a mathematical microscope tool


The wavelet theory is established in the Fourier Hilbert space framework. In order to apply the Calderón inverse formula in a Hilbert scale framework it requires the so-called admissibility condition defining a wavelet analyzing a (signal) function. "A wavelet synthesis can be performed locally as opposed to the Fourier transform which is inherently nonlocal due to the space-filling nature of the trigonometric functions" (FaM1). So the question is, how to bundle both necessary concepts into one applying it to corresponding "right" physical NSE variation representation with related energy Hilbert space.

Mathematically speaking, the L(2) based wavelets are accompanied by the distributional Hilbert space H(-1/2). The proposed quantum potential energy model H(1/2) is based on a Hilbert scale framework built on the spectrum of a considered kinematical phenomenon govenred by a corresponding self-adjoint kinetic energy operator. The "coarse grained" kinematical Hilbert space pair (H(0),H(1)) accompanied by concepts like the Shannon entropy is compactly and densely embedded into the overall Hilbert space pair (H(-1/2),H(1/2)), where its complementary closed sub-space pair defined the complementary quantum potential element and energy Hilbert space pair.

We note that L(2) functions with vanishing constant Fourier term are wavelet mother functions, and that the Hilbert transform of L(2) functions fulfill this condition.

Wavelet analysis can be used as a mathematical microscope, looking at the details that are added if one goes from a scale "a" to a scale "a+da", where " da " is infinitesimally small. We mention that an alternative model for an "a" to a scale "a+da model is the concept of the ordered field of ideal points, an extension to the ordered field of real numbers with same cardinality, but having additionally infinitesimal elements (also called non-Archimedean numbers).

The mathematical microscope wavelet tool 'unfolds' a function over the one-dimensional space R into a function over the two-dimensional half-plane of "positions" and "details". This two-dimensional parameter space may also be called the position-scale half-plane. The wavelet duality relationship provides an additional degree of freedom to apply wavelet analysis with appropriately (problem specific) defined wavelets in a (distributional) Hilbert scale framework where the "microscope observations" of two wavelet (optics) functions f and g can be compared with each other by the "reproducing" ("duality") formula.