Stable Calculation of Discrete Hahn Functions

Abstract

Generating discrete orthogonal polynomials from the recurrence or difference equation is error-prone, as it is sensitive to error propagation and dependent on highly accurate initial values. Strategies to handle this, involving control over the deviation of norm and orthogonality, have already been developed for the discrete Chebyshev and Krawtchouk functions, i.e., the orthonormal basis in ℓ2 derived from the polynomials. Since these functions are limiting cases of the discrete Hahn functions, it suggests that the strategy could also be successful there. We outline the algorithmic strategies including the specific method of generating the initial values, and show that the orthonormal basis can indeed be generated for large supports and polynomial degrees with controlled numerical error. Special attention is devoted to symmetries, as the symmetric windows are most commonly used in signal processing, allowing for simplification of the algorithm due to this prior knowledge, and leading to savings in the required computational power.

Keywords

• orthogonal polynomials
• discrete Hahn polynomials
• discrete Hahn functions
• error propagation
• difference equation
• three-term recurrence relation