scholarly journals On the Connection between Spherical Laplace Transform and Non-Euclidean Fourier Analysis

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 287 ◽  
Author(s):  
Enrico De Micheli
Keyword(s):  
Fourier Transform ◽  
Fourier Analysis ◽  
Laplace Transform ◽  
Unitary Representation ◽  
Principal Series ◽  
Holomorphic Extension ◽  
Type Condition ◽  
Jump Function ◽  
Legendre Expansion ◽  
Laplace Type

We prove that, if the coefficients of a Fourier–Legendre expansion satisfy a suitable Hausdorff-type condition, then the series converges to a function which admits a holomorphic extension to a cut-plane. Next, we introduce a Laplace-type transform (the so-called Spherical Laplace Transform) of the jump function across the cut. The main result of this paper is to establish the connection between the Spherical Laplace Transform and the Non-Euclidean Fourier Transform in the sense of Helgason. In this way, we find a connection between the unitary representation of SO ( 3 ) and the principal series of the unitary representation of SU ( 1 , 1 ) .

scholarly journals A unified mode decomposition method for physical fields in homogeneous cosmology

2014 ◽  
Vol 26 (03) ◽  
pp. 1430001 ◽  
Author(s):  
Zhirayr G. Avetisyan
Keyword(s):  
Fourier Transform ◽  
Field Equation ◽  
Scalar Field ◽  
Vector Bundle ◽  
Fourier Analysis ◽  
Homogeneous Spaces ◽  
Unified Framework ◽  
Mode Decomposition ◽  
Physical Fields ◽  
Homogeneous Vector Bundle

The methods of mode decomposition and Fourier analysis of classical and quantum fields on curved spacetimes previously available mainly for the scalar field on Friedman–Robertson–Walker (FRW) spacetimes are extended to arbitrary vector bundle fields on general spatially homogeneous spacetimes. This is done by developing a rigorous unified framework which incorporates mode decomposition, harmonic analysis and Fourier analysis. The limits of applicability and uniqueness of mode decomposition by separation of the time variable in the field equation are found. It is shown how mode decomposition can be naturally extended to weak solutions of the field equation under some analytical assumptions. It is further shown that these assumptions can always be fulfilled if the vector bundle under consideration is analytic. The propagator of the field equation is explicitly mode decomposed. A short survey on the geometry of the models considered in mathematical cosmology is given and it is concluded that practically all of them can be represented by a semidirect homogeneous vector bundle. Abstract harmonic analytical Fourier transform is introduced in semidirect homogeneous spaces and it is explained how it can be related to the spectral Fourier transform. The general form of invariant bi-distributions on semidirect homogeneous spaces is found in the Fourier space which generalizes earlier results for the homogeneous states of the scalar field on FRW spacetimes.

scholarly journals Spectral Analysis of Traffic Functions in Urban Areas

2015 ◽  
Vol 27 (6) ◽  
pp. 477-484 ◽  
Author(s):  
Florin Nemtanu ◽  
Ilona Madalina Costea ◽  
Catalin Dumitrescu
Keyword(s):  
Spectral Analysis ◽  
Fourier Transform ◽  
Fourier Analysis ◽  
Traffic Flow ◽  
Traffic Control ◽  
Urban Areas ◽  
Urban Traffic ◽  
The Fourier Transform ◽  
The City ◽  
Different Time Scales

The paper is focused on the Fourier transform application in urban traffic analysis and the use of said transform in traffic decomposition. The traffic function is defined as traffic flow generated by different categories of traffic participants. A Fourier analysis was elaborated in terms of identifying the main traffic function components, called traffic sub-functions. This paper presents the results of the method being applied in a real case situation, that is, an intersection in the city of Bucharest where the effect of a bus line was analysed. The analysis was done using different time scales, while three different traffic functions were defined to demonstrate the theoretical effect of the proposed method of analysis. An extension of the method is proposed to be applied in urban areas, especially in the areas covered by predictive traffic control.

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.

New Laplace, z and Fourier-related transforms

2007 ◽  
Vol 463 (2081) ◽  
pp. 1179-1198 ◽  
Author(s):  
Michael J Corinthios
Keyword(s):  
Fourier Transform ◽  
Laplace Transform ◽  
Large Class ◽  
Large Family ◽  
Step Function ◽  
Dirac Delta ◽  
Mellin Transforms ◽  
Generalized Distributions ◽  
The Laplace Transform ◽  
The Fourier Transform

In this paper, the author uses his recently proposed complex variable generalized distribution theory to expand the domains of existence of bilateral Laplace and z transforms, as well as a whole new class of related transforms. A vast expansion of the domains of existence of bilateral Laplace and z transforms and continuous-time and discrete-time Hilbert, Hartley and Mellin transforms, as well as transforms of multidimensional functions and sequences are obtained. It is noted that the Fourier transform and its applications have advanced by leaps and bounds during the last century, thanks to the introduction of the theory of distributions and, in particular, the concept of the Dirac-delta impulse. Meanwhile, however, the truly two-sided ‘bilateral’ Laplace and z transforms, which are more general than Fourier, remained at a standstill incapable of transforming the most basic of functions. In fact, they were reduced by half to one-sided transforms and received no more than a passing reference in the literature. It is shown that the newly proposed generalized distributions expand the domains of existence and application of Laplace and z transforms similar to and even more extensively than the expansion of the domain of Fourier transform that resulted from the introduction, nearly a century ago, of the theory of distributions and the Dirac-delta impulse. It is also shown that the new generalized distributions put an end to an anomaly that still exists today, which meant that for a large class of basic functions, the Fourier transform exists while the more general Laplace and z transforms do not. The anomaly further manifests itself in the fact that even for the one-sided causal functions, such as the Heaviside unit step function u ( t ) and the sinusoid sin βtu ( t ), the Laplace transform does not exist on the j ω -axis, and the Fourier transform which does exist cannot be deduced thereof by the substitution s =j ω in the Laplace transform, which by definition it should. The extended generalized transforms are well defined for a large class of functions ranging from the most basic to highly complex fast-rising exponential ones that have so far had no transform. Among basic applications, the solution of partial differential equations using the extended generalized transforms is provided. This paper clearly presents and articulates the significant impact of extending the domains of Laplace and z transforms on a large family of related transforms, after nearly a century during which bilateral Laplace and z transforms of even the most basic of functions were undefined, and the domains of definition of related transforms such as Hilbert, Hartley and Mellin transforms were confined to a fraction of the space they can now occupy.

scholarly journals CLASSICAL SOLUTION OF THE CAUCHY PROBLEM FOR BIWAVE EQUATION: APPLICATION OF FOURIER TRANSFORM

2012 ◽  
Vol 17 (5) ◽  
pp. 630-641 ◽  
Author(s):  
Victor Korzyuk ◽  
Nguyen Van Vinh ◽  
Nguyen Tuan Minh
Keyword(s):  
Cauchy Problem ◽  
Exact Solution ◽  
Fourier Transform ◽  
Fourier Analysis ◽  
Classical Solution ◽  
Half Space ◽  
Analysis Techniques ◽  
The Cauchy Problem

In this paper, we use some Fourier analysis techniques to find an exact solution to the Cauchy problem for the n-dimensional biwave equation in the upper half-space ℝ n × [0, +∞).

Assessing the Influence of Fourier Analysis Parameters on Short-Time Modal Parameter Extraction

2012 ◽  
Vol 134 (3) ◽  
Author(s):  
Matthew Lamb ◽  
Vincent Rouillard
Keyword(s):  
Fourier Transform ◽  
Fourier Analysis ◽  
Natural Frequency ◽  
Spectral Resolution ◽  
Parameter Extraction ◽  
Modal Parameter ◽  
Zero Padding ◽  
Spectral Enhancement ◽  
The Fourier Transform ◽  
Modal Parameter Extraction

It is sometimes necessary to determine the manner in which materials and structures deteriorate with respect to time when subjected to sustained random dynamic loads. In such cases a system’s fatigue characteristics can be obtained by continuously monitoring its modal parameters. This allows for any structural deterioration, often manifested as a loss in stiffness, to be detected. Many common structural integrity assessment techniques make use of Fourier analysis for modal parameter extraction. For continual modal parameter extraction, the Fourier transform requires that a compromise be made between the accuracy of the estimates and how frequently they can be obtained. The limitations brought forth by this compromise can be significantly reduced by selecting suitable values for the analysis parameters, mainly subrecord length and number of averages. Further improvements may also be possible by making use of spectral enhancement techniques, specifically overlapped averaging and zero padding. This paper uses the statistical analysis of results obtained from numerous physical and numerical experiments to evaluate the influence of the analysis parameters and spectral enhancement techniques on modal estimates obtained from limited data sets. This evaluation will assist analysts in selecting the most suitable inputs for parameter extraction purposes. The results presented in this paper show that when using the Fourier transform to extract modal characteristics, any variation in the parameters used for analysis can have a significant influence on the extraction of natural frequency estimates from systems subjected to random excitation. It was found that for records containing up to 10% noise, subrecord length; hence spectral resolution, has a more pronounced influence on the accuracy of modal estimates than the level of spectral averaging; therefore spectral uncertainty. It was also found that while zero padding may not increase the actual spectral resolution, it does allow for improved natural frequency estimates with the introduction of interpolated estimates at the nondescribed frequencies. Finally, it was found that for modal parameter extraction purposes (in this case natural frequency), increased amounts of overlapped averaging can significantly reduce the variance of the estimates obtained. This is particularly useful as it allows for increased precision without compromising temporal resolution.

scholarly journals Waves due to initial disturbances at the inertial surface in a stratified fluid of finite depth

2000 ◽  
Vol 24 (4) ◽  
pp. 265-276 ◽  
Author(s):  
Prity Ghosh ◽  
Uma Basu ◽  
B. N. Mandal
Keyword(s):  
Fourier Transform ◽  
Laplace Transform ◽  
Mathematical Analysis ◽  
Large Time ◽  
Finite Depth ◽  
Integral Theorem ◽  
Poisson Problem ◽  
Inertial Surface ◽  
Green’S Integral Theorem ◽  
Stratified Ocean

This paper is concerned with a Cauchy-Poisson problem in a weakly stratified ocean of uniform finite depth bounded above by an inertial surface (IS). The inertial surface is composed of a thin but uniform distribution of noninteracting materials. The techniques of Laplace transform in time and either Green's integral theorem or Fourier transform have been utilized in the mathematical analysis to obtain the form of the inertial surface in terms of an integral. The asymptotic behaviour of the inertial surface is obtained for large time and distance and displayed graphically. The effect of stratification is discussed.

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