Convolutions for the Fourier transforms with geometric variables and applications

Abstract

This paper gives a general formulation of convolutions for arbitrary linear operators from a linear space to a commutative algebra, constructs three convolutions for the Fourier transforms with geometric variables and four generalized convolutions for the Fourier-cosine, Fourier-sine transforms. With respect to applications, by using the constructed convolutions normed rings on L1(R n ) are constructed, and explicit solutions of integral equations of convolution type are obtained.

Two New Convolutions for the Fractional Fourier Transform

“…However, when α = π/2 the second one turns out to be the Fourier case as showed above in Almeida's case. -Equations ( 16) and ( 17) in [23,Theorem 1] are in fact generalized convolution and product theorems (see [27,28]). In this work, the authors use the linear canonical transform (LCT) which is a result of parameterizing the kernel of FRFT by four items.…”

“…Those convolutions are very interesting, and applicable to both theoretical and practical problems as they may be viewed as extensions of the convolution theorem of the Fourier transform. Namely, a convolution transform, mathematically, is diagonalized by another transform; and in the new (momentum) representation a convolution turns into an operator of multiplication by a function (see [27,28]). An interesting description of the history of the development of convolutions for FRFT and their potential applications was addressed in [22].…”

Operational Properties of Two Integral Transforms of Fourier Type and their Convolutions

“…One reason for this is that they have many applications in pure and applied mathematics (see ), Vladimirov [23] and references therein). Each convolution is a new transform which can be an object of study (see [4,5,6,10,20,21,22]). Moreover, convolution is a mathematical way of combining two signals to form a third signal, which is a very important technique in digital signal processing (see Smith [14]).…”

The Solvability of a Class of Convolution Equations Associated with 2D FRFT

“…In recent years, convolution-type singular integral equations have received increasing attention from many mathematicians due to the wide applications in the field of engineering mechanics, fracture mechanics, and so on. They have formed a relatively perfect theoretical system [1][2][3][4][5][6][7][8]. Solvability is one of the essential issues of equation theory, and has been studied in-depth by many researchers [9][10][11][12][13].…”