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 Heisenberg–Weyl Groups and Generalized Hermite Functions

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1060
Author(s):  
Enrico Celeghini ◽  
Manuel Gadella ◽  
Mariano A. del del Olmo
Keyword(s):  
Hilbert Space ◽  
Fourier Transform ◽  
Heisenberg Group ◽  
Real Line ◽  
Hermite Functions ◽  
Unitary Representations ◽  
Weyl Groups ◽  
The Real ◽  
Symmetry Properties ◽  
The Fourier Transform

We introduce a multi-parameter family of bases in the Hilbert space L2(R) that are associated to a set of Hermite functions, which also serve as a basis for L2(R). The Hermite functions are eigenfunctions of the Fourier transform, a property that is, in some sense, shared by these “generalized Hermite functions”. The construction of these new bases is grounded on some symmetry properties of the real line under translations, dilations and reflexions as well as certain properties of the Fourier transform. We show how these generalized Hermite functions are transformed under the unitary representations of a series of groups, including the Weyl–Heisenberg group and some of their extensions.

scholarly journals The Fourier transform of thick distributions

2020 ◽  
pp. 1-26
Author(s):  
Ricardo Estrada ◽  
Jasson Vindas ◽  
Yunyun Yang
Keyword(s):  
Fourier Transform ◽  
Dual Space ◽  
Canonical Projection ◽  
Test Functions ◽  
Tempered Distributions ◽  
Delta Functions ◽  
The Fourier Transform ◽  
The One ◽  
One Point Compactification ◽  
Logarithmic Type

We first construct a space [Formula: see text] whose elements are test functions defined in [Formula: see text] the one point compactification of [Formula: see text] that have a thick expansion at infinity of special logarithmic type, and its dual space [Formula: see text] the space of sl-thick distributions. We show that there is a canonical projection of [Formula: see text] onto [Formula: see text] We study several sl-thick distributions and consider operations in [Formula: see text] We define and study the Fourier transform of thick test functions of [Formula: see text] and thick tempered distributions of [Formula: see text] We construct isomorphisms [Formula: see text] [Formula: see text] that extend the Fourier transform of tempered distributions, namely, [Formula: see text] and [Formula: see text] where [Formula: see text] are the canonical projections of [Formula: see text] or [Formula: see text] onto [Formula: see text] We determine the Fourier transform of several finite part regularizations and of general thick delta functions.

scholarly journals Fourier Analysis with Generalized Integration

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1199
Author(s):  
Juan H. Arredondo ◽  
Manuel Bernal ◽  
María Guadalupe Morales
Keyword(s):  
Fourier Transform ◽  
Fourier Analysis ◽  
Integral Theory ◽  
Dense Subspace ◽  
Fourier Cosine ◽  
Sine Transform ◽  
Integrable Functions ◽  
Fourier Sine Transform ◽  
Fourier Cosine Transform ◽  
The Fourier Transform

We generalize the classic Fourier transform operator F p by using the Henstock–Kurzweil integral theory. It is shown that the operator equals the H K -Fourier transform on a dense subspace of L p , 1 < p ≤ 2 . In particular, a theoretical scope of this representation is raised to approximate the Fourier transform of functions on the mentioned subspace numerically. Besides, we show the differentiability of the Fourier transform function F p ( f ) under more general conditions than in Lebesgue’s theory. Additionally, continuity of the Fourier Sine transform operator into the space of Henstock-Kurzweil integrable functions is proved, which is similar in spirit to the already known result for the Fourier Cosine transform operator. Because our results establish a representation of the Fourier transform with more properties than in Lebesgue’s theory, these results might contribute to development of better algorithms of numerical integration, which are very important in applications.

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 The conjugation operator onAq(G)

2003 ◽  
Vol 2003 (37) ◽  
pp. 2345-2347
Author(s):  
Sanjiv Kumar Gupta
Keyword(s):  
Fourier Transform ◽  
Abelian Group ◽  
Compact Abelian Group ◽  
Bounded Operator ◽  
Conjugation Operator ◽  
Integrable Functions ◽  
The Fourier Transform ◽  
Space Of Integrable Functions

Letq>2. We prove that the conjugation operatorHdoes not extend to a bounded operator on the space of integrable functions defined on any compact abelian group with the Fourier transform inlq.

INVARIANT PATTERN RECOGNITION USING RIDGELET PACKETS AND THE FOURIER TRANSFORM

2009 ◽  
Vol 07 (02) ◽  
pp. 215-228 ◽  
Author(s):  
G. Y. CHEN ◽  
P. BHATTACHARYA
Keyword(s):  
Pattern Recognition ◽  
Fourier Transform ◽  
State Of The Art ◽  
Experimental Results ◽  
Noisy Environments ◽  
Rotation Invariant ◽  
Pattern Recognition Techniques ◽  
Orthonormal Bases ◽  
The Fourier Transform ◽  
Very High

In this paper, we propose two novel invariant algorithms for pattern recognition by using ridgelet packets and the Fourier transform. Ridgelet packets provide many orthonormal bases that can effectively capture directional features present in pattern images. The Fourier transform is good at eliminating the orientation differences. By combining these two tools, very efficient rotation invariant pattern recognition techniques are created. Experimental results show that the proposed methods achieve very high classification rates and they outperform other state-of-the-art methods for rotation invariant pattern recognition under both noise-free and noisy environments.

scholarly journals A NOTE ON NONCOMMUTATIVE CHERN–SIMONS MODEL ON MANIFOLDS WITH BOUNDARY

2002 ◽  
Vol 17 (03) ◽  
pp. 141-155 ◽  
Author(s):  
ADRIÁN R. LUGO
Keyword(s):  
Canonical Quantization ◽  
Quantum Hall ◽  
Simons Theory ◽  
Quantum Hall Systems ◽  
Integrable Functions ◽  
Dimensional Boundary ◽  
The One ◽  
Definition Of ◽  
Square Integrable Functions ◽  
Chern Simons

We study field theories defined in regions of the spatial noncommutative (NC) plane with a boundary present delimiting them, concentrating in particular on the U(1) NC Chern–Simons theory on the upper half-plane. We find that classical consistency and gauge invariance lead necessary to the introduction of K0-space of square integrable functions null together with all their derivatives at the origin. Furthermore the requirement of closure of K0 under the *-product leads to the introduction of a novel notion of the *-product itself in regions where a boundary is present, that in turn yields the complexification of the gauge group and to consider chiral waves in one sense or other. The canonical quantization of the theory is sketched identifying the physical states and the physical operators. These last ones include ordinary NC Wilson lines starting and ending on the boundary that yield correlation functions depending on points on the one-dimensional boundary. We finally extend the definition of the *-product to a strip and comment on possible relevance of these results to finite quantum Hall systems.

scholarly journals Rotor Fault Detection in Induction Motors Based on Time-Frequency Analysis Using the Bispectrum and the Autocovariance of Stray Flux Signals

Energies ◽  
2019 ◽  
Vol 12 (4) ◽  
pp. 597 ◽  
Author(s):  
Miguel Iglesias-Martínez ◽  
Jose Antonino-Daviu ◽  
Pedro Fernández de Córdoba ◽  
J. Conejero
Keyword(s):  
Spectral Analysis ◽  
Fourier Transform ◽  
Induction Motor ◽  
Mean Value ◽  
High Order ◽  
One Dimensional ◽  
Time Frequency ◽  
Temporal Domain ◽  
The Fourier Transform ◽  
The One

The aim of this work is to find out, through the analysis of the time and frequency domains, significant differences that lead us to obtain one or several variables that may result in an indicator that allows diagnosing the condition of the rotor in an induction motor from the processing of the stray flux signals. For this, the calculation of two indicators is proposed: the first is based on the frequency domain and it relies on the calculation of the sum of the mean value of the bispectrum of the flux signal. The use of high order spectral analysis is justified in that with the one-dimensional analysis resulting from the Fourier Transform, there may not always be solid differences at the spectral level that enable us to distinguish between healthy and faulty conditions. Also, based on the high-order spectral analysis, differences may arise that, with the classical analysis with the Fourier Transform, are not evident, since the high order spectra from the Bispectrum are immune to Gaussian noise, but not the results that can be obtained using the one-dimensional Fourier transform. On the other hand, a second indicator based on the temporal domain that is based on the calculation of the square value of the median of the autocovariance function of the signal is evaluated. The obtained results are satisfactory and let us conclude the affirmative hypothesis of using flux signals for determining the condition of the rotor of an induction motor.

scholarly journals Expansions of Functions Based on Rational Orthogonal Basis with Nonnegative Instantaneous Frequencies

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Xiaona Cui ◽  
Suxia Yao
Keyword(s):  
Fourier Transform ◽  
Fast Fourier Transform ◽  
Fast Algorithm ◽  
Orthogonal Basis ◽  
Basis Functions ◽  
Almost Everywhere ◽  
Integrable Functions ◽  
Instantaneous Frequencies ◽  
Square Integrable Functions

We consider in this paper expansions of functions based on the rational orthogonal basis for the space of square integrable functions. The basis functions have nonnegative instantaneous frequencies so that the expansions make physical sense. We discuss the almost everywhere convergence of the expansions and develop a fast algorithm for computing the coefficients arising in the expansions by combining the characterization of the coefficients with the fast Fourier transform.

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