81 research outputs found
Octonion special affine fourier transform: pitt's inequality and the uncertainty principles
The special affine Fourier transform (SAFT) is an extended version of the classical Fourier transform and incorporates various signal processing tools which include the Fourier transforms, the fractional Fourier transform, the linear canonical transform, and other related transforms. This paper aims to introduce a novel octonion special affine Fourier transform (O-SAFT) and establish several classes of uncertainty inequalities for the proposed transform. We begin by studying the norm split and energy conservation properties of the proposed (O-SAFT). Afterwards, we generalize several uncertainty relations for the (O-SAFT) which include Pitt's inequality, Heisenberg-Weyl inequality, logarithmic uncertainty inequality, Hausdorff-Young inequality, and local uncertainty inequalities. Finally, we provide an illustrative example and some possible applications of the proposed transform
A quantum wavelet uncertainty principle
The aim of this paper is to derive a new uncertainty principle for the
generalized -Bessel wavelet transform studied earlier in \cite{Rezguietal}.
In this paper, an uncertainty principle associated with wavelet transforms in
the -calculus framework has been established. A two-parameters extension of
the classical Bessel operator is applied to generate a wavelet function which
is exploited next to explore a wavelet uncertainty principle already in the
-calculus framework.Comment: 16 page
Clifford algebras, Fourier transforms and quantum mechanics
In this review, an overview is given of several recent generalizations of the
Fourier transform, related to either the Lie algebra sl_2 or the Lie
superalgebra osp(1|2). In the former case, one obtains scalar generalizations
of the Fourier transform, including the fractional Fourier transform, the Dunkl
transform, the radially deformed Fourier transform and the super Fourier
transform. In the latter case, one has to use the framework of Clifford
analysis and arrives at the Clifford-Fourier transform and the radially
deformed hypercomplex Fourier transform. A detailed exposition of all these
transforms is given, with emphasis on aspects such as eigenfunctions and
spectrum of the transform, characterization of the integral kernel and
connection with various special functions.Comment: Review paper, 39 pages, to appear in Math. Methods. Appl. Sc
Convolution theorems associated with quaternion linear canonical transform and applications
Novel types of convolution operators for quaternion linear canonical
transform (QLCT) are proposed. Type one and two are defined in the spatial and
QLCT spectral domains, respectively. They are distinct in the quaternion space
and are consistent once in complex or real space. Various types of convolution
formulas are discussed. Consequently, the QLCT of the convolution of two
quaternionic functions can be implemented by the product of their QLCTs, or the
summation of the products of their QLCTs. As applications, correlation
operators and theorems of the QLCT are derived. The proposed convolution
formulas are used to solve Fredholm integral equations with special kernels.
Some systems of second-order partial differential equations, which can be
transformed into the second-order quaternion partial differential equations,
can be solved by the convolution formulas as well. As a final point, we
demonstrate that the convolution theorem facilitates the design of
multiplicative filters
Relation between Quaternion Fourier Transform and Quaternion Wigner-Ville Distribution Associated with Linear Canonical Transform
The quaternion Wigner-Ville distribution associated with linear canonical transform (QWVD-LCT) is a nontrivial generalization of the quaternion Wigner-Ville distribution to the linear canonical transform (LCT) domain. In the present paper, we establish a fundamental relationship between the QWVD-LCT and the quaternion Fourier transform (QFT). Based on this fact, we provide alternative proof of the well-known properties of the QWVD-LCT such as inversion formula and Moyal formula. We also discuss in detail the relationship among the QWVD-LCT and other generalized transforms. Finally, based on the basic relation between the quaternion ambiguity function associated with the linear canonical transform (QAF-LCT) and the QFT, we present some important properties of the QAF-LCT
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