Methods of Functional Analysis
and Topology

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.

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.

Any abstract Gram operator is consistent with some graph. For an arbitrary operator $B_\Gamma$ that is consistent with a graph $\Gamma$, the question arises as to when it is an abstract Gram operator, i.e., whether it is nonnegative. We study this question for certain types of graphs. The simplest case is a star graph. Next, we use the results obtained for star graphs to explore a more general case, where a graph $\Gamma$ can be treated as a collection of rooted trees, with their roots connected by additional edges into a connected subgraph $\Gamma_0$. The work shows that the question about the nonnegativity of an operator $B_\Gamma$ for such a graph can be reduced to the corresponding question for some operator that is consistent with the subgraph $\Gamma_0$.

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.