A fast-transform basis with hysteretic features

Hysteretic distortion is the common feature of signals generated by magnetic quantum- and nano-systems. In light of recent advances in nano-electronics, particularly the assent of memristance, it is expected that hysteretic features will become the trademark of next-generation electronics. Consequently, signal processing tasks, such as compression or denoising, will benefit from the use of bases which adapt to the hysteretic distortion. In this paper I describe a basis that consists of periodic functions with an inherent hysteretic feature. These functions originate from a certain nonlinear eigenvalue problem which, it is demonstrated here, relates to oscillators with memristance. The discussion of the specific basis is set in the discrete framework. It is demonstrated that a signal may be represented in the said basis via a fast-transform. In other words, an algorithm is displayed, which allows the basis coefficients of an N-dimensional signal to be computed in O(N log N) arithmetical operations. Remarkably, the crucial estimate for the algorithm's efficiency originates from the canon of Analytic Number Theory.