Skip to main content
SpringerLink
Log in

Representation theory and multilevel filters

  • Original Research
  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

We present a general setting where wavelet filters and multiresolution decompositions can be defined, beyond the classical \({\mathbf {L}}^2({\mathbb {R}},dx)\) setting. This is done in a framework of iterated function system (IFS) measures; these include all cases studied so far, and in particular the Julia set/measure cases. Every IFS has a fixed order, say N, and we show that the wavelet filters are indexed by the infinite dimensional group G of functions from X into the unitary group \(U_N\). We call G the loop group because of the special case of the unit circle.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

Data sharing

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

References

  1. Adamska, M., Bezuglyi, S., Karpel, O., Kwiatkowski, J.: Subdiagrams and invariant measures on Bratteli diagrams. Ergod. Theory Dyn. Syst. 37(8), 2417–2452 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alpay, D., Ball, J., Gohberg, I., Rodman, L.: Realization and factorization of rational matrix functions with symmetries, volume 47 of Operator Theory: Advances and Applications , pp. 1–60. Birkhäuser Verlag, Basel (1990)

  3. Alpay, D., Ball, J., Gohberg, I., Rodman, L.: State space theory of automorphisms of rational matrix functions. Integr. Eq. Oper. Theory 15, 349–377 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  4. Alpay, D., Colombo, F., Lewkowiz, I., Sabadini, I.: Realization of slice hyperholomorphic generalized contractive and positive functions. In: Alpay, D., Kirstein, B. (eds.), Recent Advances in Inverse Scattering, Schur Analysis and Stochastic Processes, volume 244 of Operator Theory: Advances and Applications, pp. 67–97. Birkhäuser (2015)

  5. Alpay, D., Gohberg, I.: On orthogonal matrix polynomials, volume 34 of Operator Theory: Advances and Applications, pp. 25–46. Birkhäuser Verlag, Basel (1988)

  6. Alpay, D., Gohberg, I.: Unitary rational matrix functions. In: Gohberg, I. (ed.) Topics in Interpolation Theory of Rational Matrix-Valued Functions. Operator Theory: Advances and Applications, vol. 33, pp. 175–222. Birkhäuser Verlag, Basel (1988)

  7. Alpay, D., Jorgensen, P.: New characterizations of reproducing kernel Hilbert spaces and applications to metric geometry. Opusc. Math. 41, 283–300 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  8. Alpay, D., Jorgensen, P.: Finitely additive functions in measure theory and applications. arXiv (2022)

  9. Alpay, D., Jorgensen, P., Lewkowicz, I.: Extending wavelet filters: infinite dimensions, the nonrational case, and indefinite inner product spaces. In: Excursions in harmonic analysis. Volume 2, Appl. Numer. Harmon. Anal., pp. 69–111. Birkhäuser/Springer, New York (2013)

  10. Alpay, D., Jorgensen, P., Lewkowicz, I.: Parametrizations of all wavelet filters: input–output and state-space. Sampling Theory Signal Image Process. (STSIP) 12, 159–188 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Alpay, D., Jorgensen, P., Lewkowicz, I.: Realizations of infinite products, Ruelle operators and wavelet filters. J. Fourier Anal. Appl. 21(5), 1034–1052 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Alpay, D., Jorgensen, P., Lewkowicz, I.: Characterizations of rectangular (para)-unitary rational functions. Opusc. Math. 36(6), 695–716 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. Alpay, D., Jorgensen, P., Lewkowicz, I.: Characterizations of families of rectangular, finite impulse response, para-unitary systems. J. Appl. Math. Comput. 54(1–2), 395–423 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  14. Alpay, D., Jorgensen, P., Lewkowicz, I.: \(W\)-Markov measures, transfer operators, wavelets and multiresolutions. In: Frames and Harmonic Analysis, volume 706 of Contemp. Math., pp. 293–343. American Mathematical Society, Providence, RI (2018)

  15. Alpay, D., Jorgensen, P., Lewkowicz, I., Martziano, I.: Infinite product representations for kernels and iterations of functions. In: Recent Advances in Inverse Scattering, Schur Analysis and Stochastic Processes, volume 244 of Oper. Theory Adv. Appl., pp. 67–87. Birkhäuser/Springer, Cham (2015)

  16. Alpay, D., Jorgensen, P., Lewkowicz, I., Marziano, I.: Representation formulas for Hardy space functions through the Cuntz relations and new interpolation problems. In Shen, X., Zayed, A. (eds.), Multiscale Signal Analysis and Modeling, pp. 161–182. Springer (2013)

  17. Alpay, D., Jorgensen, P., Lewkowicz, I., Volok, D.: A new realization of rational functions, with applications to linear combination interpolation, the Cuntz relations and kernel decompositions. Complex Var. Elliptic Equ. 61(1), 42–54 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. Alpay, D., Levanony, D.: Linear stochastic systems: a white noise approach. Acta Appl. Math. 110(2), 545–572 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Barndorff-Nielsen, O.E., Shiryaev, A.: Change of time and change of measure, volume 21 of Advanced Series on Statistical Science & Applied Probability. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, second edition (2015)

  20. Bezuglyi, S., Jorgensen, P.: Transfer Operators, Endomorphisms, and Measurable Partitions. Lecture Notes in Mathematics, vol. 2217. Springer, Cham (2018)

  21. Bezuglyi, S., Jorgensen, P.: Graph Laplace and Markov operators on a measure space. In: Linear Systems, Signal Processing and Hypercomplex Analysis, volume 275 of Oper. Theory Adv. Appl., pp. 67–138. Birkhäuser/Springer, Cham (2019)

  22. Bezuglyi, S., Jorgensen, P.: Symmetric measures, continuous networks, and dynamics. In: New Directions in Function Theory: From Complex to Hypercomplex to Non-commutative, volume 286 of Oper. Theory Adv. Appl., pp. 139–197. Birkhäuser/Springer, Cham (2021)

  23. Bochner, S., Chandrasekharan, K.: Fourier transforms. Princeton University Press: Reprinted with permission of the original publishers, p. 1965. Kraus Reprint Corporation, New-York (1949)

  24. Bratteli, O., Jorgensen, P.: Isometries, shifts, Cuntz algebras and multiresolution wavelet analysis of scale \(N\). Integr. Equ. Oper. Theory 28(4), 382–443 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  25. Bratteli, O., Jorgensen, P.: Convergence of the cascade algorithm at irregular scaling functions. In: The Functional and Harmonic Analysis of Wavelets and Frames (San Antonio, TX, 1999), volume 247 of Contemp. Math., pages 93–130. American Mathematical Society, Providence, RI (1999)

  26. Bratteli, O., Jorgensen, P.: Iterated function systems and permutation representations of the Cuntz algebra. Mem. Amer. Math. Soc., 139(663):x+89 (1999)

  27. Bratteli, O., Jorgensen, P.: Wavelet filters and infinite-dimensional unitary groups. In: Wavelet Analysis and Applications (Guangzhou, 1999), volume 25 of AMS/IP Stud. Adv. Math., pages 35–65. American Mathematical Society, Providence, RI (2002)

  28. Bratteli, O., Jorgensen, P.: Wavelets Through a Looking Glass. Applied and Numerical Harmonic Analysis. Birkhäuser Boston Inc., Boston, MA (2002)

    MATH  Google Scholar 

  29. Bratteli, O., Jorgensen, P.E.T.: Endomorphisms of \({ B}({ H})\). II. Finitely Correlated States on \({ O}_n\). J. Funct. Anal., 145(2):323–373 (1997)

  30. Bratteli, O., Evans, D.E., Jorgensen, P.E.T.: Compactly supported wavelets and representations of the Cuntz relations. Appl. Comput. Harmon. Anal., 8(2):166–196 (2000)

  31. Chatterji, S.D.: Les martingales et leurs applications analytiques. In: École d’Été de Probabilités: Processus Stochastiques (Saint Flour, 1971), pp. 27–164. Lecture Notes in Math., Vol. 307. Springer Verlag (1973)

  32. Cohen, A., Ryan, R.D.: Wavelets and multiscale signal processing. Transl. from the French, volume 11 of Appl. Math. Math. Comput. London: Chapman & Hall (1995)

  33. Cuntz, J.: Simple \(C^*\)-algebras generated by isometries. Commun. Math. Phys. 57(2), 173–185 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  34. Cuntz, J.: \(K\)-theory for certain \(C^{\ast } \)-algebras. Ann. of Math. (2), 113(1):181–197 (1981)

  35. Dutkay, D., Jorgensen, P.E.T.: Spectral measures and Cuntz algebras. Math. Comput. 81(280), 2275–2301 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  36. Dutkay, D., Jorgensen, P.E.T., Picioroaga, G.: Unitary representations of wavelet groups and encoding of iterated function systems in solenoids. Ergod. Theory Dyn. Syst. 29(6), 1815–1852 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  37. Dutkay, D.E., Jorgensen, P.E.T.: Representations of Cuntz algebras associated to quasi-stationary Markov measures. Ergod. Theory Dyn. Syst. 35(7), 2080–2093 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  38. Farsi, C., Gillaspy, E., Jorgensen, P., Kang, S., Packer, J.: Purely atomic representations of higher-rank graph \(C^*\)-algebras. Integral Equations Operator Theory, 90(6):Paper No. 67, 26 (2018)

  39. Farsi, C., El, Gillaspy, Jorgensen, P., Kang, S., Packer, J.: Monic representations of finite higher-rank graphs. Ergod. Theory Dyn. Syst. 40(5), 1238–1267 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  40. Feng, D., Simon, K.: Dimension estimates for \(C^1\) iterated function systems and \(C^1\) repellers, a survey. In: Thermodynamic formalism, volume 2290 of Lecture Notes in Math., pp. 421–467. Springer, Cham (2021)

  41. Hernández-Lerma, O., Lasserre, J.B.: Markov chains and invariant probabilities. Progress in Mathematics, vol. 211. Birkhäuser Verlag, Basel (2003)

  42. Jorgensen, P.: Iterated function systems, representations, and Hilbert space. Int. J. Math. 15(8), 813–832 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  43. Jorgensen, P.: Harmonic analysis, volume 128 of CBMS Regional Conference Series in Mathematics. American Mathematical Society, Providence, RI. Smooth and non-smooth, Published for the Conference Board of the Mathematical Sciences (2018)

  44. Jorgensen, P., Song, M.-S.: Markov chains and generalized wavelet multiresolutions. J. Anal. 26(2), 259–283 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  45. Jorgensen, P., Tian, F.: Dynamical properties of endomorphisms, multiresolutions, similarity and orthogonality relations. Discrete Contin. Dyn. Syst. Ser. S 12(8), 2307–2348 (2019)

    MathSciNet  MATH  Google Scholar 

  46. Jorgensen, P., Tian, J.: Noncommutative boundaries arising in dynamics and representations of the Cuntz relations. Numer. Funct. Anal. Optim. 41(5), 571–620 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  47. Kechris, A.S.: Classical descriptive set theory. Graduate Texts in Mathematics, vol. 156. Springer-Verlag, New York (1995)

  48. Kloeckner, B.: Optimal transportation and stationary measures for iterated function systems. Math. Proc. Camb. Philos. Soc. 173(1), 163–187 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  49. Kreĭn, M.G., Langer, H.: Über die verallgemeinerten Resolventen und die charakteristische Funktion eines isometrischen Operators im Raume \({\Pi _k}\). In Hilbert space operators and operator algebras (Proc. Int. Conf. Tihany, 1970), pp. 353–399. North-Holland, Amsterdam,: Colloquia Math. Soc, János Bolyai (1972)

  50. Natanson, I.P.: Theory of functions of a real variable. Volume I. Frederick Ungar Publishing Co., New York. Translated by Leo F. Boron with the collaboration of Edwin Hewitt (1955)

  51. Nazir, T., Khumalo, M., Makhoshi, V.: Iterated function system of generalized contractions in partial metric spaces. Filomat 35(15), 5161–5180 (2021)

    Article  MathSciNet  Google Scholar 

  52. Neveu, J.: Bases mathématiques du calcul des probabilités. Masson et Cie, Éditeurs, Paris (1964)

    MATH  Google Scholar 

  53. Papadakis, M., Šikić, H., Weiss, G.: The characterization of low pass filters and some basic properties of wavelets, scaling functions and related concepts. J. Fourier Anal. Appl. 5(5), 495–521 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  54. Riesz, F.: Untersuchungen über Systeme integrierbarer Funktionen. Math. Ann. 69(4), 449–497 (1910)

    Article  MathSciNet  MATH  Google Scholar 

  55. Simmons, D.: Conditional measures and conditional expectation; Rohlin’s disintegration theorem. Discrete Contin. Dyn. Syst. 32(7), 2565–2582 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  56. Takahashi, Y.: Invariant measures for iterated function systems with inverses. J. Fractal Geom. 9(1), 129–149 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  57. Zhang, Z., Jorgensen, P.: Hermite-wavelet transforms of multivariate functions on \([0,1]^d\). Acta Appl. Math. 170, 773–788 (2020)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Daniel Alpay thanks the Foster G. and Mary McGaw Professorship in Mathematical Sciences, which supported this research. The authors wish to thank Professor Sergey Bezuglyi for discussions on disentegration of measures.

Author information

Authors and Affiliations

  1. Schmid College of Science and Technology, Chapman University, One University Drive, Orange, CA, 92866, USA

    Daniel Alpay

  2. Department of Mathematics, 14 MLH, The University of Iowa, Iowa City, IA, 52242-1419, USA

    Palle Jorgensen

  3. School of Electrical and Computer Engineering, Ben-Gurion University of the Negev, P.O.B. 653, 84105, Beer-Sheva, Israel

    Izchak Lewkowicz

Authors
  1. Daniel Alpay
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Palle Jorgensen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Izchak Lewkowicz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Daniel Alpay.

Ethics declarations

Conflict of interest

The authors have no conflicts of interest to declare that are relevant to the content of this article.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Alpay, D., Jorgensen, P. & Lewkowicz, I. Representation theory and multilevel filters. J. Appl. Math. Comput. 69, 1599–1657 (2023). https://doi.org/10.1007/s12190-022-01805-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-022-01805-z

Keywords

Mathematics Subject Classification

Search

Navigation