arXiv subjects

Anand Ganesh

Publications and source records attributed to Anand Ganesh.

4 recordsLinked to original sources

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↗

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↗

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↗

arXiv2025

A Singular Integral for a Simplified Clairaut Equation

This expository article on the Lagrange singular integral contains two novelties. The first novelty involves a connection between the Lagrange singular integral for a simplified Clairaut equation, and Euler's homogeneous function theorem. The paper presents a formal derivation of Euler's solution from Lagrange's complete integral, though with some caveats, and also constructs more general surfaces from the complete integral which go beyond Euler's solutions. The first rather complicated construction is based directly on Goursat's definition of a general integral, while the subsequent simpler constructions are based on a suitably expanded notion of the general integral. This generalized general integral is our second novelty. It bridges some of the gap between the the general integral, and the complete integral, partially addressing Evans' remarks (Partial Differential Equations, AMS Graduate Studies in Mathematics, 1998) on the limitations of the general integral. Finally we discuss some subtleties around complete integrals as noted by Chojnacki (Proceedings of the AMS, 1995) and some around general integrals as noted by Evans, and how they apply to our examples. We aim to present these classical PDE concepts to readers with a basic knowledge of multivariable calculus.

math.CA↗