arXiv subjects

Babhrubahan Bose

Publications and source records attributed to Babhrubahan Bose.

8 recordsLinked to original sources

arXiv2026

The Shift Operator Calculus for Stationary Time Series Analysis

The article establishes a rigorous shift operator calculus for stationary time series modeling, addressing a certain gap in the literature. It provides proofs of existence and isometry for the transfer function operators $f(B)$ and $f(T)$ where $B$ is the bilateral shift operator and $T$ is the unilateral shift operator for different families of functions $f$. The article establishes convergence of the power series of $f(B)$ and $f(T)$ under the operator norm for the Wiener algebra $\mathbb{W}_+$, and convergence under strong operator topology for $f$ in $H^{\infty}$, based on the use of Abel sums. Based on this calculus, it unifies the notion of stationary process invertibility with the operator invertibility of the transfer function $f(T)$.

math.FA↗

arXiv2026

On an $L^2$ norm for stationary ARMA processes

We propose an $L^2$ norm for stationary Autoregressive Moving Average (ARMA) models. We look at ARMA models within the Hilbert space of the past with present of a true purely linearly non-deterministic stationary process $X_t$, and compute the $L^2$ norm based on its Wold decomposition. As an application of this $L^2$ norm, we derive bounds on the mean square prediction error for AR(1) models of MA(1) processes, and verify these bounds empirically for sample data.

cs.LG↗

arXiv2025

Schauder Bases for $C[0, 1]$ Using ReLU, Softplus and Two Sigmoidal Functions

We construct four Schauder bases for the space $C[0,1]$, one using ReLU functions, another using Softplus functions, and two more using sigmoidal versions of the ReLU and Softplus functions. This establishes the existence of a basis using these functions for the first time, and improves on the universal approximation property associated with them. We also show an $O(\frac{1}{n})$ approximation bound based on our ReLU basis, and a negative result on constructing multivariate functions using finite combinations of ReLU functions.

cs.LG↗

arXiv2023

Geometry of $\ell_p$-direct sums of normed linear spaces

We consider $\ell_p$-direct sums ($1\leq p<\infty$) and $c_0$-direct sums of countably many normed spaces and find the duals of these spaces. We characterize the support functionals of arbitrary elements in these spaces to characterize smoothness and approximate smoothness, both locally and globally. These results let us obtain examples of spaces that are not approximately smooth but where every non-zero element is approximately smooth. We also characterize Birkhoff-James orthogonality and its pointwise symmetry in these spaces.

math.FA↗

arXiv2022

Geometry of nowhere vanishing, point separating sub-algebras of $\mathcal{H}ol(Γ\cup\text{Int}(Γ))$ and zeros of Holomorphic functions

We study $ \mathcal{H}ol(Γ\cup\text{Int}(Γ)) $, the normed algebra of all holomorphic functions defined on some simply connected neighbourhood of a simple closed curve $Γ$ in $\mathbb{C} $, equipped with the supremum norm on $ Γ$. We explore the geometry of nowhere vanishing, point separating sub-algebras of $ \mathcal{H}ol(Γ\cup \text{Int}(Γ)) $. We characterize the extreme points and the exposed points of the unit balls of the said sub-algebras for $Γ$ analytic. We also characterize the smoothness of an element in these sub-algebras by using Birkhoff-James orthogonality techniques without any restriction on $Γ$. As a culmination of our study, we assimilate the geometry of the aforesaid sub-algebras with some classical concepts of complex analysis and establish a connection between Birkhoff-James orthogonality and zeros of holomorphic functions.

math.FA↗

arXiv2022

Point-wise Symmetry of Birkhoff-James Orthogonality and Geometry of $\mathbb{B}(\ell_\infty^n,\ell_1^m)$

We study the relationship between the point-wise symmetry of Birkhoff-James orthogonality and the geometry of the space of operators $\mathbb{B}(\ell_\infty^n,\ell_1^m)$. We show that any non-zero left-symmetric point in this space is a smooth point. We also show that for $n\geq4$, any unit norm right-symmetric point of this space is an extreme point of the closed unit ball. This marks the first step towards characterizing the extreme points of these unit balls and finding the Grothendieck constants $G(m,n)$ using Birkhoff-James orthogonality techniques.

math.FA↗

arXiv2022

Birkhoff-James Orthogonality and Its Pointwise Symmetry in Some Function Spaces

We study Birkhoff-James orthogonality and its pointwise symmetry in commutative $C^*$ algebras, i.e., the space of all continuous functions defined on a locally compact Hausdorff space that vanish at infinity. We use this characterization to obtain the characterization of Birkhoff-James orthogonality on $L_\infty$ space defined on any arbitrary measure space. We also do the same for the $L_p$ spaces for $1\leq p<\infty$.

math.FA↗

arXiv2022

Birkhoff-James Orthogonality and Its Local Symmetry in Some Sequence Spaces

We study Birkhoff-James orthogonality and its local symmetry in some sequence spaces namely $\ell_p,$ for $1\leq p\leq\infty$, $p\neq2$, $c$, $c_0$ and $c_{00}$. Using the characterization of the local symmetry of Birkhoff-James orthogonality, we characterize isometries of each of these spaces onto itself and obtain the Banach-Lamperti theorem for onto operators on the sequence spaces.

math.FA↗