Orthogonal Polynomial

Orthogonal polynomials are sets of polynomials that are mutually orthogonal with respect to a specific inner product, finding applications across diverse scientific fields. Current research emphasizes their use in constructing efficient and accurate models for various tasks, including solving differential equations (via Kolmogorov-Arnold Networks and physics-informed neural networks), analyzing time series data (using novel state-space models), and improving machine learning algorithms (e.g., by enhancing minibatch sampling in stochastic gradient descent). This renewed interest stems from orthogonal polynomials' ability to provide stable, computationally efficient, and often highly accurate approximations of complex functions, leading to advancements in areas such as survival analysis, image processing, and signal processing.