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Operator-Like Wavelet Bases of $L_{2}(\mathbb{R}^{d})$
Springer Science and Business Media LLC - Tập 19 - Trang 1294-1322 - 2013
The connection between derivative operators and wavelets is well known. Here we generalize the concept by constructing multiresolution approximations and wavelet basis functions that act like Fourier multiplier operators. This construction follows from a stochastic model: signals are tempered distributions such that the application of a whitening (differential) operator results in a realization of...... hiện toàn bộ
The Plancherel Formula for an Inhomogeneous Vector Group
Springer Science and Business Media LLC - Tập 25 - Trang 2837-2876 - 2019
We give a concrete realization of the Plancherel measure for a semi-direct product $$N \rtimes H$$ where N and H are vector groups for which the linear action of H on N is almost everywhere regular. A procedure using matrix reductions produces explicit (orbital) parameters by which a continuous field of unitary irreducible representations is realized and the almost all of the dual space of $$N \rt...... hiện toàn bộ
A New Scheme for the Tensor Representation
Springer Science and Business Media LLC - Tập 15 - Trang 706-722 - 2009
The paper presents a new scheme for the representation of tensors which is well-suited for high-order tensors. The construction is based on a hierarchy of tensor product subspaces spanned by orthonormal bases. The underlying binary tree structure makes it possible to apply standard Linear Algebra tools for performing arithmetical operations and for the computation of data-sparse approximations. In...... hiện toàn bộ
Admissible Measurements and Robust Algorithms for Ptychography
Springer Science and Business Media LLC - Tập 27 - Trang 1-39 - 2021
We study an approach to solving the phase retrieval problem as it arises in a phase-less imaging modality known as ptychography. In ptychography, small overlapping sections of an unknown sample (or signal, say
$$x_0\in \mathbb {C}^{d}$$
) are illuminated one at a time, often with a physical mask be...... hiện toàn bộ
The Turán Problem and Its Dual for Positive Definite Functions Supported on a Ball in $${\mathbb {R}}^d$$
Springer Science and Business Media LLC - Tập 30 - Trang 1-31 - 2024
The Turán problem for an open ball of radius r centered at the origin in
$${\mathbb {R}}^d$$
consists in computing the supremum of the integrals of positive definite functions compactly supported on that ball and taking the value 1 at the origin. Siegel proved, in the 1930s that this supremum is e...... hiện toàn bộ
The Stability of Wavelet-Like Expansions in $$A_\infty $$ Weighted Spaces
Springer Science and Business Media LLC - Tập 25 - Trang 2877-2898 - 2019
We prove $$L^p$$ boundedness in $$A_\infty $$ weighted spaces for operators defined by almost-orthogonal expansions indexed over the dyadic cubes. The constituent functions in the almost-orthogonal families satisfy weak decay, smoothness, and cancellation conditions. We prove that these expansions are stable (with respect to the $$L^p$$ operator norm) when the constituent functions suffer small di...... hiện toàn bộ
Bounds on the Segal-Bargmann transform ofL p functions
Springer Science and Business Media LLC - - 2001
This article gives necessary conditions and slightly stronger sufficient conditions for a holomorphic function to be the Segal-Bargmann transform of a function inL
p
(ℝ
d
, ρ) where ρ is a Gaussian measure. The proof relies on a family of inversion formulas for the Segal-Bargmann transform, which can be “tuned” t...... hiện toàn bộ
Pseudodifferential Operators on Spaces of Distributions Associated with Non-negative Self-Adjoint Operators
Springer Science and Business Media LLC - Tập 23 - Trang 344-378 - 2016
We consider Hörmander type symbols on a family of spaces associated with non-negative self-adjoint operators, and we prove boundedness of the corresponding pseudodifferential operators on both classical and non-classical Besov and Triebel–Lizorkin spaces. Consequently, this also covers the case of Sobolev spaces. As an application, we obtain boundedness of spectral multipliers on the mentioned spa...... hiện toàn bộ
On Eigenmeasures Under Fourier Transform
Springer Science and Business Media LLC - Tập 29 - Trang 1-33 - 2023
Several classes of tempered measures are characterised that are eigenmeasures of the Fourier transform, the latter viewed as a linear operator on (generally unbounded) Radon measures on
$$\mathbb {R}\hspace{0.5pt}^d$$
. In particular, we classify all periodic eigenmeasures on
...... hiện toàn bộ
An Analogue of Slepian Vectors on Boolean Hypercubes
Springer Science and Business Media LLC - Tập 25 - Trang 2004-2020 - 2018
Analogues of Slepian vectors are defined for finite-dimensional Boolean hypercubes. These vectors are the most concentrated in neighborhoods of the origin among bandlimited vectors. Spaces of bandlimited vectors are defined as spans of eigenvectors of the Laplacian of the hypercube graph with lowest eigenvalues. A difference operator that almost commutes with space and band limiting is used to ini...... hiện toàn bộ
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