Haarzellfunktion

Haarzellfunktion is a term sometimes encountered in German mathematical literature to denote the basic building blocks of the Haar wavelet system, or, more broadly, functions related to the Haar family. In standard terminology, the fundamental object is the Haar function (Haar-Funktion) or Haar wavelet, notationally used to construct simple, efficient representations of functions and signals.

The classic Haar mother wavelet h is a real-valued, piecewise constant function with compact support on the interval [0,1). It is defined by h(t) = 1 for t in [0, 1/2), h(t) = -1 for t in [1/2, 1), and h(t) = 0 elsewhere. Dilated and translated versions, called Haar functions, are obtained by h_{j,k}(t) = 2^{j/2} h(2^j t − k) for integers j and k. Alongside these wavelets, the corresponding scaling functions φ(t) = 1 on [0,1) and φ_{j,k}(t) = 2^{j/2} φ(2^j t − k) play a role in forming a multiresolution analysis of L^2(R).

Key properties include orthonormality of the family {h_{j,k}} in L^2(R), compact support on dyadic intervals, and zero mean. These features yield a simple, fast transform, the Haar transform, with linear computational complexity in the signal length. The Haar system provides the most elementary example of a wavelet basis and is valued for its ease of implementation and interpretability in time-domain analysis.

Note: the term Haarzellfunktion is not widely standardized; most sources refer to Haar-Funktion or Haar-Wavelet. When precision is required, it is advisable to specify whether one means the mother wavelet, a scaled/shifts version, or the related scaling function.