Colloquium

The Least-Squares Property of the Lanczos Derivative by Paul Fishback, Professor of Mathematics, Grand Valley State University

In his text Applied Analysis, Cornelius Lanczos presented a technique by which one can compute the derivative of a function by an intriguing integration method. His results led to an extension of the usual derivative, which now bears his name. On the surface, Lanczos' formula bears no resemblance to the usual derivative. However, a closer look reveals deep connections between his formula and the areas of probability, statistics, and orthogonal polynomials. These connections demonstrate that a common thread links the Lanczos Derivative to a wide class of derivative extensions and, most importantly, to the usual derivative itself.

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