Related papers: Fourier transforms, fractional derivatives, and a …
A new definition of a fractional derivative has recently been developed, making use of a fractional Dirac delta function as its integral kernel. This derivative allows for the definition of a distributional fractional derivative, and as…
In the present article the author extends the Fourier transform to a more general class of functions; First to power-law functions with integer and half-integer exponents then to the widely used quantum statistics function (Fermi-Dirac and…
In the present article, a new method for the evaluation of fractional derivatives of arbitrary real order is proposed. Numerous but inequivalent formulations have been given in the past. Some of them exhibit unsatisfactory properties such…
The Fourier transform is naturally defined for integrable functrions. Otherwise, it should be stipulated in which sense the Fourier transform is understood. We consider some class of radial and, generally saying, nonintegrable functions.…
Definitions of fractional derivatives as fractional powers of derivative operators are suggested. The Taylor series and Fourier series are used to define fractional power of self-adjoint derivative operator. The Fourier integrals and Weyl…
A new derivative, called deformable derivative, is introduced here which is equivalent to ordinary derivative in the sense that one implies other. The deformable derivative is defined using limit approach like that of ordinary one but with…
In this paper, a Fourier series in fractional dimensional space is introduced for an arbitrarily periodic function $f(t;\alpha)$. We call it fractional Fourier series of the order $\alpha$. Extending the basis functions of the linear space…
We find a formula that relates the Fourier transform of a radial function on $\mathbf{R}^n$ with the Fourier transform of the same function defined on $\mathbf{R}^{n+2}$. This formula enables one to explicitly calculate the Fourier…
The Fourier transform of a bounded measurable function, $f$, on the real line is shown to be the second distributional derivative of a H\"older continuous function. The Fourier transform is written as the difference of $\int_{-1}^1…
We consider the set of power functions defined on the set of positive real number, and their linear combinations. After recalling some properties of the gamma function, we give two general definitions of derivatives of positive and negative…
Fractional analysis is applied to describe classical dynamical systems. Fractional derivative can be defined as a fractional power of derivative. The infinitesimal generators {H, .} and L=G(q,p) \partial_q+F(q,p) \partial_p, which are used…
We study Fourier theory on quantum Euclidean space. A modified version of the general definition of the Fourier transform on a quantum space is used and its inverse is constructed. The Fourier transforms can be defined by their Bochner's…
Fractals equipped with intrinsic arithmetic lead to a natural definition of differentiation, integration and complex numbers. Applying the formalism to the problem of a Fourier transform on fractals we show that the resulting transform has…
We give a rigorous derivation of the Fourier transform of the Heaviside function within a framework for tempered distributions that is suitable for undergraduate engineering and mathematics students. The proofs rely on fundamental concepts…
The transformation of the partial fractional derivatives under spatial rotation in $R^2$ are derived for the Riemann-Liouville and Caputo definitions. These transformation properties link the observation of physical quantities, expressed…
Fourier and fractional-Fourier transformations are widely used in theoretical physics. In this paper we make quantum perspectives and generalization for the fractional Fourier transformation (FrFT). By virtue of quantum mechanical…
In this article we study the basic theoretical properties of Mellin-type fractional integrals, known as generalizations of the Hadamard-type fractional integrals. We give a new approach and version, specifying their semigroup property,…
The transformation property of the Caputo fractional derivative operator of a scalar function under rotation in two dimensional space is derived. The study of the transformation property is essential for the formulation of fractional…
In this article, we introduce a new general definition of fractional derivative and fractional integral, which depends on an unknown kernel. By using these definitions, we obtain the basic properties of fractional integral and fractional…
The paper deals with a fractional derivative introduced by means of the Fourier transform. The explicit form of the kernel of general derivative operator acting on the functions analytic on a curve in complex plane is deduced and the…