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

Guy Battle

Open Mathematics, Tome 11 (2013), p. 1317-1333 / Harvested from The Polish Digital Mathematics Library

Résumé

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.)

Détail

Publié le : 2013-01-01

Zbl 1268.42068

EUDML-ID : urn:eudml:doc:269050

Bibliographie

[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] 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] | Zbl 0694.46006

[3] Battle G., Wavelets and Renormalization, Ser. Approx. Decompos., 10, World Scientific, River Edge, 1999 http://dx.doi.org/10.1142/3066[Crossref]

[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] | Zbl 0644.42026

[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] 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] 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] Haar A., Zur Theorie der Orthogonalen Funktionensysteme, Math. Ann., 1910, 69(3), 331–371 http://dx.doi.org/10.1007/BF01456326[Crossref]

[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] | Zbl 0108.09301

[10] Kahane J.-P., Lemarié-Rieusset P.-G., Fourier Series and Wavelets, Stud. Develop. Modern Math., 3, Gordon and Breach, London, 1996 | Zbl 0966.42002

[11] Lemarié P. G., Ondelettes à localisation exponentielle, J. Math. Pures Appl., 1988, 67(3), 227–236

[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] Mallat S., A Wavelet Tour of Signal Processing, Academic Press, San Diego, 1998 | Zbl 1125.94306

[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] Reed M., Simon B., Methods of Modern Mathematical Physics. II. Functional Analysis Academic Press, New York-London, 1975

[16] Schauder J., Eine Eigenschaft des Haarschen Orthogonalsystems, Math. Z., 1928, 28(1), 317–320 http://dx.doi.org/10.1007/BF01181164[Crossref] | Zbl 54.0324.02

[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] | Zbl 1236.82017

[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] | Zbl 1236.82016