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

• Autorzy: Guy Battle
• Języki publikacji: EN

Abstrakty

EN

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 \(\left\| {\nabla ( \cdot )} \right\|_{L^2 (\mathbb{R}^n )} \) . 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.)

Słowa kluczowe

EN

Fourier transform; Wavelets

Kategorie tematyczne

• 42B99: None of the above, but in this section
• 41A63: Multidimensional problems (should also be assigned at least one other classification number in this section)

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2013
• Tom: 11
• Numer: 7
• Strony: 1317-1333

Daty

wydano

2013-07-01

online

2013-04-26

Twórcy

autor

Guy Battle

• Mathematics Department, Texas A&M University, College Station, Texas, 77843-3368, USA

Bibliografia

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

Identyfikatory

DOI

10.2478/s11533-013-0245-5