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Czasopismo

2011 | 9 | 3 | 627-639

Tytuł artykułu

Haar system on a product of zero-dimensional compact groups

Treść / Zawartość

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Języki publikacji

EN

Abstrakty

EN
In this work, we study the problem of constructing Haar bases on a product of arbitrary compact zero-dimensional Abelian groups. A general scheme for the construction of Haar functions is given for arbitrary dimension. For dimension d=2, we describe all Haar functions.

Twórcy

  • Saratov State University

Bibliografia

  • [1] Agaev G.N., Vilenkin N.Ya., Dzhafarli G.M., Rubinshteĭn A.I., Multiplicative Systems and Harmonic Analysis on Zero-Dimensional Groups, ELM, Baku, 1981 (in Russian)
  • [2] Albeverio S., Khrennikov A.Yu., Shelkovich V.M., Theory of p-adic Distributions: Linear and Nonlinear Models, London Math. Soc. Lecture Note Ser., 370, Cambridge University Press, Cambridge, 2010
  • [3] Albeverio S., Evdokimov S., Skopina M., p-adic nonorthogonal wavelet bases, Tr. Mat. Inst. Steklova, 2009, 265, Izbrannye Voprosy Matematicheskoi Fiziki i p-adicheskogo Analiza, 7–18
  • [4] Albeverio S., Evdokimov S., Skopina M., p-adic multiresolution analysis and wavelet frames, J. Fourier Anal. Appl., 2010, 16(5), 693–714 http://dx.doi.org/10.1007/s00041-009-9118-5
  • [5] Benedetto J.J., Benedetto R.L., A wavelet theory for local fields and related groups, J. Geom. Anal., 2004, 14(3), 423–456
  • [6] Benedetto R.L., Examples of wavelets for local fields, In: Wavelets, Frames and Operator Theory, Contemp. Math., 345, American Mathematical Society, Providence, 2004, 27–47
  • [7] Evdokimov S.A., Skopina M.A., 2-adic wavelet bases, Trudy Inst. Mat. i Meh. Ural. Otd. Ross. Akad. Nauk, 2009, 15(1), 135–146 (in Russian)
  • [8] Farkov Yu.A., Orthogonal wavelets on direct products of cyclic groups, Math. Notes, 2007, 82(6), 843–859 http://dx.doi.org/10.1134/S0001434607110296
  • [9] Farkov Yu.A., Orthogonal wavelets with compact support on locally compact abelian groups, Izv. Math., 2005, 69(3), 623–650 http://dx.doi.org/10.1070/IM2005v069n03ABEH000540
  • [10] Golubov B.I., On one class of complete orthogonal systems, Sibirsk. Mat. Zh., 1968, 9(2), 297–314 (in Russian)
  • [11] Haar A., Zur Theorie der orthogonalen Funktionen systeme, Math. Ann., 1910, 69(3), 331–371 http://dx.doi.org/10.1007/BF01456326
  • [12] Khrennikov A.Y., Shelkovich V.M., Non-Haar p-adic wavelets and their application to pseudo-differential operators and equations, Appl. Comput. Harmon. Anal., 2010, 28(1), 1–23 http://dx.doi.org/10.1016/j.acha.2009.05.007
  • [13] Khrennikov A.Yu., Shelkovich V.M., An infinite family of p-adic non-Haar wavelet bases and pseudo-differential operators, P-Adic Numbers Ultrametric Anal. Appl., 2010, 1(3), 204–216 http://dx.doi.org/10.1134/S2070046609030030
  • [14] King E.J., Skopina M.A., Quincunx multiresolution analysis for L 2(ℚ 22), P-Adic Numbers Ultrametric Anal. Appl., 2010, 2(3), 222–231 http://dx.doi.org/10.1134/S2070046610030040
  • [15] Kozyrev S.V., Wavelet theory as p-adic spectral analysis, Izv. Math., 2002, 66(2), 367–376 http://dx.doi.org/10.1070/IM2002v066n02ABEH000381
  • [16] Lang W.C., Orthogonal wavelets on the Cantor dyadic group, SIAM J. Math. Anal., 1996, 27(1), 305–312 http://dx.doi.org/10.1137/S0036141093248049
  • [17] Lang W.C., Wavelet analysis on the Cantor dyadic group, Houston J. Math., 1998, 24(3), 533–544
  • [18] Lang W.C., Fractal multiwavelets related to the Cantor dyadic group, Internat. J. Math. Math. Sci., 1998, 21(2), 307–314 http://dx.doi.org/10.1155/S0161171298000428
  • [19] Lukomskii S.F., On Haar series on compact zero-dimensional groups, Izv. Saratov. Univ. Mat. Mekh. Inform., 2009, 9(1), 14–19 (in Russian)
  • [20] Lukomskii S.F., Multiresolution analysis on zero-dimensional groups, and wavelet bases, Sb. Math., 2010, 201(5), 669–691 http://dx.doi.org/10.1070/SM2010v201n05ABEH004088
  • [21] Lukomskii S.F., On Haar system on the product of groups of p-adic integer, Math. Notes (in press)
  • [22] Khrennikov A.Yu., Shelkovich V.M., Skopina M.A., p-adic refinable functions and MRA-based wavelets, J. Approx. Theory, 2009, 161(1), 226–238 http://dx.doi.org/10.1016/j.jat.2008.08.008
  • [23] Protasov V.Yu., Farkov Yu.A., Dyadic wavelets and refinable functions on a half-line, Sb. Math., 2006, 197(10), 1529–1558 http://dx.doi.org/10.1070/SM2006v197n10ABEH003811
  • [24] Shelkovich V.M., Skopina M.A., p-adic Haar multiresolution analysis and pseudo-differential operators, J. Fourier Anal. Appl., 2009, 15(3), 366–393 http://dx.doi.org/10.1007/s00041-008-9050-0

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Bibliografia

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