Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On wavelets interpolated from a pair of wavelet sets


Authors: Ziemowit Rzeszotnik and Darrin Speegle
Journal: Proc. Amer. Math. Soc. 130 (2002), 2921-2930
MSC (2000): Primary 42C40
DOI: https://doi.org/10.1090/S0002-9939-02-06416-X
Published electronically: May 8, 2002
MathSciNet review: 1908915
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that any wavelet, with the support of its Fourier transform small enough, can be interpolated from a pair of wavelet sets. In particular, the support of the Fourier transform of such wavelets must contain a wavelet set, answering a special case of an open problem of Larson. The interpolation procedure, which was introduced by X. Dai and D. Larson, allows us also to prove the extension property.


References [Enhancements On Off] (What's this?)

  • [B] Bownik, M., On Characterizations of multiwavelets in $L^{2}(\mathbb{R} ^{n})$, Proc. Amer. Math. Soc. 129 (2001), 3263-3274. CMP 2001:16
  • [BCM] Baggett, L., Courter, J. and Merrill, K., The construction of wavelets from generalized conjugate mirror filters in $L^{2}(\mathbb{R} ^{n})$, preprint.
  • [BGRW] Brandolini L., Garrigós G., Rzeszotnik Z. and Weiss G., The behaviour at the origin of a class of band-limited wavelets, Contemporary Mathematics 247 (1999), 75-91. MR 2001b:42045
  • [BMM] Baggett, L.W., Medina, H.A. and Merrill, K.D., Generalized multiresolution analyses, and a construction procedure for all wavelet sets in ${\mathbb{R} ^{n}}$, J. Fourier. Anal. Appl. 5 (no.6) (1999), 563-573. MR 2001f:42055
  • [BSW] Bonami, A., Soria, F. and Weiss, G., Band-limited wavelets, Jour. of Geom. Anal. 3 no.6 (1993), 543-578. MR 94k:42046
  • [DL] Dai, X. and Larson, D.R., Wandering vectors for unitary systems and orthogonal wavelets, Mem. Amer. Math. Soc. 134 (no. 640) (1998). MR 98m:47067
  • [DLS] Dai, X., Larson, D.R. and Speegle, D., Wavelet sets in ${\mathbb{R} }^{n}$, II, Contemp. Math. 216 (1998), 15-40. MR 99d:42054
  • [Gr] Gripenberg, G., A necessary and sufficient condition for the existence of a father wavelet, Studia Math. 114 (1995), 207-226. MR 96d:42049
  • [Gu] Gu, Q., On interpolation families of wavelet sets, Proc. Amer. Math. Soc. 128 (2000), 2973-2979. MR 2000m:42026
  • [HKLS] Ha, Y., Kang, H., Lee, J. and Seo, J., Unimodular wavelets for $L^{2}$ and the Hardy space $H^{2}$, Michigan Math. J. 41 (1994), 345-361. MR 95g:42050
  • [HW] Hernández, E. and Weiss, G., A first course on wavelets, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1996. MR 97i:42015
  • [HWW] Hernández, E., Wang, X. and Weiss, G., Smoothing minimally supported frequency wavelets. I, J. Fourier Anal. Appl. 2 no. 4 (1996), 329-340. MR 97h:42015
  • [IP] Ionascu, E.J. and Pearcy, C.M., On subwavelet sets, Proc. Amer. Math. Soc. 126 (1998), 3549-3552. MR 99b:42040
  • [La] Larson, D.R., Von Neumann algebras and wavelets, Operator algebras and applications (Samos, 1996), 267-312, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 495, Kluwer Acad. Publ., Dordrecht, 1997. MR 98g:46091
  • [Le] Lemarié-Rieusset, P.G., Ondolettes à localisation exponentielle, J. Math. Pure et Appl 67 (1988), 227-236. MR 89m:42024
  • [WW] Weiss G. and Wilson E.N., The Mathematical Theory of Wavelets, preprint.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 42C40

Retrieve articles in all journals with MSC (2000): 42C40


Additional Information

Ziemowit Rzeszotnik
Affiliation: Institute of Mathematics, University of Wroclaw, pl Grunwaldzki 2/4, 50-384 Wroclaw, Poland
Email: zioma@math.uni.wroc.pl

Darrin Speegle
Affiliation: Department of Mathematics & Computer Science, Saint Louis University, St. Louis, Missouri 63103
Email: speegled@slu.edu

DOI: https://doi.org/10.1090/S0002-9939-02-06416-X
Keywords: Orthonormal wavelets, MSF wavelets, interpolated wavelets
Received by editor(s): September 19, 2000
Received by editor(s) in revised form: March 22, 2001
Published electronically: May 8, 2002
Communicated by: David R. Larson
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society