Methods of Functional Analysis
and Topology

Using the new definition of the $\mathcal{N}^{\alpha}_F$-derivative function introduced by Juan E. Nápoles Valdés and al. (2020), we provide a new definition for the $\mathcal{N}^{\alpha}_F$-Sumudu transform, $\mathcal{N}^{\alpha}_F$-Sumudu conformable transform. Additionally, we establish several important results related to these new transforms. We also give a new definition of convolution related to this $\mathcal{N}^{\alpha}_F$-derivative and we show that it is commutative and associative.

This paper is devoted to the study of hyperstability phenomena in the context of convex modular spaces. In particular, we investigate the hyperstability of three fundamental functional equations: the quadratic equation \begin{equation} \varphi(x+y)+\varphi(x-y)=2\varphi(x)+2\varphi(y) \end{equation} the general linear equation \begin{equation} \varphi(ax+by)=A\varphi(x)+B\varphi(y)+C \end{equation} and the $n$-dimensional quadratic equation \begin{equation} f\left(\sum_{i=1}^{m}x_{i}\right)+\sum_{1\leq i < j\leq m}f\big(x_{i}-x_{j}\big)=m\sum_{i=1}^{m}f(x_{i}). \end{equation} Using the direct method, we establish sufficient conditions under which every approximate solution of these equations in modular spaces coincides exactly with an exact solution. Our results extend earlier contributions obtained in Banach spaces via fixed point techniques, and provide new insights into the stability of functional equations in the broader context of modular spaces.

In this paper, the Laguerre-Bessel wavelet packets transform is defined and studied. The scale discrete scaling function and the associated Plancherel and inversion formulas are given and established. Furthermore, the Calderón reproducing formula is given and proved for the proposed transform.

The paper is devoted to the study of conditions for the Hausdorff-Besicovitch faithfulness of the family of cylinders generated by Cantor series expansions. We show that there exist subgeometric Cantor series expansions for which the corresponding families of cylinders are not faithful for the Hausdorff-Besicovitch dimension on the unit interval. On the other hand we found a rather wide subfamily of subgeometric Cantor series expansions generating faithful families of cylinders.

We also study conditions for the Hausdorff-Besicovitch dimension preservation on [0;1] by probability distribution functions of random variables with independent symbols of arithmetic Cantor series expansions.