Refinement equations with nonnegative coefficients

  • Vladimir Protasov
Article

Abstract

In this paper we analyze solutions of the n-scale functional equation Ф(x) = Σk∈ℤPk Ф(nx−k), where n≥2 is an integer, the coefficients {Pk} are nonnegative and Σpk = 1. We construct a sharp criterion for the existence of absolutely continuous solutions of bounded variation. This criterion implies several results concerning the problem of integrable solutions of n-scale refinement equations and the problem of absolutely continuity of distribution function of one random series. Further we obtain a complete classification of refinement equations with positive coefficients (in the case of finitely many terms) with respect to the existence of continuous or integrable compactly supported solutions.

Math Subjec Classifications

15A24 26C10 30B20 42A32 42A38 

Keywords and Phrases

refinement equation Fourier series tree distributions joint spectral radius 

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References

  1. [1]
    Berger, M.A. and Wang, Y. (1992). Bounded semi-groups of matrices.Linear Alg. Appl.,166, 21–27.MATHMathSciNetCrossRefGoogle Scholar
  2. [2]
    Cavaretta, A., Dahmen, W., and Micchelli, C. (1991). Stationary subdivision,Mem. Am. Math. Soc.,93, 1–186.MathSciNetGoogle Scholar
  3. [3]
    Cohen, A. and Daubechies, I. (1992). A sStability criterion for the orthogonal wavelet bases and their related subband coding scheme,Duke Math. J.,68(2), 313–335.MATHMathSciNetCrossRefGoogle Scholar
  4. [4]
    Collela, D. and Heil, C. (1994). Characterization of scaling functions, I. Continuous solutions,SIAM. J. Matrix Anal. Appl.,15, 496–518.MathSciNetCrossRefGoogle Scholar
  5. [5]
    Gelfand, I.M. and Shilov, G.E. (1964).Generalized Functions, Academic Press, New York.Google Scholar
  6. [6]
    Daubechies, I. (1988). Orthonormal bases of wavelets with compact support,Comm. Pure Appl. Math.,41, 909–996.MATHMathSciNetGoogle Scholar
  7. [7]
    Daubechies, I. and Lagarias, J. (1991). Two-scale difference equations, I. Global regularity of solutions,SIAM. J. Math. Anal.,22, 1388–1410.MATHMathSciNetCrossRefGoogle Scholar
  8. [8]
    Daubechies, I. and Lagarias, J. (1992). Two-scale difference equations, II. Local regularity, infinite products of matrices and fractals,SIAM. J. Math. Anal.,23, 1031–1079.MATHMathSciNetCrossRefGoogle Scholar
  9. [9]
    Derfel, G.A. (1989). Probabilistic method for a class of functional-differential equations,Ukrain. Math. J.,41(19).Google Scholar
  10. [10]
    Derfel, G.A., Dyn, N., and Levin, D. (1995). Generalized refinement equations and subdivision processes,J. Approx. Theory,80, 272–297.MATHMathSciNetCrossRefGoogle Scholar
  11. [11]
    Deslauriers, G. and Dubuc, S. (1989). Symmetric iterative interpolation processes,Constr. Approx.,5, 49–68.MATHMathSciNetCrossRefGoogle Scholar
  12. [12]
    Dyn, N., Gregory, J.A., and Levin, D. (1987). A four-point interpolatory subdivision scheme for curve design,Comput. Aided Geom. Design,4, 257–268.MATHMathSciNetCrossRefGoogle Scholar
  13. [13]
    Dyn, N., Gregory, J.A., and Levin, D. (1991). Analysis of linear binary subdivision schemes for curve design,Constr. Approx.,7, 127–147.MATHMathSciNetCrossRefGoogle Scholar
  14. [14]
    Deliu, A. and Spruill, M.C. (1994). Dilation equations and absolute continuity of random expansions, School of Mathematics Technical Rept. No 103194-025, Georgia Tech.Google Scholar
  15. [15]
    Derfel, G. and Schilling, R. (1996). Spatially chaotic configurations and functional equations with rescaling,J. Phys. A.,29, 4537–4547.MATHMathSciNetCrossRefGoogle Scholar
  16. [16]
    Erdös, P. (1939). On a family of symmetric Bernuolli convolutions,Am. J. Math.,61, 974–975.MATHCrossRefGoogle Scholar
  17. [17]
    Erdös, P. (1940). On the smoothness properties of Bernuolli convolutions,Am. J. Math.,62, 180–186.MATHCrossRefGoogle Scholar
  18. [18]
    Garsia, A.M. (1962). Arithmetic properties of Bernuolli convolutions,Trans. Am. Math. Soc.,102, 409–432.MATHMathSciNetCrossRefGoogle Scholar
  19. [19]
    Heil, C. and Collela, D. (1993). Dilation equations and the smoothness of compactly supported wavelets, inWavelets: Mathematics and Applications. Benedetto, J.J. and Frazier, M., Eds., CRC Press, Boca Raton, FL, 161–200.Google Scholar
  20. [20]
    Lagarias, J.C. and Wang, Y. (1995). The finiteness conjecture for the generalized spectral radius of a set of matrices,Linear Alg. Appl.,214, 17–42.MATHMathSciNetCrossRefGoogle Scholar
  21. [21]
    Lau, K.S., Ma, M.-F., and Wang, J. (1996). On some sharp regularity estimations ofL 2-scaling functions,SIAM. J. Math. Anal.,27, 835–864.MATHMathSciNetCrossRefGoogle Scholar
  22. [22]
    Lau, K.S. and Wang, J. (1995). Characterization ofL p-solutions for two-scale dilation equations,SIAM. J. Math. Anal.,26, 1018–1046.MATHMathSciNetCrossRefGoogle Scholar
  23. [23]
    Micchelli, C.A. (1986). Subdivision algorithms for curves and surfaces,Proc. SIGGRAPH, Dallas, TX.Google Scholar
  24. [24]
    Micchelli, C.A. and Prautzsch, H. (1987). Refinement and subdivision for spaces of integer translates of a compactly supported function, inNumerical Analysis, Griffiths, D.F. and Watson, G.A., Eds., 192–222.Google Scholar
  25. [25]
    Micchelli, C.A. and Prautzsch, H. (1989). Uniform refinement of curves,Linear Alg. Appl.,114/115, 841–870.MathSciNetCrossRefGoogle Scholar
  26. [26]
    Peres, Y. and Solomyak, B. (1996). Absolute continuity of Bernuolli convolution, a simple proof,Math. Res. Lett.,3(2), 231–239.MATHMathSciNetGoogle Scholar
  27. [27]
    Protasov, V. (1996). The joint spectral radius and invariant sets of the several linear operators,Fundamentalnaya i Prikladnaya Matematika,2(1), 205–231.MATHMathSciNetGoogle Scholar
  28. [28]
    Protasov, V. (1997). The generalized joint spectral radius. The geometric approach,Izvestiya Akademii Nauk. Seriya Matematicheskaya,61(5), 99–136.MATHMathSciNetGoogle Scholar
  29. [29]
    Protasov, V. A complete solution characterizing smooth refinable functions,SIAM J. Math. Anal., to appear.Google Scholar
  30. [30]
    Rota, G.C. and Strang, G. (1960). A note on the joint spectral radius,Kon. Nederl. Acad. Wet. Proc.,63, 379–381.MATHMathSciNetGoogle Scholar
  31. [31]
    Salem, R. (1963).Algebraic Numbers and Fourier Analysis. D.C. Heath and Co., Boston, MA.MATHGoogle Scholar
  32. [32]
    Schumaker, L.L. (1981).Spline Functions: Basic Theory, John Wiley & Sons, New York.Google Scholar
  33. [33]
    Solomyak, B. (1995). On the random series Σ±λi (an Erdös problem),Ann. Math.,142, 611–625.MATHMathSciNetCrossRefGoogle Scholar
  34. [34]
    Wang, Y. (1995). Two-scale dilation equations and the cascade algorithm,Random Comput. Dynamic,3(4), 289–307.Google Scholar
  35. [35]
    Wang, Y. (1996). Two-scale dilation equations and the mean spectral radius,Random Comput. Dynamic,4(1), 49–72.MATHGoogle Scholar
  36. [36]
    Zakusilo, O.K. (1975). On classes of limit distributions in some scheme of summing up,Teoria Veroyatnosti i Mat. Statistika,12, 44–48.MATHMathSciNetGoogle Scholar
  37. [37]
    Zakusilo, O.K. (1976). Some properties of classesL p of limit distribution,Teoria Veroyatnosti i Mat. Statistika,15, 68–73.MATHMathSciNetGoogle Scholar

Copyright information

© Birkhäuser 2000

Authors and Affiliations

  • Vladimir Protasov
    • 1
  1. 1.Department of Mechanics and MathematicsMoscow State UniversityMoscowRussia

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