Quasi-Analytic Collections of Square Integrable Functions

1988 ◽  
Vol 20 (4) ◽  
pp. 321-326 ◽  
Author(s):  
J. W. Neuberger
Keyword(s):  
Integrable Functions ◽  
Square Integrable Functions

scholarly journals A New Upper Bound for Sampling Numbers

2021 ◽  
Author(s):  
Nicolas Nagel ◽  
Martin Schäfer ◽  
Tino Ullrich
Keyword(s):  
Upper Bound ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Sampling Procedure ◽  
Compact Embedding ◽  
Recovery Method ◽  
Grid Sampling ◽  
Integrable Functions ◽  
Random Draw ◽  
Square Integrable Functions

AbstractWe provide a new upper bound for sampling numbers $$(g_n)_{n\in \mathbb {N}}$$ ( g n ) n ∈ N associated with the compact embedding of a separable reproducing kernel Hilbert space into the space of square integrable functions. There are universal constants $$C,c>0$$ C , c > 0 (which are specified in the paper) such that $$\begin{aligned} g^2_n \le \frac{C\log (n)}{n}\sum \limits _{k\ge \lfloor cn \rfloor } \sigma _k^2,\quad n\ge 2, \end{aligned}$$ g n 2 ≤ C log ( n ) n ∑ k ≥ ⌊ c n ⌋ σ k 2 , n ≥ 2 , where $$(\sigma _k)_{k\in \mathbb {N}}$$ ( σ k ) k ∈ N is the sequence of singular numbers (approximation numbers) of the Hilbert–Schmidt embedding $$\mathrm {Id}:H(K) \rightarrow L_2(D,\varrho _D)$$ Id : H ( K ) → L 2 ( D , ϱ D ) . The algorithm which realizes the bound is a least squares algorithm based on a specific set of sampling nodes. These are constructed out of a random draw in combination with a down-sampling procedure coming from the celebrated proof of Weaver’s conjecture, which was shown to be equivalent to the Kadison–Singer problem. Our result is non-constructive since we only show the existence of a linear sampling operator realizing the above bound. The general result can for instance be applied to the well-known situation of $$H^s_{\text {mix}}(\mathbb {T}^d)$$ H mix s ( T d ) in $$L_2(\mathbb {T}^d)$$ L 2 ( T d ) with $$s>1/2$$ s > 1 / 2 . We obtain the asymptotic bound $$\begin{aligned} g_n \le C_{s,d}n^{-s}\log (n)^{(d-1)s+1/2}, \end{aligned}$$ g n ≤ C s , d n - s log ( n ) ( d - 1 ) s + 1 / 2 , which improves on very recent results by shortening the gap between upper and lower bound to $$\sqrt{\log (n)}$$ log ( n ) . The result implies that for dimensions $$d>2$$ d > 2 any sparse grid sampling recovery method does not perform asymptotically optimal.

scholarly journals A generalization of the Itô formula

2002 ◽  
Vol 31 (8) ◽  
pp. 477-496
Author(s):  
Said Ngobi
Keyword(s):  
Sobolev Space ◽  
White Noise ◽  
Itô Formula ◽  
Ito Formula ◽  
Integrable Functions ◽  
Square Integrable Functions ◽  
White Noise Space

The classical Itô formula is generalized to some anticipating processes. The processes we consider are in a Sobolev space which is a subset of the space of square integrable functions over a white noise space. The proof of the result uses white noise techniques.

scholarly journals Hermite Functions and Fourier Series

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 853
Author(s):  
Enrico Celeghini ◽  
Manuel Gadella ◽  
Mariano del Olmo
Keyword(s):  
Fourier Transform ◽  
Hermite Functions ◽  
Quantum Oscillator ◽  
Linear Combinations ◽  
Ladder Operators ◽  
Orthonormal Bases ◽  
Integrable Functions ◽  
The Fourier Transform ◽  
The One ◽  
Square Integrable Functions

Using normalized Hermite functions, we construct bases in the space of square integrable functions on the unit circle (L2(C)) and in l2(Z), which are related to each other by means of the Fourier transform and the discrete Fourier transform. These relations are unitary. The construction of orthonormal bases requires the use of the Gramm–Schmidt method. On both spaces, we have provided ladder operators with the same properties as the ladder operators for the one-dimensional quantum oscillator. These operators are linear combinations of some multiplication- and differentiation-like operators that, when applied to periodic functions, preserve periodicity. Finally, we have constructed riggings for both L2(C) and l2(Z), so that all the mentioned operators are continuous.

scholarly journals Bound states of an inverse-cube singular potential: A candidate for electron-quadrupole binding

2019 ◽  
Vol 34 (28) ◽  
pp. 1950223 ◽  
Author(s):  
A. D. Alhaidari
Keyword(s):  
Bound States ◽  
Recursion Relation ◽  
Singular Potential ◽  
Electric Quadrupole ◽  
Exact Bound ◽  
Short Range Potential ◽  
Integrable Functions ◽  
Solution Energy ◽  
Expansion Coefficients ◽  
Square Integrable Functions

We use the Tridiagonal Representation Approach (TRA) to obtain exact bound states solution (energy spectrum and wave function) of the Schrödinger equation for a three-parameter short-range potential with [Formula: see text], [Formula: see text] and [Formula: see text] singularities at the origin. The solution is a finite series of square-integrable functions with expansion coefficients that satisfy a three-term recursion relation. The solution of the recursion is a non-conventional orthogonal polynomial with discrete spectrum. The results of this work could be used to study the binding of an electron to a molecule with an effective electric quadrupole moment which has the same [Formula: see text] singularity.

Lie methods for color robot vision

Robotica ◽  
2008 ◽  
Vol 26 (4) ◽  
pp. 453-464 ◽  
Author(s):  
Reiner Lenz
Keyword(s):  
Color Image ◽  
Robot Vision ◽  
Image Sequences ◽  
Color Changes ◽  
Integrable Functions ◽  
Conical Section ◽  
Lie Methods ◽  
Color Signals ◽  
Half Axis ◽  
Square Integrable Functions

SUMMARYWe describe how Lie-theoretical methods can be used to analyze color related problems in machine vision. The basic observation is that the nonnegative nature of spectral color signals restricts these functions to be members of a limited, conical section of the larger Hilbert space of square-integrable functions. From this observation, we conclude that the space of color signals can be equipped with a coordinate system consisting of a half-axis and a unit ball with the Lorentz groups as natural transformation group. We introduce the theory of the Lorentz group SU(1, 1) as a natural tool for analyzing color image processing problems and derive some descriptions and algorithms that are useful in the investigation of dynamical color changes. We illustrate the usage of these results by describing how to compress, interpolate, extrapolate, and compensate image sequences generated by dynamical color changes.

scholarly journals Toeplitz and slant Toeplitz operators on the polydisk

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Munmun Hazarika ◽  
Sougata Marik
Keyword(s):  
Toeplitz Operators ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Integrable Functions ◽  
Necessary And Sufficient ◽  
Square Integrable Functions

For n ≥ 1, let Dn be the polydisk in ℂn, and let Tn be the n-torus. L2(Tn) denotes the space of Lebesgue square integrable functions on Tn. In this paper we define slant Toeplitz operators on L2(Tn). Besides giving a necessary and sufficient condition for an operator on L2(Tn) to be slant Toeplitz, we also establish several properties of slant Toeplitz operators.

scholarly journals Resolution of the Identity of the Classical Hardy Space by Means of Barut-Girardello Coherent States

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Zouhaïr Mouayn
Keyword(s):  
Fourier Transform ◽  
Hardy Space ◽  
Real Line ◽  
Coherent States ◽  
Positive Real ◽  
Integrable Functions ◽  
Resolution Of The Identity ◽  
Square Integrable Functions ◽  
Complex Valued ◽  
Classical Hardy Space

We construct a one-parameter family of coherent states of Barut-Girdrardello type performing a resolution of the identity of the classical Hardy space of complex-valued square integrable functions on the real line, whose Fourier transform is supported by the positive real semiaxis.

scholarly journals On some smooth approximations on thick periodic multi-structures

2021 ◽  
Vol 18 ◽  
pp. 158
Author(s):  
P.I. Kogut ◽  
T.N. Rudyanova
Keyword(s):  
Asymptotic Behaviour ◽  
Bounded Function ◽  
Approximation Properties ◽  
Smooth Functions ◽  
Norm Estimates ◽  
Integrable Functions ◽  
Oscillating Boundary ◽  
Square Integrable Functions ◽  
Smooth Approximations

In this paper we study the approximation properties of measurable and square-integrable functions. In particular we show that any $L^2$-bounded function can be approximated in $L^2$-norm by smooth functions defined on a highly oscillating boundary of thick multi-structures in ${\mathbb{R}}^n$. We derive the norm estimates for the approximating functions and study their asymptotic behaviour.

scholarly journals On the Parity Under Metapletic Operators and an Extension of a Result of Lyubarskii and Nes

2019 ◽  
Vol 75 (1) ◽  
Author(s):  
Markus Faulhuber
Keyword(s):  
Hilbert Space ◽  
Real Line ◽  
Gabor Frame ◽  
Window Function ◽  
Gabor System ◽  
The Real ◽  
Metaplectic Group ◽  
Integrable Functions ◽  
Square Integrable Functions

AbstractIn this work we show that if the frame property of a Gabor frame with window in Feichtinger’s algebra and a fixed lattice only depends on the parity of the window, then the lattice can be replaced by any other lattice of the same density without losing the frame property. As a byproduct we derive a generalization of a result of Lyubarskii and Nes, who could show that any Gabor system consisting of an odd window function from Feichtinger’s algebra and any separable lattice of density $$\tfrac{n+1}{n}$$n+1n, $$n \in \mathbb {N}_+$$n∈N+, cannot be a Gabor frame for the Hilbert space of square-integrable functions on the real line. We extend this result by removing the assumption that the lattice has to be separable. This is achieved by exploiting the interplay between the symplectic and the metaplectic group.

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