THE QUANTUM QUARTER PLANE AND THE REAL QUANTUM PLANE

2002 ◽  
Vol 13 (03) ◽  
pp. 279-321 ◽  
Author(s):  
KONRAD SCHMÜDGEN
Keyword(s):  
Fourier Transforms ◽  
Pseudodifferential Operators ◽  
Quantum Plane ◽  
Left Module ◽  
The Real ◽  
Weyl Calculus ◽  
Positive Linear Functionals ◽  
Quarter Plane ◽  
The Fourier Transform ◽  
Module Algebras

Suppose q≠±1 is a complex number of modulus one. Let [Formula: see text] be the *-algebra with two hermitean generators x and y satisfying the relation xy=qyx. Using Hilbert space representations of [Formula: see text] and the Weyl calculus of pseudodifferential operators we construct *-algebras of "functions" on the quantum quarter plane [Formula: see text] and on the real quantum plane [Formula: see text] which are left module *-algebras for the Hopf *-algebra [Formula: see text]. We define covariant positive linear functionals hk, k∈ℤ2, and study the actions of the *-algebras [Formula: see text] and [Formula: see text] on the associated Hilbert spaces. Quantum analogs of the partial Fourier transforms and the Fourier transform are found. A differential calculus on the "function" *-algebras is also developed and investigated.

scholarly journals On the Fourier transforms

1985 ◽  
Vol 31 (2) ◽  
pp. 171-179
Author(s):  
Hwai-chiuan Wang
Keyword(s):  
Fourier Transform ◽  
Convex Function ◽  
Real Line ◽  
Integrable Function ◽  
Fourier Transforms ◽  
The Real ◽  
Factorization Problem ◽  
Several Variables ◽  
The Fourier Transform

In this article we give a new proof of the theorem that a positive even convex function on the real line, which vanishes at infinity, is the Fourier transform of an integrable function. Related results in several variables are also proved. As an application of our results we solve the factorization problem of Sobolev algebras.

Fourier Transforms of Unbounded Measures

1979 ◽  
Vol 31 (6) ◽  
pp. 1281-1292 ◽  
Author(s):  
James Stewart
Keyword(s):  
Fourier Transform ◽  
Real Line ◽  
Fourier Transforms ◽  
Dual Group ◽  
Locally Compact Abelian Group ◽  
Stieltjes Transform ◽  
The Real ◽  
Bounded Complex ◽  
The Fourier Transform ◽  
Image Position

1. Introduction. One of the basic objects of study in harmonic analysis is the Fourier transform (or Fourier-Stieltjes transform) μ of a bounded (complex) measure μ on the real line R:(1.1)More generally, if μ is a bounded measure on a locally compact abelian group G, then its Fourier transform is the function(1.2)where Ĝ is the dual group of G and One answer to the question “Which functions can be represented as Fourier transforms of bounded measures?” was given by the following criterion due to Schoenberg [11] for the real line and Eberlein [5] in general: f is a Fourier transform of a bounded measure if and only if there is a constant M such that(1.3)for all ϕ ∈ L1(G) where

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.

Wavelet packets associated with linear canonical transform on spectrum

2021 ◽  
pp. 2150030
Author(s):  
M. Younus Bhat ◽  
Aamir H. Dar
Keyword(s):  
Fourier Transform ◽  
Multiresolution Analysis ◽  
Fourier Transforms ◽  
Fractional Fourier Transform ◽  
Integral Transforms ◽  
Wavelet Packets ◽  
Linear Canonical Transform ◽  
Fresnel Transform ◽  
The Fourier Transform ◽  
Vector Valued

The linear canonical transform (LCT) provides a unified treatment of the generalized Fourier transforms in the sense that it is an embodiment of several well-known integral transforms including the Fourier transform, fractional Fourier transform, Fresnel transform. Using this fascinating property of LCT, we, in this paper, constructed associated wavelet packets. First, we construct wavelet packets corresponding to nonuniform Multiresolution analysis (MRA) associated with LCT and then those corresponding to vector-valued nonuniform MRA associated with LCT. We investigate their various properties by means of LCT.

An analysis of the spectrum of a gravity anomaly over an inclined dike

Geophysics ◽  
1985 ◽  
Vol 50 (9) ◽  
pp. 1500-1501
Author(s):  
B. N. P. Agarwal ◽  
D. Sita Ramaiah
Keyword(s):  
Fourier Transform ◽  
Gravity Anomaly ◽  
Fourier Spectrum ◽  
Fourier Transforms ◽  
Weighted Average ◽  
Window Function ◽  
Average Spectrum ◽  
Amplitude Spectra ◽  
Distance Data ◽  
The Fourier Transform

Bhimasankaram et al. (1977) used Fourier spectrum analysis for a direct approach to the interpretation of gravity anomaly over a finite inclined dike. They derived several equations from the real and imaginary components and from the amplitude and phase spectra to relate various parameters of the dike. Because the width 2b of the dike (Figure 1) appears only in sin (ωb) term—ω being the angular frequency—they determined its value from the minima/zeroes of the amplitude spectra. The theoretical Fourier spectrum uses gravity field data over an infinite distance (length), whereas field observations are available only for a limited distance. Thus, a set of observational data is viewed as a product of infinite‐distance data with an appropriate window function. Usually, a rectangular window of appropriate distance (width) and of unit magnitude is chosen for this purpose. The Fourier transform of the finite‐distance and discrete data is thus represented by convolution operations between Fourier transforms of the infinite‐distance data, the window function, and the comb function. The combined effect gives a smooth, weighted average spectrum. Thus, the Fourier transform of actual observed data may differ substantially from theoretic data. The differences are apparent for low‐ and high‐frequency ranges. As a result, the minima of the amplitude spectra may change considerably, thereby rendering the estimate of the width of the dike unreliable from the roots of the equation sin (ωb) = 0.

FOURIER TRANSFORMS OF FINITE LENGTH THEORETICAL GRAVITY ANOMALIES

Geophysics ◽  
1977 ◽  
Vol 42 (7) ◽  
pp. 1450-1457 ◽  
Author(s):  
Robert D. Regan ◽  
William J. Hinze
Keyword(s):  
Fourier Transform ◽  
Analytical Solution ◽  
Finite Length ◽  
Fourier Transforms ◽  
Gravity Anomalies ◽  
Mathematical Structure ◽  
Practical Application ◽  
Convolution Integrals ◽  
The Fourier Transform ◽  
Interpretation Method

The mathematical structure of the Fourier transformations of theoretical gravity anomalies of several geometrically simple bodies appears to have distinct advantages in the interpretation of these anomalies. However, the practical application of this technique is dependent upon the transformation of an observed gravity anomaly of finite length. Ideally, interpretation methods similar to those for the transformations of the theoretical gravity anomalies should be developed for anomalies of a finite length. However, the mathematical complexity of the convolution integrals in the transform calculations of theoretical anomaly segments indicate that no general closed analytical solution useful for interpretation is available. Thus, in order to utilize the Fourier transform interpretation method, the data must be of sufficient length for the finite transform to closely approximate the theoretical transforms.

Two notes on spectral synthesis for discrete Abelian groups

1965 ◽  
Vol 61 (3) ◽  
pp. 617-620 ◽  
Author(s):  
R. J. Elliott
Keyword(s):  
Fourier Transform ◽  
Dual Space ◽  
Fourier Transforms ◽  
Spectral Synthesis ◽  
Division Problem ◽  
Exponential Functions ◽  
Translation Invariant ◽  
Annihilator Ideal ◽  
The Fourier Transform ◽  
Translation Invariant Subspace

Introduction. Spectral synthesis is the study of whether functions in a certain set, usually a translation invariant subspace (a variety), can be synthesized from certain simple functions, exponential monomials, which are contained in the set. This problem is transformed by considering the annihilator ideal in the dual space, and after taking the Fourier transform the problem becomes one of deciding whether a function is in a certain ideal, that is, we have a ‘division problem’. Because of this we must take into consideration the possibility of the Fourier transforms of functions having zeros of order greater than or equal to 1. This is why, in the original situation, we study whether varieties are generated by their exponential monomials, rather than just their exponential functions. This viewpoint of the problem as a division question, of course, perhaps throws light on why Wiener's Tauberian theorem works, and is implicit in the construction of Schwartz's and Malliavin's counter examples to spectral synthesis in L1(G) (cf. Rudin ((4))).

scholarly journals A Resolvent Approach to the Real Quantum Plane

2014 ◽  
Vol 79 (4) ◽  
pp. 451-476
Author(s):  
Vasyl Ostrovskyi ◽  
Konrad Schmüdgen
Keyword(s):  
Quantum Plane ◽  
The Real ◽  
Resolvent Approach

Optimization of Stepsize in X-Ray Powder Diffractogram Collection

1988 ◽  
Vol 3 (1) ◽  
pp. 32-38 ◽  
Author(s):  
David G. Cameron ◽  
Ernest E. Armstrong
Keyword(s):  
Fourier Transform ◽  
Fourier Transforms ◽  
X Ray Diffraction ◽  
Step Size ◽  
Reasonable Estimate ◽  
X Ray ◽  
Fourier Transform Methods ◽  
Step Interval ◽  
The Fourier Transform ◽  
Selection Of

AbstractFourier transform methods of smoothing and interpolation are applied to X-ray diffraction data. It is shown that, frequently, too small a step size is used. Major gains are to be expected by selection of the optimum step size and use of these methods.A comparison of Fourier transforms of diffractograms of quartz measured between 67 and 69° 2θ, collected at varying step intervals (0.1 to 0.01° 2θ) was used to illustrate these applications. By examining the Fourier transform of the diffractogram and noting where it decays to die baseline, a reasonable estimate of the optimal step interval can be obtained. In addition, Fourier interpolation can be used to enhance the appearance of the diffractogram, approximating a continuous plot.

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