scholarly journals The Generalized Fourier Transform: A Unified Framework for the Fourier, Laplace, Mellin and Z Transforms

2021 ◽  
Author(s):  
Pushpendra Singh ◽  
Anubha Gupta ◽  
Shiv Dutt Joshi
Keyword(s):  
Fourier Transform ◽  
Random Variable ◽  
Integral Transforms ◽  
Exponential Map ◽  
Generalized Fourier Transform ◽  
Unified Framework ◽  
Special Cases ◽  
The Fourier Transform ◽  
Generalized Moments ◽  
Generalized Gamma Function

<div>This paper introduces Generalized Fourier transform (GFT) that is an extension or the generalization of the Fourier transform (FT). The Unilateral Laplace transform (LT) is observed to be the special case of GFT. GFT, as proposed in this work, contributes significantly to the scholarly literature. There are many salient contribution of this work. Firstly, GFT is applicable to a much larger class of signals, some of which cannot be analyzed with FT and LT. For example, we have shown the applicability of GFT on the polynomially decaying functions and super exponentials. Secondly, we demonstrate the efficacy of GFT in solving the initial value problems (IVPs). Thirdly, the generalization presented for FT is extended for other integral transforms with examples shown for wavelet transform and cosine transform. Likewise, generalized Gamma function is also presented. One interesting application of GFT is the computation of generalized moments, for the otherwise non-finite moments, of any random variable such as the Cauchy random variable. Fourthly, we introduce Fourier scale transform (FST) that utilizes GFT with the topological isomorphism of an exponential map. Lastly, we propose Generalized Discrete-Time Fourier transform (GDTFT). The DTFT and unilateral $z$-transform are shown to be the special cases of the proposed GDTFT. The properties of GFT and GDTFT have also been discussed. </div><div><br></div>

scholarly journals The Generalized Fourier Transform: A Unified Framework for the Fourier, Laplace, Mellin and Z Transforms

2021 ◽  
Author(s):  
Pushpendra Singh ◽  
Anubha Gupta ◽  
Shiv Dutt Joshi
Keyword(s):  
Fourier Transform ◽  
Random Variable ◽  
Integral Transforms ◽  
Exponential Map ◽  
Generalized Fourier Transform ◽  
Unified Framework ◽  
Special Cases ◽  
The Fourier Transform ◽  
Generalized Moments ◽  
Generalized Gamma Function

<div>This paper introduces Generalized Fourier transform (GFT) that is an extension or the generalization of the Fourier transform (FT). The Unilateral Laplace transform (LT) is observed to be the special case of GFT. GFT, as proposed in this work, contributes significantly to the scholarly literature. There are many salient contribution of this work. Firstly, GFT is applicable to a much larger class of signals, some of which cannot be analyzed with FT and LT. For example, we have shown the applicability of GFT on the polynomially decaying functions and super exponentials. Secondly, we demonstrate the efficacy of GFT in solving the initial value problems (IVPs). Thirdly, the generalization presented for FT is extended for other integral transforms with examples shown for wavelet transform and cosine transform. Likewise, generalized Gamma function is also presented. One interesting application of GFT is the computation of generalized moments, for the otherwise non-finite moments, of any random variable such as the Cauchy random variable. Fourthly, we introduce Fourier scale transform (FST) that utilizes GFT with the topological isomorphism of an exponential map. Lastly, we propose Generalized Discrete-Time Fourier transform (GDTFT). The DTFT and unilateral $z$-transform are shown to be the special cases of the proposed GDTFT. The properties of GFT and GDTFT have also been discussed. </div><div><br></div>

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.

scholarly journals Uncertainty principles invariant under the fractional Fourier transform

1991 ◽  
Vol 33 (2) ◽  
pp. 180-191 ◽  
Author(s):  
David Mustard
Keyword(s):  
Fourier Transform ◽  
Fourier Transforms ◽  
Fractional Fourier Transform ◽  
Intrinsic Property ◽  
Continuous Group ◽  
Uncertainty Principles ◽  
New Family ◽  
Special Cases ◽  
Fractional Fourier Transforms ◽  
The Fourier Transform

AbstractUncertainty principles like Heisenberg's assert an inequality obeyed by some measure of joint uncertainty associated with a function and its Fourier transform. The more groups under which that measure is invariant, the more that measure represents an intrinsic property of the underlying object represented by the given function. The Fourier transform is imbedded in a continuous group of operators, the fractional Fourier transforms, but the Heisenberg measure of overall spread turns out not to be invariant under that group. A new family is developed of measures that are invariant under the group of fractional Fourier transforms and that obey associated uncertainty principles. The first member corresponds to Heisenberg's measure but is generally smaller than his although equal to it in special cases.

scholarly journals A New Representation for Srivastava’s λ-Generalized Hurwitz-Lerch Zeta Functions

Symmetry ◽  
2018 ◽  
Vol 10 (12) ◽  
pp. 733 ◽  
Author(s):  
Asifa Tassaddiq
Keyword(s):  
Fourier Transform ◽  
Series Representation ◽  
Zeta Functions ◽  
Special Cases ◽  
Duality Property ◽  
The Fourier Transform

Taking inspiration principally from some of the latest research, we develop a new series representation for the λ-generalized Hurwitz-Lerch zeta functions. This representation led to important new results. The Fourier transform played a foundational role in this work. The duality property of the Fourier transform became significant for checking the consistency of the results. Some known data has been verified as special cases of the results obtained in this investigation.

scholarly journals A modified convolution and product theorem for the linear canonical transform derived by representation transformation in quantum mechanics

2013 ◽  
Vol 23 (3) ◽  
pp. 685-695 ◽  
Author(s):  
Navdeep Goel ◽  
Kulbir Singh
Keyword(s):  
Fourier Transform ◽  
Quantum Mechanics ◽  
Fractional Fourier Transform ◽  
Integral Transform ◽  
Linear Canonical Transform ◽  
Product Theorem ◽  
Special Cases ◽  
Fresnel Transform ◽  
The Fourier Transform ◽  
Representation Transformation

Abstract The Linear Canonical Transform (LCT) is a four parameter class of integral transform which plays an important role in many fields of signal processing. Well-known transforms such as the Fourier Transform (FT), the FRactional Fourier Transform (FRFT), and the FreSnel Transform (FST) can be seen as special cases of the linear canonical transform. Many properties of the LCT are currently known but the extension of FRFTs and FTs still needs more attention. This paper presents a modified convolution and product theorem in the LCT domain derived by a representation transformation in quantum mechanics, which seems a convenient and concise method. It is compared with the existing convolution theorem for the LCT and is found to be a better and befitting proposition. Further, an application of filtering is presented by using the derived results.

On integral transforms whose kernels are solutions of singular Sturm–Liouville problems

1988 ◽  
Vol 108 (3-4) ◽  
pp. 201-228 ◽  
Author(s):  
Ahmed I. Zayed
Keyword(s):  
Fourier Transform ◽  
Asymptotic Expansions ◽  
Spectral Measure ◽  
Liouville Problem ◽  
Integral Transforms ◽  
Tempered Distributions ◽  
Sturm Liouville ◽  
The Fourier Transform ◽  
Image Position ◽  
Special Case

SynopsisIn this paper we investigate integral transforms of type , where φ(x, s) is the solution of the singular Sturm–Liouville problem: y″ + (s2 – q(x))y = 0, 0≦x <∞ with y(0) cos α + y′(0)sin α = 0, y(x) is bounded at ∞, and dp is the spectral measure. If F(s) = sk for some k = 0, 1, 2, …, then f(x) may not exist since, in general, φ(x, s) is not even in . One aim of this paper is to investigate the Abel summability of these integrals. In the special case where q(x) = 0 and α = π/2, then φ(x, s) = cos sx and dp = ds, while if α = 0, then φ(x, s) = −sin sx/s and dp = s2ds. It is known thatwhere the values of these integrals are interpreted as the Abel limits of these integrals or as the Fourier transform of some tempered distributions. Another aim of this paper is to derive the analogue of these results for the general kernel φ(x, s), and then apply that to the theory of asymptotic expansions.

scholarly journals The class Bp for weighted generalized Fourier transform inequalities

2015 ◽  
Vol 14 (1) ◽  
pp. 121-133
Author(s):  
Chokri Abdelkefi ◽  
Mongi Rachdi
Keyword(s):  
Fourier Transform ◽  
Dunkl Transform ◽  
Weighted Inequalities ◽  
Generalized Fourier Transform ◽  
Class B ◽  
Power Weights ◽  
Pitt’S Inequality ◽  
The Fourier Transform

Abstract In the present paper, we prove weighted inequalities for the Dunkl transform (which generalizes the Fourier transform) when the weights belong to the well-known class Bp. As application, we obtain the Pitt’s inequality for power weights.

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