arXiv subjects

Mostafa Nasri

Publications and source records attributed to Mostafa Nasri.

3 recordsLinked to original sources

arXiv2025

Functional Calculi, Positivity, and Convolution of Matrices

Convolution admits a natural formulation as a functional operation on matrices. Motivated by the functional and entrywise calculi, this leads to a framework in which convolution defines a matrix transform that preserves positivity. Within this setting, we establish results parallel to the classical theories of P\'olya--Szeg\H{o}, Schoenberg, Rudin, Loewner, and Horn in the context of entrywise calculus. The structure of our transform is governed by a Cayley--Hamilton-type theory valid in commutative rings of characteristic zero, together with a novel polynomial-matrix identity specific to convolution. Beyond these analytic aspects, we uncover an intrinsic connection between convolution and the Bruhat order on the symmetric group, illuminating the combinatorial aspect of this functional operation. This work extends the classical theory of entrywise positivity preservers and operator monotone functions into the convolutional setting.

math.FA↗

arXiv2025

Schauder Basis with Finite Blaschke Products

We construct a Schauder basis for the space $Hol(\mathbb D)$, the space of holomorphic functions on the closed unit disk, consisting entirely of finite Blaschke products. The expansion coefficients are given explicitly. Our result remains valid when $Hol(\mathbb D)$ is equipped with a broader class of norms satisfying natural structural conditions. These conditions are satisfied by norms of classical function spaces such as the Hardy spaces $H^p$ ($1\leq p\leq \infty$), the weighted Bergman spaces $A_\alpha^p$ ($1\leq p\leq \infty$, $\alpha>-1$), and BMOA. We also establish the optimality of this framework by proving that such a basis cannot exist in larger spaces, such as the Hardy space $H^p$ and the disc algebra $A(\mathbb D)$.

math.CV↗

arXiv2024

The Ces\`{a}ro operator on local Dirichlet spaces

The family of Ces\`{a}ro operators $\sigma_n^\alpha$, $n \geq 0$ and $\alpha \in [0,1]$, consists of finite rank operators on Banach spaces of analytic functions on the open unit disc. In this work, we investigate these operators as they act on the local Dirichlet spaces $\mathcal{D}_\zeta$. It is well-established that they provide a linear approximation scheme when $\alpha > \frac{1}{2}$, with the threshold value $\alpha = \frac{1}{2}$ being optimal. We strengthen this result by deriving precise asymptotic values for the norm of these operators when $\alpha \leq \frac{1}{2}$, corresponding to the breakdown of approximation schemes. Additionally, we establish upper and lower estimates for the norm when $\alpha > \frac{1}{2}$.

math.FA↗