbb{R}^{d}$
. Find conditions on
$D$
for which
$E(\mathbb{Z}^{d})$
is a frame, a Riesz sequence, or a Riesz basis for
$L^{2}(D)$
.
We prove that an
$L^{\infty }$
potential in the Schrödinger equation in three and higher dimensions can be uniquely determined from a finite number of boundary measurements, provided it belongs to a known finite dimensional subspace
${\mathcal{W}}$
. As a corollary, we obtain a similar result for Calderón’s inverse conductivity problem. Lipschitz stability estimates and a globally convergent nonlinear reconstruction algorithm for both inverse problems are also presented. These are the first results on global uniqueness, stability and reconstruction for nonlinear inverse boundary value problems with finitely many measurements. We also discuss a few relevant examples of finite dimensional subspaces
${\mathcal{W}}$
, including bandlimited and piecewise constant potentials, and explicitly compute the number of required measurements as a function of
$\dim {\mathcal{W}}$
.
We study the heat semigroup maximal operator associated with a well-known orthonormal system in the
$d$
-dimensional ball. The corresponding heat kernel is shown to satisfy Gaussian bounds. As a consequence, we can prove weighted
$L^{p}$
estimates, as well as some weighted inequalities in mixed norm spaces, for this maximal operator.
We consider upper‒lower (UL) (and lower‒upper (LU)) factorizations of the one-step transition probability matrix of a random walk with the state space of nonnegative integers, with the condition that both upper and lower triangular matrices in the factorization are also stochastic matrices. We provide conditions on the free parameter of the UL factorization in terms of certain continued fractions such that this stochastic factorization is possible. By inverting the order of the factors (also known as a Darboux transformation) we obtain a new family of random walks where it is possible to state the spectral measures in terms of a Geronimus transformation. We repeat this for the LU factorization but without a free parameter. Finally, we apply our results in two examples; the random walk with constant transition probabilities, and the random walk generated by the Jacobi orthogonal polynomials. In both situations we obtain urn models associated with all the random walks in question.
We give a simple argument which shows that Gabor systems consisting of odd
functions of
$d$
variables and symplectic lattices of density
$2^{d}$
cannot constitute a Gabor frame. In the one-dimensional,
separable case, this follows from a more general result of Lyubarskii and
Nes [‘Gabor frames with rational density’, Appl. Comput. Harmon.
Anal.34(3) (2013), 488–494]. We use a different
approach exploiting the algebraic relation between the ambiguity function
and the Wigner distribution as well as their relation given by the
(symplectic) Fourier transform. Also, we do not need the assumption that the
lattice is separable and, hence, new restrictions are added to the full
frame set of odd functions.
We prove Hardy-type inequalities for a fractional Dunkl–Hermite operator, which incidentally gives Hardy inequalities for the fractional harmonic oscillator as well. The idea is to use h-harmonic expansions to reduce the problem in the Dunkl–Hermite context to the Laguerre setting. Then, we push forward a technique based on a non-local ground representation, initially developed by Frank et al. [‘Hardy–Lieb–Thirring inequalities for fractional Schrödinger operators, J. Amer. Math. Soc.21 (2008), 925–950’] in the Euclidean setting, to obtain a Hardy inequality for the fractional-type Laguerre operator. The above-mentioned method is shown to be adaptable to an abstract setting, whenever there is a ‘good’ spectral theorem and an integral representation for the fractional operators involved.
We show that if A is a compact C*-algebra without identity that has a faithful *-representation in the C*-algebra of all compact operators on a separable Hilbert space and its multiplier algebra admits a minimal central projection p such that pA is infinite-dimensional, then there exists a Hilbert A1-module admitting no frames, where A1 is the unitization of A. In particular, there exists a frame-less Hilbert C*-module over the C*-algebra
$K(\ell^2) \dotplus \mathbb{C}I_{\ell^2}$
.
Two measurable sets
$S,\unicode[STIX]{x1D6EC}\subseteq \mathbb{R}^{d}$
form a Heisenberg uniqueness pair, if every bounded measure
$\unicode[STIX]{x1D707}$
with support in
$S$
whose Fourier transform vanishes on
$\unicode[STIX]{x1D6EC}$
must be zero. We show that a quadratic hypersurface and the union of two hyperplanes in general position form a Heisenberg uniqueness pair in
$\mathbb{R}^{d}$
. As a corollary we obtain a new, surprising version of the classical Cramér–Wold theorem: a bounded measure supported on a quadratic hypersurface is uniquely determined by its projections onto two generic hyperplanes (whereas an arbitrary measure requires the knowledge of a dense set of projections). We also give an application to the unique continuation of eigenfunctions of second-order PDEs with constant coefficients.
In this paper we consider the algorithm for recovering sparse orthogonal polynomials using stochastic collocation via ℓq minimization. The main results include: 1) By using the norm inequality between ℓq and ℓ2 and the square root lifting inequality, we present several theoretical estimates regarding the recoverability for both sparse and non-sparse signals via ℓq minimization; 2) We then combine this method with the stochastic collocation to identify the coefficients of sparse orthogonal polynomial expansions, stemming from the field of uncertainty quantification. We obtain recoverability results for both sparse polynomial functions and general non-sparse functions. We also present various numerical experiments to show the performance of the ℓq algorithm. We first present some benchmark tests to demonstrate the ability of ℓq minimization to recover exactly sparse signals, and then consider three classical analytical functions to show the advantage of this method over the standard ℓ1 and reweighted ℓ1 minimization. All the numerical results indicate that the ℓq method performs better than standard ℓ1 and reweighted ℓ1 minimization.
Image inpainting methods recover true images from partial noisy observations. Natural images usually have two layers consisting of cartoons and textures. Methods using simultaneous cartoon and texture inpainting are popular in the literature by using two combined tight frames: one (often built from wavelets, curvelets or shearlets) provides sparse representations for cartoons and the other (often built from discrete cosine transforms) offers sparse approximation for textures. Inspired by the recent development on directional tensor product complex tight framelets (
$\text{TP}\text{-}\mathbb{C}\text{TF}$
s) and their impressive performance for the image denoising problem, we propose an iterative thresholding algorithm using tight frames derived from
$\text{TP}\text{-}\mathbb{C}\text{TF}$
s for the image inpainting problem. The tight frame
$\text{TP}\text{-}\mathbb{C}\text{TF}_{6}$
contains two classes of framelets; one is good for cartoons and the other is good for textures. Therefore, it can handle both the cartoons and the textures well. For the image inpainting problem with additive zero-mean independent and identically distributed Gaussian noise, our proposed algorithm does not require us to tune parameters manually for reasonably good performance. Experimental results show that our proposed algorithm performs comparatively better than several well-known frame systems for the image inpainting problem.