On the HAAR and Walsh Systems on a Triangle.

Abstract : A number of papers have been concerned with developing the theories of discontinuous orthonormal systems and their applications. In particular, the Haar and Walsh systems are presently the most important examples of nonsinusoidal functions, and have proved most useful in communication. Some authors have studied the properties of approximation from the mathematical point of view. It seems interesting and helpful for both theory and practice to investigate the Haar and Walsh functions for a multivariate setting. In fact, many signals in communications and other functions are of several variables (for instance, TV signals have two space variables and the time variable). If the domain of definition of the system is tensor product, then the existing systems are readily extended to several variables. The problem is how to construct an orthonormal system on a triangular domain in the plane, or more generally, on a simplex in n-dimensional space. This paper defines the Haar and Walsh system on a triangle domain, proves the orthogonality and completeness in L sub 2. Also the uniform convergence for the Haar-Fourier series, uniform convergence by group for the Walsh-Fourier series are studied. All of these results can be generalized easily to n dimensions.