×

Tauberian theorems for the wavelet transform. (English) Zbl 1219.42030

The goal of this paper is to study, via Abelian-Tauberian results, asymptotic properties of distributions using wavelet transforms
\[ W_\psi f(b,a)=\langle \bar\psi(x), f(b+ax)\rangle \qquad(f\in{\mathcal S}'({\mathbb R})) \]
that admit a reconstruction wavelet and defined by functions \(\psi\in {\mathcal S}_0({\mathbb R})\).
A basic Tauberian proposition characterizes in terms of the behavior of such a wavelet \(W_\psi\) at approaching points of the boundary the existence of a distribution \(g\) defined by a quasi-asymptotic behavior
\[ \langle\varphi, g\rangle = \lim_{\varepsilon \downarrow 0} {1\over \varepsilon L(\varepsilon)} \langle f(x_0+\varepsilon x),\varphi(x)(x)\rangle, \]
if \(f\in{\mathcal S}'_0({\mathbb R})\) and \(L\) is a given slowly varying function at \(0\).
This proposition allows the authors to prove their main Tauberian theorems for quasi-asymptotics at points. Also corresponding Tauberian theorems for quasi-asymptotics at infinity are presented.
A number of clarifying examples and remarks are also included in this interesting paper.

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
26A12 Rate of growth of functions, orders of infinity, slowly varying functions
40E05 Tauberian theorems
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
46F10 Operations with distributions and generalized functions
PDF BibTeX XML Cite
Full Text: DOI Link

References:

[1] Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Encyclopedia of Mathematics and Its Applications, vol. 27. Cambridge University Press, Cambridge (1989) · Zbl 0667.26003
[2] Bogoliubov, N.N., Shirkov, D.V.: Introduction to the Theory of Quantized Fields. Wiley, New York (1959) · Zbl 0088.21701
[3] Bony, J.M.: Second microlocalization and propagation of singularities for semilinear hyperbolic equations. In: Tanaguchi Symp. HERT, Katata, pp. 11–49 (1984)
[4] Christensen, O.: An Introduction to Frames and Riesz Bases. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston (2003) · Zbl 1017.42022
[5] Conway, J.B.: Functions of One Complex Variable II. Springer, New York (1995) · Zbl 0887.30003
[6] Daubechies, I.: Ten Lectures on Wavelets. SIAM, Philadelphia (1992) · Zbl 0776.42018
[7] Drozhzhinov, Y.N., Zavialov, B.I.: Tauberian-type theorems for a generalized multiplicative convolution. Izv. Math. 64, 35–92 (2000) · Zbl 0973.46031
[8] Drozhzhinov, Y.N., Zavialov, B.I.: Tauberian theorems for generalized functions with values in Banach spaces. Izv. Math. 66, 701–769 (2002) · Zbl 1029.46048
[9] Drozhzhinov, Y.N., Zavialov, B.I.: Asymptotically homogeneous generalized functions. Izv. Nats. Akad. Nauk Arm. Mat. 41, 23–32 (2006) · Zbl 1240.46070
[10] Drozhzhinov, Y.N., Zavialov, B.I.: Asymptotically quasihomogeneous generalized functions. Dokl. Akad. Nauk 421, 157–161 (2008)
[11] Durán, A.L., Estrada, R.: Strong moment problems for rapidly decreasing smooth functions. Proc. Am. Math. Soc. 120, 529–534 (1994) · Zbl 0795.44005
[12] Estrada, R.: Characterization of the Fourier series of distributions having a value at a point. Proc. Am. Math. Soc. 124, 1205–1212 (1996) · Zbl 0843.46024
[13] Estrada, R.: The nonexistence of regularization operators. J. Math. Anal. Appl. 286, 1–10 (2003) · Zbl 1038.46028
[14] Estrada, R., Kanwal, R.P.: Singular Integral Equations. Birkhäuser, Boston (2000) · Zbl 0945.45001
[15] Estrada, R., Kanwal, R.P.: A Distributional Approach to Asymptotics. Theory and Applications, 2nd edn. Birkhäuser, Boston (2002) · Zbl 1033.46031
[16] Feichtinger, H.G., Gröchenig, K.: Gabor frames and time-frequency analysis of distributions. J. Funct. Anal. 146, 464–495 (1997) · Zbl 0887.46017
[17] Gröchenig, K.: Foundation of Time-Frequency Analysis. Birkhäuser, Boston (2001) · Zbl 0966.42020
[18] Holschneider, M.: Wavelets. An Analysis Tool. Clarendon/Oxford University Press, New York (1995) · Zbl 0874.42020
[19] Holschneider, M., Tchamitchian, Ph.: Pointwise analysis of Riemann’s ”nondifferentiable” function. Invent. Math. 105, 157–175 (1991) · Zbl 0741.26004
[20] Hörmander, L.: Lectures on Nonlinear Hyperbolic Differential Equations. Mathematiques & Applications, vol. 26. Springer, Berlin (1997) · Zbl 0881.35001
[21] Jaffard, S.: The spectrum of singularities of Riemann’s function. Rev. Mat. Iberoam. 12, 441–460 (1996) · Zbl 0889.26005
[22] Jaffard, S.: Wavelet expansions, function spaces and multifractal analysis. In: Twentieth Century Harmonic Analysis–A Celebration (Il Ciocco, 2000). NATO Sci. Ser. II Math. Phys. Chem., vol. 33, pp. 127–144. Kluwer Academic, Dordrecht (2001)
[23] Jaffard, S., Meyer, Y.: Wavelet methods for pointwise regularity and local oscillations of functions. Mem. Am. Math. Soc. 123, 587 (1996) · Zbl 0873.42019
[24] Lazzarini, S., Gracia-Bondía, J.M.: Improved Epstein-Glaser renormalization. II. Lorentz invariant framework. J. Math. Phys. 44, 3863–3875 (2003) · Zbl 1062.81114
[25] Littlewood, J.E.: The converse of Abel’s theorem on power series. Proc. Lond. Math. Soc. 9, 434–448 (1911) · JFM 42.0276.01
[26] Łojasiewicz, S.: Sur la valeur et la limite d’une distribution en un point. Stud. Math. 16, 1–36 (1957) · Zbl 0086.09405
[27] Meyer, Y.: Wavelets and Operators. Cambridge University Press, Cambridge (1992) · Zbl 0776.42019
[28] Meyer, Y.: Wavelets, Vibrations and Scalings. CRM Monograph Series, vol. 9. Am. Math. Soc., Providence (1998) · Zbl 0893.42015
[29] Pilipović, S.: Quasiasymptotics and S-asymptotics in \(\mathcal{S}'\) and \(\mathcal{D}'\) . Publ. Inst. Math. (Beograd) 72, 13–20 (1995) · Zbl 0889.46038
[30] 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
[31] Pilipović, S., Stanković, B., Takači, A.: Asymptotic Behaviour and Stieltjes Transformation of Distributions. Texte zuer Mathematik. Teubner, Leipzig (1990) · Zbl 0756.46020
[32] Saneva, K., Bučkovska, A.: Asymptotic expansion of distributional wavelet transform. Integral Transforms Spec. Funct. 17, 85–91 (2006) · Zbl 1101.46025
[33] Saneva, K., Bučkovska, A.: Tauberian theorems for distributional wavelet transform. Integral Transforms Spec. Funct. 18, 359–368 (2007) · Zbl 1132.46030
[34] Schwartz, L.: Théorie des Distributions. Hermann, Paris (1966)
[35] Seuret, S., Véhel, J.L.: The local Hölder function of a continuous function. Appl. Comput. Harmon. Anal. 13, 263–276 (2002) · Zbl 1023.26003
[36] Seuret, S., Véhel, J.L.: A time domain characterization of 2-microlocal spaces. J. Fourier Anal. Appl. 9, 473–495 (2003) · Zbl 1064.42029
[37] Sohn, B.K., Pahk, D.H.: Pointwise convergence of wavelet expansion of \({\mathcal{K}_{r}^{M}}'(\mathbb{R})\) . Bull. Korean Math. Soc. 38, 81–91 (2001) · Zbl 0988.46034
[38] Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970) · Zbl 0207.13501
[39] Trèves, F.: Topological Vector Spaces, Distributions and Kernel. Academic Press, New York (1967) · Zbl 0171.10402
[40] Triebel, H.: Wavelet frames for distributions; local and pointwise regularity. Stud. Math. 154, 59–88 (2003) · Zbl 1047.46027
[41] Vindas, J.: Structural theorems for quasiasymptotics of distributions at infinity. Publ. Inst. Math. (Beograd) (N.S.) 84(98), 159–174 (2008) · Zbl 1199.46094
[42] Vindas, J.: The structure of quasiasymptotics of Schwartz distributions. In: Linear and Non-linear Theory of Generalized Functions and Its Applications. Banach Center Publ., vol. 88, pp. 297–314. Polish Acad. Sc. Inst. Math., Warsaw (2010) · Zbl 1202.46049
[43] Vindas, J., Estrada, E.: Distributional point values and convergence of Fourier series and integrals. J. Fourier Anal. Appl. 13, 551–576 (2007) · Zbl 1138.46030
[44] Vindas, J., Estrada, R.: On the jump behavior of distributions and logarithmic averages. J. Math. Anal. Appl. 347, 597–606 (2008) · Zbl 1163.46027
[45] Vindas, J., Pilipović, S.: Structural theorems for quasiasymptotics of distributions at the origin. Math. Nachr. 282, 1584–1599 (2009) · Zbl 1189.46032
[46] Vladimirov, V.S., Zavialov, B.I.: On the Tauberian theorems in quantum field theory. Theor. Mat. Fiz. 40, 155–178 (1979)
[47] Vladimirov, V.S., Zavialov, B.I.: Tauberian theorems in quantum field theory. In: Current Problems in Mathematics, vol. 15, pp. 95–130, 228. Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow (1980) (in Russian)
[48] Vladimirov, V.S., Drozhzhinov, Y.N., Zavialov, B.I.: Tauberian Theorems for Generalized Functions. Kluwer Academic, Dordrecht (1988)
[49] Vladimirov, V.S., Drozhzhinov, Y.N., Zavialov, B.I.: Tauberian theorems for generalized functions in a scale of regularly varying functions and functionals, dedicated to Jovan Karamata. Publ. Inst. Math. (Beograd) (N.S.) 71, 123–132 (2002) (in Russian) · Zbl 1037.46040
[50] Wagner, P.: On the quasi-asymptotic expansion of the casual fundamental solution of hyperbolic operators and systems. Z. Anal. Anwend. 10, 159–167 (1991) · Zbl 0763.35018
[51] Walter, G.: Pointwise convergence of wavelet expansions. J. Approx. Theory 80, 108–118 (1995) · Zbl 0821.42019
[52] Walter, G., Shen, X.: Wavelets and Other Orthogonal Systems, 2nd edn. Studies in Advanced Mathematics. Chapman & Hall/CRC, Boca Raton (2001) · Zbl 1005.42018
[53] Zavialov, B.I.: Scaling 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)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.