A gradient-projective basis of compactly supported wavelets in dimension n > 1

Guy Battle

Open Mathematics (2013)

  • Volume: 11, Issue: 7, page 1317-1333
  • ISSN: 2391-5455

Abstract

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A given set W = W X of n-variable class C 1 functions is a gradient-projective basis if for every tempered distribution f whose gradient is square-integrable, the sum ∑ χ ( ∫ ℝ n ∇ f · ∇ W χ * ) W χ converges to f with respect to the norm ∇ ( · ) L 2 ( ℝ n ) . The set is not necessarily an orthonormal set; the orthonormal expansion formula is just an element of the convex set of valid expansions of the given function f over W. We construct a gradient-projective basis W = W x of compactly supported class C 2−ɛ functions on ℝn such that [...] where X has the structure χ = ( χ ˜ , ν ) , ν ∈ ℤ. W is a wavelet set in the sense that the functions indexed by χ ˜ are generated by an averaging of lattice translations with wave propagations, and there are two additional discrete parameters associated with the latter. These functions indexed by χ ˜ are the unit-scale wavelets. The support volumes of our unit-scale wavelets are not uniformly bounded, however. While the practical value of this construction is doubtful, our motivation is theoretical. The point is that a gradient-orthonormal basis of compactly supported wavelets has never been constructed in dimension n > 1. (In one dimension the construction of such a basis is easy - just anti-differentiate the Haar functions.)

How to cite

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Guy Battle. "A gradient-projective basis of compactly supported wavelets in dimension n > 1." Open Mathematics 11.7 (2013): 1317-1333. <http://eudml.org/doc/269050>.

@article{GuyBattle2013,
abstract = {A given set W = W X of n-variable class C 1 functions is a gradient-projective basis if for every tempered distribution f whose gradient is square-integrable, the sum $\sum \limits _\chi \{(\int _\{\mathbb \{R\}^n \} \{\nabla f \cdot \} \nabla W_\chi ^* )\} W_\chi $ converges to f with respect to the norm $\left\Vert \{\nabla ( \cdot )\} \right\Vert _\{L^2 (\mathbb \{R\}^n )\} $ . The set is not necessarily an orthonormal set; the orthonormal expansion formula is just an element of the convex set of valid expansions of the given function f over W. We construct a gradient-projective basis W = W x of compactly supported class C 2−ɛ functions on ℝn such that [...] where X has the structure $\chi = (\tilde\{\chi \},\nu )$ , ν ∈ ℤ. W is a wavelet set in the sense that the functions indexed by $\tilde\{\chi \}$ are generated by an averaging of lattice translations with wave propagations, and there are two additional discrete parameters associated with the latter. These functions indexed by $\tilde\{\chi \}$ are the unit-scale wavelets. The support volumes of our unit-scale wavelets are not uniformly bounded, however. While the practical value of this construction is doubtful, our motivation is theoretical. The point is that a gradient-orthonormal basis of compactly supported wavelets has never been constructed in dimension n > 1. (In one dimension the construction of such a basis is easy - just anti-differentiate the Haar functions.)},
author = {Guy Battle},
journal = {Open Mathematics},
keywords = {Fourier transform; Wavelets; wavelets},
language = {eng},
number = {7},
pages = {1317-1333},
title = {A gradient-projective basis of compactly supported wavelets in dimension n > 1},
url = {http://eudml.org/doc/269050},
volume = {11},
year = {2013},
}

TY - JOUR
AU - Guy Battle
TI - A gradient-projective basis of compactly supported wavelets in dimension n > 1
JO - Open Mathematics
PY - 2013
VL - 11
IS - 7
SP - 1317
EP - 1333
AB - A given set W = W X of n-variable class C 1 functions is a gradient-projective basis if for every tempered distribution f whose gradient is square-integrable, the sum $\sum \limits _\chi {(\int _{\mathbb {R}^n } {\nabla f \cdot } \nabla W_\chi ^* )} W_\chi $ converges to f with respect to the norm $\left\Vert {\nabla ( \cdot )} \right\Vert _{L^2 (\mathbb {R}^n )} $ . The set is not necessarily an orthonormal set; the orthonormal expansion formula is just an element of the convex set of valid expansions of the given function f over W. We construct a gradient-projective basis W = W x of compactly supported class C 2−ɛ functions on ℝn such that [...] where X has the structure $\chi = (\tilde{\chi },\nu )$ , ν ∈ ℤ. W is a wavelet set in the sense that the functions indexed by $\tilde{\chi }$ are generated by an averaging of lattice translations with wave propagations, and there are two additional discrete parameters associated with the latter. These functions indexed by $\tilde{\chi }$ are the unit-scale wavelets. The support volumes of our unit-scale wavelets are not uniformly bounded, however. While the practical value of this construction is doubtful, our motivation is theoretical. The point is that a gradient-orthonormal basis of compactly supported wavelets has never been constructed in dimension n > 1. (In one dimension the construction of such a basis is easy - just anti-differentiate the Haar functions.)
LA - eng
KW - Fourier transform; Wavelets; wavelets
UR - http://eudml.org/doc/269050
ER -

References

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  1. [1] Battle G., A block spin construction of ondelettes. II. The QFT connection, Commun. Math. Phys., 1988, 114(1), 93–102 http://dx.doi.org/10.1007/BF01218290[Crossref] 
  2. [2] Battle G., Phase space localization theorem for ondelettes, J. Math. Phys., 1989, 30(10), 2195–2196 http://dx.doi.org/10.1063/1.528544[Crossref] Zbl0694.46006
  3. [3] Battle G., Wavelets and Renormalization, Ser. Approx. Decompos., 10, World Scientific, River Edge, 1999 http://dx.doi.org/10.1142/3066[Crossref] 
  4. [4] Daubechies I., Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math., 1988, 41(7), 909–996 http://dx.doi.org/10.1002/cpa.3160410705[Crossref] Zbl0644.42026
  5. [5] Federbush P., Williamson C., A phase cell approach to Yang-Mills theory. II. Analysis of a mode, J. Math. Phys., 1987, 28(6), 1416–1419 http://dx.doi.org/10.1063/1.527495[Crossref] 
  6. [6] Gawedzki K., Kupiainen A., A rigorous block spin approach to massless lattice theories, Comm. Math. Phys., 1980, 77(1), 31–64 http://dx.doi.org/10.1007/BF01205038[Crossref] 
  7. [7] Glimm J., Jaffe A., Quantum Physics, 2nd ed., Springer, New York, 1987 http://dx.doi.org/10.1007/978-1-4612-4728-9[Crossref] 
  8. [8] Haar A., Zur Theorie der Orthogonalen Funktionensysteme, Math. Ann., 1910, 69(3), 331–371 http://dx.doi.org/10.1007/BF01456326[Crossref] 
  9. [9] Hormander L., Linear Partial Differential Operators, Grundlehren Math. Wiss., 116, Academic Press/Springer, New York/Berlin, 1963 http://dx.doi.org/10.1007/978-3-642-46175-0[Crossref] Zbl0108.09301
  10. [10] Kahane J.-P., Lemarié-Rieusset P.-G., Fourier Series and Wavelets, Stud. Develop. Modern Math., 3, Gordon and Breach, London, 1996 Zbl0966.42002
  11. [11] Lemarié P. G., Ondelettes à localisation exponentielle, J. Math. Pures Appl., 1988, 67(3), 227–236 
  12. [12] Lemarié-Rieusset P.-G., Projecteurs invariants, matrices de dilatation, ondelettes et analyses multi-résolutions, Rev. Mat. Iberoamericana, 1994, 10(2), 283–347 http://dx.doi.org/10.4171/RMI/153[Crossref] 
  13. [13] Mallat S., A Wavelet Tour of Signal Processing, Academic Press, San Diego, 1998 Zbl1125.94306
  14. [14] Meyer Y., Principe d’incertitude, bases hilbertiennes et algèbres d’opérateurs, In: Séminaire Bourbaki, 1985–1986, 662, Astérisque, 1987, 145–146(4), 209–223 
  15. [15] Reed M., Simon B., Methods of Modern Mathematical Physics. II. Functional Analysis Academic Press, New York-London, 1975 
  16. [16] Schauder J., Eine Eigenschaft des Haarschen Orthogonalsystems, Math. Z., 1928, 28(1), 317–320 http://dx.doi.org/10.1007/BF01181164[Crossref] Zbl54.0324.02
  17. [17] Wilson K., Renormalization group and critical phenomena. I. Renormalization group and the Kadanoff scaling picture, Phys. Rev. B, 1971, 4(9), 3174–3183 http://dx.doi.org/10.1103/PhysRevB.4.3174[Crossref] Zbl1236.82017
  18. [18] Wilson K., Renormalization group and critical phenomena. II. Phase-space cell analysis of critical behavior, Phys. Rev. B, 1971, 4(9), 3183–3205 [Crossref] Zbl1236.82016

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