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Wavelet expansions and asymptotic behavior of distributions. (English) Zbl 1196.42031
Author’s abstract: We develop a distribution wavelet expansion theory for the space of highly time-frequency localized test functions over the real line \(\mathcal S_0(\mathbb R)\subset \mathcal S (\mathbb R)\) and its dual space \(\mathcal S'_0(\mathbb R)\), namely, the quotient of the space of tempered distributions modulo polynomials. We prove that the wavelet expansions of tempered distributions converge in \(\mathcal S'_0(\mathbb R)\). A characterization of boundedness and convergence in \(\mathcal S'_0(\mathbb R)\) is obtained in terms of wavelet coefficients. Our results are then applied to study local and non-local asymptotic properties of Schwartz distributions via wavelet expansions. We provide Abelian and Tauberian type results relating the asymptotic behavior of tempered distributions with the asymptotics of wavelet coefficients.

MSC:
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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[1] Bingham, N.H.; Goldie, C.M.; Teugels, J.L., Regular variation, Encyclopedia math. appl., vol. 27, (1989) · Zbl 0667.26003
[2] Daubechies, I., Ten lectures of wavelets, (1992), SIAM Philadelphia · Zbl 0776.42018
[3] Durán, A.L.; Estrada, R., Strong moment problems for rapidly decreasing smooth functions, Proc. amer. math. soc., 120, 529-534, (1994) · Zbl 0795.44005
[4] Estrada, R., The nonexistence of regularization operators, J. math. anal. appl., 286, 1-10, (2003) · Zbl 1038.46028
[5] Estrada, R.; Kanwal, R.P., A distributional approach to asymptotics. theory and applications, (2002), Birkhäuser Boston · Zbl 0836.34056
[6] Hernández, E.; Weiss, G., A first course of wavelets, (1996), CRC Press Boca Raton · Zbl 0885.42018
[7] Holschneider, M.; Tchamitchian, Ph., Pointwise analysis of Riemann’s “nondifferentiable” function, Invent. math., 105, 157-175, (1991) · Zbl 0741.26004
[8] Hölschneider, M., Wavelets. an analysis tool, (1995), The Clarendon Press, Oxford University Press New York · Zbl 0874.42020
[9] Jaffard, S.; Meyer, Y., Wavelet methods for pointwise regularity and local oscillations of functions, Mem. amer. math. soc., 123, 587, (1996) · Zbl 0873.42019
[10] Kelly, S.E.; Kon, M.A.; Raphael, L.A., Pointwise convergence of wavelet expansions, Bull. amer. math. soc., 30, 87-94, (1994) · Zbl 0788.42014
[11] Korevaar, J., Tauberian theory. A century of developments, (2004), Springer-Verlag Berlin · Zbl 1056.40002
[12] Kyriazis, G.C., Wavelet coefficients measuring smoothness in \(H^p(\mathbb{R}^d)\), Appl. comput. harmon. anal., 3, 2, 100-119, (1996) · Zbl 0877.42017
[13] Lemarié, P.G.; Meyer, Y., Ondeletles et bases hilbertiennes, Rev. mat. iberoamericana, 2, 1-18, (1986) · Zbl 0657.42028
[14] Łojasiewicz, S., Sur la valeur et la limite d’une distribution en un point, Studia math., 16, 1-36, (1957) · Zbl 0086.09405
[15] Mallat, S.G., A wavelet tour of signal processing, (1998), Academic Press San Diego · Zbl 0937.94001
[16] Y. Meyer, Principe d’incertitude, bases hilbertiennes et algèbres d’opérateurs, in: Séminare Bourbaki, 1985-1986, 38 année, no. 662
[17] Meyer, Y., Wavelets and operators, (1992), Cambridge Univ. Press Cambridge
[18] Meyer, Y., Wavelets, vibrations and scalings, CRM monogr. ser., vol. 9, (1998), American Mathematical Society Providence · Zbl 0893.42015
[19] Pilipović, S.; Stanković, B.; Takači, A., Asymptotic behaviour and Stieltjes transformation of distributions, (1990), Teubner-Texte zur Mathematik Leipzig · Zbl 0756.46020
[20] Pilipović, S.; Takači, A.; Teofanov, N., Wavelets and quasiasymptotics at a point, J. approx. theory, 97, 40-52, (1999) · Zbl 0928.42021
[21] Pilipović, S.; Teofanov, N., Multiresolution expansion, approximation order and quasiasymptotic behavior of tempered distributions, J. math. anal. appl., 331, 455-471, (2007) · Zbl 1122.46020
[22] Saneva, K.; Bučkovska, A., Asymptotic behaviour of the distributional wavelet transform at 0, Math. balkanica, 18, 437-441, (2004) · Zbl 1077.42032
[23] Saneva, K.; Bučkovska, A., Asymptotic expansion of distributional wavelet transform, Integral transforms spec. funct., 17, 85-91, (2006) · Zbl 1101.46025
[24] Saneva, K.; Bučkovska, A., Tauberian theorems for distributional wavelet transform, Integral transforms spec. funct., 18, 359-368, (2007) · Zbl 1132.46030
[25] Saneva, K., Asymptotic behaviour of wavelet coefficients, Integral transforms spec. funct., 20, 3-4, 333-339, (2009) · Zbl 1181.46031
[26] Schwartz, L., Théorie des distributions, (1966), Hermann Paris
[27] Seneta, E., Regularly varying functions, (1976), Springer-Verlag Berlin · Zbl 0324.26002
[28] Shelkovich, V.M., Associated and quasi associated homogeneous distributions (generalized functions), J. math. anal. appl., 338, 48-70, (2008) · Zbl 1146.46021
[29] Sohn, B.K.; Pahk, D.H., Pointwise convergence of wavelet expansion of \(\mathcal{K}_r^M^\prime(\mathbb{R})\), Bull. Korean math. soc., 38, 81-91, (2001)
[30] Teofanov, N., Convergence of multiresolution expansions in the Schwartz class, Math. balkanica, 20, 101-111, (2006) · Zbl 1200.42024
[31] Trèves, F., Topological vector spaces, distributions and kernel, (1967), Academic Press New York
[32] Triebel, H., Wavelet frames for distributions; local and pointwise regularity, Studia math., 154, 59-88, (2003) · Zbl 1047.46027
[33] Vindas, J., Structural theorems for quasiasymptotics of distributions at infinity, Publ. inst. math. (beograd) (N.S.), 84, 98, 159-174, (2008) · Zbl 1199.46094
[34] J. Vindas, Local behavior of distributions and applications, Dissertation, Louisiana State University, 2009
[35] Vindas, J., The structure of quasiasymptotics of Schwartz distributions, (), 297-314 · Zbl 1202.46049
[36] Vindas, J.; Estrada, R., Distributional point values and convergence of Fourier series and integrals, J. Fourier anal. appl., 13, 551-576, (2007) · Zbl 1138.46030
[37] Vindas, J.; Pilipović, S., Structural theorems for quasiasymptotics of distributions at the origin, Math. nachr., 282, 1584-1599, (2009) · Zbl 1189.46032
[38] J. Vindas, S. Pilipović, D. Rakić, Tauberian theorems for the wavelet transform, submitted for publication
[39] Vladimirov, V.S.; Drozhzhinov, Yu.N.; Zavialov, B.I., Tauberian theorems for generalized functions, (1988), Kluwer Academic Publishers Group Dordrecht · Zbl 0636.40003
[40] Vladimirov, V.S.; Zavialov, B.I., Tauberian theorems in quantum field theory, Teoret. mat. fiz., 40, 155-178, (1979)
[41] Walter, G.G., Pointwise convergence of wavelet expansions, J. approx. theory, 80, 108-118, (1995) · Zbl 0821.42019
[42] Walter, G.G.; Shen, X., Wavelets and other orthogonal systems, Stud. adv. math., (2001), Chapman & Hall/CRC Boca Raton · Zbl 1005.42018
[43] Zavialov, B.I., Aling of electromagnetic form factors and the behavior of their Fourier transforms in the neighborhood of the light cone, Teoret. mat. fiz., 17, 178-188, (1973)
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