Convergence Properties of Haar Series.
Abstract
TThe report is primarily a review of Haar functions. The functions are defined, and it is shown that they form an orthonormal basis of (L sup 2)(0,1). A basic property of Haar series is then proved (the proof given is different from HAAR's). This property states that the Nth Haar-series approximation to a function f(x) belongs to (L sup 2)(0,1) is a step function of (2 sup N) steps, each of width 1/(2 sup N). The value of the approximation on each step is the mean value of the function f(x) in the interval covered by the step. This property is taken as the basis for an investigation of the convergence properties of Haar series for various classes of functions. Estimates of approximation accuracy are derived. Haar-series convergence is compared with the convergence of trigonometric Fourier series. The class of functions for which uniform convergence of Haar series can be proved is larger than that class of functions whose trigonometric Fourier series is uniformly convergent to the given function. (Author)
Document Details
- Document Type
- Technical Report
- Publication Date
- Jan 31, 1973
- Accession Number
- AD0756173
Entities
People
- John E. Shore
- Robert L. Berkowitz
Organizations
- United States Naval Research Laboratory