Multicenter molecular integrals of spherical Gaussian functions by Fourier transform convolution theorem

The aim of this article is to introduce a new definition for the Fourier transform. This new definition will be considered as one of the generalizations of the usual (classical) Fourier transform. We employ the new Katugampola derivative to obtain some properties of the Katugampola Fourier transform and find the relation between the Katugampola Fourier transform and the usual Fourier transform. The inversion formula and the convolution theorem for the Katugampola Fourier transform are considered.

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The linear canonical wavelet transform on some function spaces

International Journal of Wavelets Multiresolution and Information Processing ◽

10.1142/s0219691318500108 ◽

2018 ◽

Vol 16 (01) ◽

pp. 1850010 ◽

Cited By ~ 12

Author(s):

Yong Guo ◽

Bing-Zhao Li

Keyword(s):

Fourier Transform ◽

Wavelet Transform ◽

Continuous Operator ◽

Function Spaces ◽

Linear Continuous Operator ◽

Convolution Theorem ◽

Linear Canonical Transform ◽

Equivalent Definition ◽

New Space ◽

Definition Of

It is well known that the domain of Fourier transform (FT) can be extended to the Schwartz space [Formula: see text] for convenience. As a generation of FT, it is necessary to detect the linear canonical transform (LCT) on a new space for obtaining the similar properties like FT on [Formula: see text]. Therefore, a space [Formula: see text] generalized from [Formula: see text] is introduced firstly, and further we prove that LCT is a homeomorphism from [Formula: see text] onto itself. The linear canonical wavelet transform (LCWT) is a newly proposed transform based on the convolution theorem in LCT domain. Moreover, we propose an equivalent definition of LCWT associated with LCT and further study some properties of LCWT on [Formula: see text]. Based on these properties, we finally prove that LCWT is a linear continuous operator on the spaces of [Formula: see text] and [Formula: see text].

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Two-Dimensional Quaternion Linear Canonical Transform: Properties, Convolution, Correlation, and Uncertainty Principle

Journal of Mathematics ◽

10.1155/2019/1062979 ◽

2019 ◽

Vol 2019 ◽

pp. 1-13 ◽

Cited By ~ 1

Author(s):

Mawardi Bahri ◽

Ryuichi Ashino

Keyword(s):

Fourier Transform ◽

Uncertainty Principle ◽

Linear Canonical Transform ◽

Two Dimensional ◽

Quaternion Fourier Transform ◽

Gaussian Functions ◽

The Fourier Transform ◽

Definition Of ◽

Quaternion Linear Canonical Transform

A definition of the two-dimensional quaternion linear canonical transform (QLCT) is proposed. The transform is constructed by substituting the Fourier transform kernel with the quaternion Fourier transform (QFT) kernel in the definition of the classical linear canonical transform (LCT). Several useful properties of the QLCT are obtained from the properties of the QLCT kernel. Based on the convolutions and correlations of the LCT and QFT, convolution and correlation theorems associated with the QLCT are studied. An uncertainty principle for the QLCT is established. It is shown that the localization of a quaternion-valued function and the localization of the QLCT are inversely proportional and that only modulated and shifted two-dimensional Gaussian functions minimize the uncertainty.

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