Spectral Methods for Time-Dependent Problems: A Survey

In this article, we explore a novel approach to quickly compute expansions in ultraspherical polynomials. These expansions are useful in various applications, including numerical analysis and signal processing. Our method leverages the properties of these polynomials to develop an efficient algorithm that reduces computational complexity while maintaining accuracy.
Ultraspherical polynomials have several advantages over traditional polynomial approximations. They are orthogonal with respect to a weight function, which means that they can be used to represent signals in a more accurate manner. Additionally, their coefficients are easy to compute and analyze, making them a valuable tool for signal processing applications.
However, computing these expansions can be computationally expensive, especially when dealing with large datasets. To address this challenge, we propose a new approach that combines the properties of ultraspherical polynomials with fast algebraic techniques. Our method takes advantage of the fact that these polynomials are semi-separable, meaning that their coefficients can be computed using a series of simpler calculations.
We demonstrate the effectiveness of our approach through several numerical examples. In each case, we show how our algorithm can efficiently compute the expansions in ultraspherical polynomials, while maintaining accuracy and reducing computational complexity. Our results illustrate the potential of this method for signal processing and other applications where accurate polynomial approximations are required.
In summary, this article presents a novel approach to rapidly computing expansions in ultraspherical polynomials. By leveraging their properties and combining them with fast algebraic techniques, we develop an efficient algorithm that can handle large datasets while maintaining accuracy. Our results demonstrate the potential of this method for various applications, including signal processing and numerical analysis.