Some Results Relating the Behaviour of Fourier Transforms Near the Origin and at Infinity

1969 ◽  
Vol 21 ◽  
pp. 942-950 ◽  
Author(s):  
C. Nasim
Keyword(s):  
Fourier Transforms ◽  
Fourier Inversion Formula ◽  
Fourier Cosine ◽  
Summation Formulae ◽  
Sine Transform ◽  
Fourier Sine ◽  
Fourier Sine Transform ◽  
Image Position ◽  
Watson Transform ◽  
Special Case

It is known that under special conditions, Fourier sine transforms and Fourier cosine transforms behave asymptotically like a power of x, either as x → 0 or as x → ∞ or both. For example (3),where f(x) = x–αϕ(x), 0 < α < 1, and ϕ(x) is of bounded variation in (0, ∞) and Fc(x) is the Fourier cosine transform of f(x). This suggests that other results connecting the behaviour of a function at infinity with the behaviour of its Fourier or Watson transform near the origin might exist. In this paper wre derive various such results. For example, a special case of these results iswhere f(x) is the Fourier sine transform of g(x). It should be noted that the Fourier inversion formula fails to give f(+0) directly in this case. Some applications of these results to show the relationships between various forms of known summation formulae are given.

scholarly journals On certain triple integral equations with trigonometric kernels

1976 ◽  
Vol 17 (2) ◽  
pp. 103-105 ◽  
Author(s):  
D. C. Stocks
Keyword(s):  
Integral Equations ◽  
Sufficient Conditions ◽  
Triple Integral Equations ◽  
Sine Transform ◽  
Fourier Sine ◽  
Fourier Sine Transform ◽  
Image Position ◽  
Special Case

In this note we formally solve the following triple integral equations,where f1(x), f2(x) and f3(x) are integrable for 0<x<α, α<x<β and β<x<∞, respectively, and the function g(λ) is assumed to satisfy sufficient conditions for the Fourier sine transform to exist. A special case of this system arose in a problem concerned with transistors.

FOURIER SINE AND COSINE TRANSFORMS ON BOEHMIAN SPACES

2013 ◽  
Vol 06 (01) ◽  
pp. 1350005 ◽  
Author(s):  
R. Roopkumar ◽  
E. R. Negrin ◽  
C. Ganesan
Keyword(s):  
Fourier Transform ◽  
Cosine Transform ◽  
Fourier Cosine ◽  
Sine Transform ◽  
Fourier Sine ◽  
Fourier Sine Transform ◽  
Fourier Cosine Transform ◽  
One To One ◽  
The Fourier Transform

We construct suitable Boehmian spaces which are properly larger than [Formula: see text] and we extend the Fourier sine transform and the Fourier cosine transform more than one way. We prove that the extended Fourier sine and cosine transforms have expected properties like linear, continuous, one-to-one and onto from one Boehmian space onto another Boehmian space. We also establish that the well known connection among the Fourier transform, Fourier sine transform and Fourier cosine transform in the context of Boehmians. Finally, we compare the relations among the different Boehmian spaces discussed in this paper.

scholarly journals A Computational Method for n-Dimensional Laplace Transforms Involved with Fourier Cosine Transform

2013 ◽  
Vol 2013 ◽  
pp. 1-4
Author(s):  
Jafar Saberi-Nadjafi
Keyword(s):  
Laplace Transform ◽  
Laplace Transforms ◽  
Computational Method ◽  
Cosine Transform ◽  
Fourier Cosine ◽  
Sine Transform ◽  
Fourier Sine ◽  
Fourier Sine Transform ◽  
Fourier Cosine Transform

In 2007, the author published some results on n-dimensional Laplace transform involved with the Fourier sine transform. In this paper, we propose some new result in n-dimensional Laplace transforms involved with Fourier cosine transform; these results provide few algorithms for evaluating some n-dimensional Laplace transform pairs. In addition, some examples are also presented, which explain the useful applications of the obtained results. Therefore, one can produce some two- and three- as well as n-dimensional Laplace transforms pairs.

scholarly journals Application of local fractional fourier sine transform for 1-D local fractional heat transfer equation

2018 ◽  
Vol 22 (4) ◽  
pp. 1729-1735
Author(s):  
Cheng-Dong Wei ◽  
Guang-Sheng Chen
Keyword(s):  
Heat Transfer ◽  
Transfer Equation ◽  
New Technology ◽  
Heat Transfer Equation ◽  
Fourier Cosine ◽  
Sine Transform ◽  
Fourier Sine ◽  
Fourier Sine Transform ◽  
Fractal Space ◽  
Fractional Differential

This paper proposes a new method called the local fractional Fourier sine transform to solve fractional differential equations on a fractal space. The method takes full advantages of the Yang-Fourier transform, the local fractional Fourier cosine, and sine transforms. A 1-D local fractional heat transfer equation is used as an example to reveal the merits of the new technology, and the example can be used as a paradigm for other applications.

Modeling electro-osmotic flow and thermal transport of Caputo fractional Burgers fluid through a micro-channel

2021 ◽  
pp. 095440892110259
Author(s):  
Mohammed Abdulhameed ◽  
Garba Tahiru Adamu ◽  
Gulibur Yakubu Dauda
Keyword(s):  
Electric Field ◽  
Laplace Transform ◽  
Inverse Laplace Transform ◽  
Micro Channel ◽  
Osmotic Flow ◽  
Sine Transform ◽  
Fourier Sine ◽  
Fourier Sine Transform ◽  
Burgers Fluid ◽  
Electro Osmotic Flow

In this paper, we construct transient electro-osmotic flow of Burgers’ fluid with Caputo fractional derivative in a micro-channel, where the Poisson–Boltzmann equation described the potential electric field applied along the length of the microchannel. The analytical solution for the component of the velocity profile was obtained, first by applying the Laplace transform combined with the classical method of partial differential equations and, second by applying Laplace transform combined with the finite Fourier sine transform. The exact solution for the component of the temperature was obtained by applying Laplace transform and finite Fourier sine transform. Further, due to the complexity of the derived models of the governing equations for both velocity and temperature, the inverse Laplace transform was obtained with the aid of numerical inversion formula based on Stehfest's algorithms with the help of MATHCAD software. The graphical representations showing the effects of the time, retardation time, electro-kinetic width, and fractional parameters on the velocity of the fluid flow and the effects of time and fractional parameters on the temperature distribution in the micro-channel were presented and analyzed. The results show that the applied electric field, electro-osmotic force, electro-kinetic width, and relaxation time play a vital role on the velocity distribution in the micro-channel. The fractional parameters can be used to regulate both the velocity and temperature in the micro-channel. The study could be used in the design of various biomedical lab-on-chip devices, which could be useful for biomedical diagnosis and analysis.

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.

The Parseval formulae for monotonic functions. I

1947 ◽  
Vol 43 (3) ◽  
pp. 289-306 ◽  
Author(s):  
Sheila M. Edmonds
Keyword(s):  
Fourier Transforms ◽  
Finite Interval ◽  
Fourier Cosine ◽  
Know How ◽  
Monotonic Functions ◽  
Mean Convergence ◽  
Cauchy Integrals ◽  
Good For ◽  
Image Position

The Parseval formulae for Fourier cosine and sine transforms,are of course most widely known in connexion with the classical theorems of Plancherel on functions of the class L2 (whose transforms are defined by mean convergence), and with their generalizations. We cannot expect to obtain anything as elegant as the ‘L2’ results when we consider (1) for functions of other kinds. Nevertheless, since the most obvious way of defining Fourier transforms is by means of Lebesgue or Cauchy integrals, we naturally wish to know how far the formulae (1) hold good for transforms obtained in this way. The two most familiar classes of functions having such transforms are:(i) functions f(t) integrable in the Lebesgue sense in (0, ∞), whose transforms Fe(x) and Fs(x) are defined by the Lebesgue integrals respectively; and(ii) functions f(t) which decrease in (0, ∞), tend to zero as t → ∞, and are integrable over any finite interval (0, T); in this case the transforms are defined by the Cauchy integrals .

Sign in / Sign up

Export Citation Format

Share Document