# A Geometric Construction of Multiwavelet Sets of L^2(R) and H^2(R)

Keywords:
symmetric multiwavelet sets, functions, Hardy space.

### Abstract

In the present article we construct symmetric multiwavelet sets of finite order in L^2(R) and multiwavelet sets in H^2(R) by considering the geometric construction determining wavelet sets provided by N. Arcozzi, B. Behera and S. Madan for large classes of minimally supported frequency wavelets of L^2(R) and H^2(R).

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### References

[1] N. Arcozzi, B. Behera and S. Madan, Large classes of minimally supported frequency wavelets of L 2 (R) and H2(R), J. Geom. Anal., 13 (2003), 557–579.

[2] R. Ashino and M. Kametani, A lemma on matrices and a construction of multiwavelets, Math. Japon., 45 (1997), 267–287.

[3] L. W. Baggett, H. A. Medina and K. D. Merrill, Generalized multi-resolution analyses and a construction procesure for all wavelet sets in Rn, J. Fourier Anal. Appl., 5 (1999), 563–573.

[4] J. J. Benedetto and M. T. Leon, The construction of multiple dyadic minimally supported frequency wavelets on Rd, The functional and harmonic analysis of wavelets and frames (San Antonio, TX, 1999), 4374, Amer. Math. Soc., Providence, RI.

[5] M. Bownik, A characterization of affine dual frames in L2 (Rn), Appl. Comput. Harmon. Anal., 8 (2000), 203–221.

[6] M. Bownik, On characterizations of multiwavelets in L 2 (Rn), Proc. Amer. Math. Soc., 129 (2001), 3265–3274.

[7] M. Bownik, Z. Rzeszotnik and D. Speegle, A characterization of dimension functions of wavelets, Appl. Comput. Harmon. Anal., 10 (2001), 71–92.

[8] C. A. Cabrelli and M. L. Gordillo, Existence of multiwavelets in Rn, Proc. Amer. Math. Soc.,130 (2002), 1413–1424.

[9] A. Calogero, Wavelets on general lattices, associated with general expanding maps of Rn, Electron. Res. Announc. Amer. Math. Soc., 5 (1999), 1–10.

[10] A. Calogero, A characterization of wavelets on general lattices, J. Geom. Anal., 10 (2000), 597–622.

[11] X. Dai, D. R. Larson and D. M. Speegle, Wavelet sets in Rn, J. Fourier Anal. Appl., 3 (1997), 451–456.

[12] X. Dai, D. R. Larson and D. M. Speegle, Wavelet sets in R^n II, Contemp. Math., 3 (1997), 15–40.

[13] L. De Michele and P. M. Soardi, On multiresolution analysis of multiplicity d, Mh. Math., 124 (1997), 255–272.

[14] M. Frazier, G. Garrig´os, K. Wang and G. Weiss, A characterization of functions that generate wavelet and related expansion, J. Fourier Anal. Appl., 3 (1997), 883–906.

[15] T. N. T. Goodman and S. L. Lee, Wavelets of multiplicity r, Trans. Amer. Math. Soc., 342 (1994), 307–324.

[16] L. Herv´e, Multi-resolution analysis of multiplicity d: applications to dyadic interpolation, Appl. and Comput. Harmonic Anal., 1 (1994), 299–315.

[17] K. D. Merrill, Simple wavelet sets for scalar dilations in R^2, Representations, Wavelets and Frames: A celebration of the mathematical work of Lawrence W. Baggett, Birkh¨auser, Boston (2009), 177–192.

[18] S. Mittal, A construction of multiwavelet sets in the Euclidean plane, Real Anal. Exchange, 38 (2012), 17–32

[19] N. K. Shukla and G. C. S. Yadav, A characterization of three-interval scaling sets, Real Anal. Exchange, 35 (2009), 121–138.

[20] A. Vyas, Construction of non-MSF non-MRA wavelets for L^2(R) and H^2(R) from MSF wavelets, Bull. Polish Acad. Sci. Math., 57 (2009), 33–40.

[2] R. Ashino and M. Kametani, A lemma on matrices and a construction of multiwavelets, Math. Japon., 45 (1997), 267–287.

[3] L. W. Baggett, H. A. Medina and K. D. Merrill, Generalized multi-resolution analyses and a construction procesure for all wavelet sets in Rn, J. Fourier Anal. Appl., 5 (1999), 563–573.

[4] J. J. Benedetto and M. T. Leon, The construction of multiple dyadic minimally supported frequency wavelets on Rd, The functional and harmonic analysis of wavelets and frames (San Antonio, TX, 1999), 4374, Amer. Math. Soc., Providence, RI.

[5] M. Bownik, A characterization of affine dual frames in L2 (Rn), Appl. Comput. Harmon. Anal., 8 (2000), 203–221.

[6] M. Bownik, On characterizations of multiwavelets in L 2 (Rn), Proc. Amer. Math. Soc., 129 (2001), 3265–3274.

[7] M. Bownik, Z. Rzeszotnik and D. Speegle, A characterization of dimension functions of wavelets, Appl. Comput. Harmon. Anal., 10 (2001), 71–92.

[8] C. A. Cabrelli and M. L. Gordillo, Existence of multiwavelets in Rn, Proc. Amer. Math. Soc.,130 (2002), 1413–1424.

[9] A. Calogero, Wavelets on general lattices, associated with general expanding maps of Rn, Electron. Res. Announc. Amer. Math. Soc., 5 (1999), 1–10.

[10] A. Calogero, A characterization of wavelets on general lattices, J. Geom. Anal., 10 (2000), 597–622.

[11] X. Dai, D. R. Larson and D. M. Speegle, Wavelet sets in Rn, J. Fourier Anal. Appl., 3 (1997), 451–456.

[12] X. Dai, D. R. Larson and D. M. Speegle, Wavelet sets in R^n II, Contemp. Math., 3 (1997), 15–40.

[13] L. De Michele and P. M. Soardi, On multiresolution analysis of multiplicity d, Mh. Math., 124 (1997), 255–272.

[14] M. Frazier, G. Garrig´os, K. Wang and G. Weiss, A characterization of functions that generate wavelet and related expansion, J. Fourier Anal. Appl., 3 (1997), 883–906.

[15] T. N. T. Goodman and S. L. Lee, Wavelets of multiplicity r, Trans. Amer. Math. Soc., 342 (1994), 307–324.

[16] L. Herv´e, Multi-resolution analysis of multiplicity d: applications to dyadic interpolation, Appl. and Comput. Harmonic Anal., 1 (1994), 299–315.

[17] K. D. Merrill, Simple wavelet sets for scalar dilations in R^2, Representations, Wavelets and Frames: A celebration of the mathematical work of Lawrence W. Baggett, Birkh¨auser, Boston (2009), 177–192.

[18] S. Mittal, A construction of multiwavelet sets in the Euclidean plane, Real Anal. Exchange, 38 (2012), 17–32

[19] N. K. Shukla and G. C. S. Yadav, A characterization of three-interval scaling sets, Real Anal. Exchange, 35 (2009), 121–138.

[20] A. Vyas, Construction of non-MSF non-MRA wavelets for L^2(R) and H^2(R) from MSF wavelets, Bull. Polish Acad. Sci. Math., 57 (2009), 33–40.

Published

2020-08-20

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*Journal of Progressive Research in Mathematics*,

*16*(4), 3122-3132. Retrieved from http://www.scitecresearch.com/journals/index.php/jprm/article/view/1892

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