We study the compactness of the embedding between Besov spaces defined on some type of isotropic fractal sets in the Euclidean space. The “degree of compactness” of such an embedding is expressed in terms of its entropy numbers.

We consider the question of the regularity of solutions of one model equation related to a fractal set in the scale of Besov spaces. The theorem of existence of solution with the optimal regularity is proved.

We prove, in standard notation from spectral theory, the following asymptotic formulas: $$\sum_{\kappa_{j}\leqslant K} \alpha_{j}H^3_j(\frac{1}{2}) = K^2 P_3 (\log K) + O(K^{5/4} \log^{37/4}K)$$ and $$\sum_{\kappa_{j}\leqslant K} \alpha_{j}H^4_j(\frac{1}{2}) = K^2 P_6 (\log K) + O(K^{3/2} \log^{25/2}K),$$ where $P_3 (x)$ and $P_6 (x)$ are polynomials of degree three and six, whose coefficients may be explicitly evaluated.

We consider algebras endowed with a generalized involution. We show that $\vert \cdot \vert ^{\frac{1}{p}}_{p}$ is a $C^*$-seminorm, for every a $p$-seminorm $\vert \cdot \vert_p$, $0 \lt p \leqslant 1$, which satisfies the $C^*$-property.

In this paper, we study certain density theorems of the zeros of the symmetric square $L$-functions (attached to a holomorphic cuspform defined over the full modular group) to the right of the critical line. We also obtain an analogous density theorem of Halasz and Turan for the zeros of the symmetric square $L$-functions.

Let $\mathcal{M}$ be the class of all multiplicatively convex homological algebras. Let $A$ be in $\mathcal{M}$. We show that an element $x \in A \in \mathcal{M}$ is permanently singular in the class $\mathcal{M}$ if and only if $x$ is an m-bounding (Déf. 4). An m-bounding element is not necessarily a bounding element. This shows that the concept of $\mathcal{M}$-singularity is of relative character.

We discuss the st.ructure of the sets where the best $m$-term approximation with respect to a natural biorthogonal system is attained.