Tag Archives: complete orthonormal system

Walsh functions I. Orthonormality and completeness

In this post I summarize some useful properties of Walsh functions. These functions were introduced by Joseph Walsh in

  1. J. L. Walsh, A closed set of normal orthogonal functions. Amer. J. Math., 45, 5-24, 1923.

Another paper where many ideas can be found is by Nathan Fine

  1. N. J. Fine, On the Walsh functions. Trans. Amer. Math. Soc., 65, 372-414, 1949.

In this exposition here we only concentrate on the simplest case of base b = 2 and dimension s = 1.

We write \mathbb{N} for the set of natural numbers 1,2,3, \ldots and \mathbb{N}_0 for the set of nonnegative integers 0,1,2,\ldots.

A table of contents for the posts on Walsh functions can be found here.

Definition of Walsh functions Continue reading

Math2111: Chapter 1: Fourier series. Additional Material: L_2 convergence of Fourier series.

In this blog entry you can find lecture notes for Math2111, several variable calculus. See also the table of contents for this course. This blog entry printed to pdf is available here.

In this post I present some ideas which shed light on the question why one can expect the Fourier series to converge to the function (under certain assumptions). Continue reading