In this paper, we consider the well-known Rudin-Shapiro polynomials as a class of constant multiples of low-pass filters to construct a sequence of compactly supported wavelets.
@article{bwmeta1.element.doi-10_2478_s11533-010-0094-4, author = {Abdolaziz Abdollahi and Jahangir Cheshmavar and Mohsen Taghavi}, title = {Wavelets generated by the Rudin-Shapiro polynomials}, journal = {Open Mathematics}, volume = {9}, year = {2011}, pages = {441-448}, zbl = {1213.42127}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-010-0094-4} }
Abdolaziz Abdollahi; Jahangir Cheshmavar; Mohsen Taghavi. Wavelets generated by the Rudin-Shapiro polynomials. Open Mathematics, Tome 9 (2011) pp. 441-448. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-010-0094-4/
[1] Brillhart J., Carlotz L., Note on the Shapiro polynomial, Proc. Amer. Math. Soc., 1970, 25, 114–118 http://dx.doi.org/10.1090/S0002-9939-1970-0260955-6
[2] Billhart J., Lomont J.S., Morton P., Cyclotomic properties of the Rudin-Shapiro polynomials, J. Reine Angew. Math., 1976, 288, 37–65 | Zbl 0335.12003
[3] Butzer P.L., Fischer A., Rückforth K., Scaling functions and wavelets with vanishing moments, Comput. Math. Appl., 1994, 27(3), 33–39 http://dx.doi.org/10.1016/0898-1221(94)90044-2 | Zbl 0852.42022
[4] Byrnes J.S., Quadrature mirror filter, low crest factor arrays, functions achieving optimal uncertainty principle bounds, and complete orthonormal sequences - a unified approach, Appl. Comput. Harmon. Anal., 1994, 1(3), 261–266 http://dx.doi.org/10.1006/acha.1994.1013 | Zbl 0802.42023
[5] Byrnes J.S., Moran W., Saffari B., Smooth PONS, J. Fourier Anal. Appl., 2000, 6(6), 663–674 http://dx.doi.org/10.1007/BF02510701
[6] Chui C.K., An Introduction to Wavelets, Wavelet Anal. Appl., 1, Academic Press, Boston, 1992 | Zbl 0925.42016
[7] Cohen A., Ondelettes, analysis multirésolutions et filtres miroirs en quadrature, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1990, 7(5), 439–459 | Zbl 0736.42021
[8] la Cour-Harbo A., On the Rudin-Shapiro transform, Appl. Comput. Harmon. Anal., 2008, 24(3), 310–328 http://dx.doi.org/10.1016/j.acha.2007.05.003 | Zbl 1143.42029
[9] 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 | Zbl 0644.42026
[10] Daubechies I., Ten Lectures on Wavelets, CBMS-NSF Regional Conf. Ser. in Appl. Math., 61, SIAM, Philadelphia, 1992
[11] Gripenberg G., Computing the joint spectral radius, Linear Algebra Appl., 1996, 234, 43–60 http://dx.doi.org/10.1016/0024-3795(94)00082-4 | Zbl 0863.65017
[12] Haar A., Zur Theorie der orthogonalen Funktionensysteme, Math. Ann., 1910, 69(3), 331–371 http://dx.doi.org/10.1007/BF01456326 | Zbl 41.0469.03
[13] Hernández E., Weiss G., A First Course on Wavelets, Stud. Adv. Math., CRC Press, Boca Raton, 1996
[14] Hong D., Wang J., Gardner R., Real Analysis with an Introduction to Wavelets and Applications, Elsevier Academic Press, Burlington-San Diego-London, 2005
[15] Lawton W.M., Necessary and sufficient conditions for constructing orthonormal wavelet bases, J. Math. Phys., 1991, 32(1), 57–61 http://dx.doi.org/10.1063/1.529093 | Zbl 0757.46012
[16] Lebedeva E.A., Protasov V.Yu., Meyer wavelets with least uncertainty constant, Math. Notes, 2008, 84(5–6), 680–687 http://dx.doi.org/10.1134/S0001434608110096 | Zbl 1219.42029
[17] Li D.F., Peng S.L., Chen H.L., Local properties of periodic cardinal interpolatory wavelets, Acta Math. Sinica (Chinese Ser.), 2001, 44(5), 947–960 | Zbl 1021.42016
[18] Mallat S.G., Multiresolution approximations and wavelet orthonormal bases of L 2(ℝ), Trans. Amer. Math. Soc., 1989, 315(1), 69–87 | Zbl 0686.42018
[19] Meyer Y., Wavelets and Operators, Cambridge Stud. Adv. Math., 37, Cambridge University Press, Cambridge, 1992
[20] Novikov I.Ya., Stechkin S.B., Basic wavelet theory, Russian Math. Surveys, 1998, 53(6), 1159–1231 http://dx.doi.org/10.1070/RM1998v053n06ABEH000089 | Zbl 0955.42019
[21] Rudin W., Some theorems on Fourier coefficients, Proc. Amer. Math. Soc., 1959, 10, 855–859 http://dx.doi.org/10.1090/S0002-9939-1959-0116184-5 | Zbl 0091.05706
[22] Shapiro H.S., Extremal Problems for Polynomials and Power Series, M.I.T. Master’s Thesis, Cambridge, 1951, available at http://hdl.handle.net/1721.1/12198
[23] Villemoes L.F., Wavelet analysis of refinement equations, SIAM J. Math. Anal., 1994, 25(5), 1433–1460 http://dx.doi.org/10.1137/S0036141092228179 | Zbl 0809.42016