In this manuscript, we review Fractal calculus and the analogues of both local Fourier transform with its related properties and Fourier convolution theorem are proposed with proofs in Fractal calculus. The Fractal Dirac delta with its derivative and the Fractal Fourier transform of the Dirac delta is also defined. In addition, some important applications of the local Fractal Fourier transform are presented in this paper such as the Fractal electric current in a simple circuit, the Fractal second order ordinary differential equation, and the Fractal Bernoulli-Euler beam equation. All discussed applications are closely related to the fact that, in Fractal calculus, a useful local Fractal derivative is a generalized local derivative in the standard calculus sense. In addition, a comparative analysis is also carried out to explain the benefits of this Fractal calculus parameter on the basis of the additional alpha parameter, which is the dimension of the Fractal set, such that when α,= 1, we obtain the same results in the standard calculus.

In this paper, using a generalized translation operator, we prove the estimates for the generalized Fourier-Bessel transform in the space L2a,n on certain classes of functions.

The weighted semigroup algebra Mb (S, w) is studied via its identification with Mb (S) together with a weighted algebra product *w so that (Mb (S, w), *) is isometrically isomorphic to (Mb (S), *w). This identification enables us to study the relation between regularity and amenability of Mb (S, w) and Mb (S), and improve some old results from discrete to general case.

Let G be a locally compact group and w be a symmetric weight function on G. We define a co-product Gw on the weighted algebra L¥ (G, w-1) of essentially w-bounded Borel measurable functions on G and show that L¥ (G, w-1) becomes a Kac algebra with natural co-inverse kw and Haar weight  fw. We use the machinery of Kac algebras to introduce the weighted Fourier and Fourier-Stieltjes algebra A(G, w-1) and B(G, w-1) of G.

In this work, the set of all functions that are Fourier transformable with regard to their structure both algebraic and topological is taken into account. Certain topological properties of the set of Fourier transformable functions with the help of a metric are described. Also determines the proofs of the statements that the set of all Fourier transformable functions is a commutative semi-group with respect to the convolution operation as well as Abelian group with respect to the operation of addition. Metric for two functions belonging to the set of all Fourier transformable functions is defined and the proof that the Fourier transformable functions space is complete with our metric is given. The separability theorem and that the Fourier transformable functions space is disconnected are also discussed.

Let G be a locally compact Hausdorff topological group and H be a compact subgroup of G. Then, the homogeneous space G/H possesses a specific Radon measure, which is called a relatively invariant measure. We show that the concepts of convolution and involution can be extended to the integrable functions defined on this homogeneous space. We study the properties of convolution and prove that the space of integrable functions is an involutive Banach algebra with an approximate identity. We also find a necessary and sufficient condition on a closed subspace of this Banach algebra to make it a left ideal.