Scalets

Frehner, M., Scalet, M. and Corm, E.E. (1990) Pattern of the cyanide-potential in developing fruits. Plant Physiol., 94, 28-34. 

Turning Point Quantization and Scalet-Wavelet Analysis (C. R. Handy)... 

Noma M, Huber J, Phareis RP (1979) Agric Biol Chem 43 1793 Noma M, Huber J, Ernst D. Pharis RP (1982) Planta 155 369 Pharis RP, King R (1985) Annu Rev Plant Physiol 36 517 Piaggesi A. Picciarelli P, Lorenzi R, Alpi A (1989) Plant Physiol 91 362 Picciarelli P, Alpi A (1985) Plant Cell Physiol 26 1233 Picciarelli P, Alpi A (1986) Plant Physiol 82 298 Picciarelli P, Alpi A (1987) Phytochemistry 26 329 Picciarelli P, Alpi A, Pistelli L, Scalet M (1984) Planta 162 566... 

Thirning Point Quantization and Scalet- Wavelet Analysis 199... 

A similar equation exists, in the translation variable space, dbjt(a, b) = e a, b)j (a, b) however, it will be implicitly incorporated within our scalet equation analysis when we specify the form of the initial, infinite scale, scalet configurations, (0,6). 

Instead of these conditions, from the scalet equation s (and CWT s) perspective, the more natural, and asymptotically equivalent, quantization conditions correspond to... 

A casual inspection of Eq.(1.9) or Eq.(l.lO) would suggest that a TPQ analysis can be done totally within a scalet representation (i.e. within the solution space corresponding to Eq.(l.S)), without the need to explicitly make use of wavelets. This is true, provided only approximate estimates for the discrete state energies are required. It turns out that for any E value (physical or not), the scalet equation admits a large subset of scalet solutions which, in the zero scale limit, become a solution of the Schrodinger equation. 

Since any solution to the Schrodinger equation will have zero kinetic energy at its turning points, it becomes clear that a purely TPQ-scalet... 

One can realize an exact TPQ analysis, capable of achieving arbitrary quantization precision, if one works within a combined scalet-wavelet representation. In this case, one uses the scalet equation as a generator for the required wavelet transform coefficients within the dual-wavelet basis expansion for the wavefunction. Imposing the TPQ conditions (either Eq.(1.9) or Eq.(l.lO)) on such an expansion defines an exact quantization formalism. This is discussed in Sec. 1.5. 

The second approach implements a Turning Point Quantization (TPQ) analysis within the MRF basis representation. This serves to validate the analogous formalism to be presented within the CWT discnission in the following sections (i.e. scalet-wavelet TPQ). In addition, the combined TPQ-MRF formalism allows one to identify the individual turning point contributions to the quantization process (through an analysis that explicitly incorporates all of the (complex) turning points). 

Within a pure scalet representation, the quantization scale is defined differently to that indicated above. This is discussed in Sec. I.4.4.2. 

It is more convenient to work with the equivalent moments (i.e. scalets)... 

In principle, the scalet equation can be solved in order to generate , provided a convenient scaling function is chosen (i.e. one whose algebraic structure generates a scalet equation). 

Alternatively, the scalet equation can be used to generate the wavelet transform for mother wavelets that C6in be generated from corresponding scaling functions (i.e. derivatives of such expressions, etc.). In this case, appropriate differences of the scalets generate the wavelet transform. For instance, in the case of the Mexican hat wavelet, the wavelet transform corresponds to the difference between the zeroth order and second order scalet functions ... 

In the earlier works by Handy and Murenzi, the initial value problem character of the scalet equation was used to recover the pointwise structure of the physical solution based upon EMM or MRF estimates for the physical energy and the associated (infinite scale) missing moments. Thus, given the physical missing moments, /if(a = oo, = 0) = /i( ) 0 < C < m , one can then generate all the infinite scale, i>-dependent, moments through the relation... 

Clearly, for theoretical completeness, one prefers a self consistent scalet theory. That is, we want to understand how to quantize purely within a scalet representation, if possible. One such approach is to impose the TPQ condition that demands that at the turning points, the scalet representation for the kinetic energy become zero, in the zero scale limit. 

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