Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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A NUMBER SYSTEM IN ℝn

  • Published : 2004.11.01
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Abstract

In this paper, we establish a number system in $R^n$ which arises from a Haar wavelet basis in connection with decompositions of certain Cuntz algebra representations on $L^2$( $R^n$). Number systems in $R^n$ are also of independent interest [9]. We study radix-representations of $\chi$ $\in$ $R^n$: $\chi$:$\alpha$$_{ι}$ $\alpha$$_{ι-1}$…$\alpha$$_1$$\alpha$$_{0}$ ㆍ$\alpha$$_{-1}$ $\alpha$$_{-2}$ … as $\chi$= $M^{ι}$$\alpha$$_{ι}$ $\alpha$＋…M$\alpha$$_1$＋$\alpha$$_{0}$ ＋ $M^{-1}$ $\alpha$$_{-1}$ ＋ $M^{-2}$ $\alpha$$_{-2}$ ＋… where each $\alpha$$_{k}$ $\in$ D, and D is some specified digit set. Our analysis uses iteration techniques of a number-theoretic flavor. The view-point is a dual one which we term fractals in the large vs. fractals in the small,illustrating the number theory of integral lattice points vs. fractions.s vs. fractions.

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References

  1. Christoph Bandt, Self-similar sets V: Integer matrices and fractal tilings of $R^n$, Proc. Amer. Math. Soc. 112 (1991), 549–562. https://doi.org/10.1090/S0002-9939-1991-1036982-1
  2. Ola Bratteli and P. E. T. Jorgensen, Iterated function system and permutation representations of the Cuntz algebra, Mem. Amer. Math. Soc., to appear.
  3. Ola Bratteli and P. E. T. Jorgensen, Isometries, shifts, Cuntz algebras multiresolution wavelet analysis of scale N, Integral Equ. Operator Theory 28 (1997), 382-443. https://doi.org/10.1007/BF01309155
  4. O. Bratteli, P. E. T. Jorgensen, and G. L. Price, Endomorpism of B(H), Proceedings of Symposia in prime Mathematics 59 (1996), 93–138.
  5. Joachim Cuntz, Simple $C^{\ast}$-algebras generated by isometries, Comm. Math. Phys. 57 (1977), 173–185. https://doi.org/10.1007/BF01625776
  6. Ingrid Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conf. Ser. in Appl. Math. 61 (1992).
  7. K. Grochenig and W. R. Madych, Multiresolution analysis, Haar bases, and self-similar tilings of $R^n$, IEEE Trans. Inform. Theory 38 (1992), 556–568. https://doi.org/10.1109/18.119723
  8. Eui-Chai Jeong, Irreducible representations of the cuntz algebra $O_n$, Proc. Amer. Math. Soc. 127 (1999), 3582–3590. https://doi.org/10.1090/S0002-9939-99-05018-2
  9. Donald E. Knuth, The art of computer programming: Vol. 2: Seminumerical algorithms, 2nd ed., Addison-Wesley Publishing Co., Reading, Mass., 1969.
  10. M. Laca, Endomorphisms of B(H) and Cuntz algebras, J. Operator Theory 30 (1993), 85–180.
  11. J. C. Lagarias and Yang Wang, Integral self-affine tiles in $R^n$, II: Lattice tilings, J. Fourier Anal. Appl. 3 (1997), 83–102. https://doi.org/10.1007/BF02647948
  12. A. M. Odlyzko, non-negative digit sets, Proc. London Math. Soc. 37 (1978), no. 3, 213–229. https://doi.org/10.1112/plms/s3-37.2.213
  13. B. M. Stewart, Theory of Numbers, The Macmillan Co., New York, 1964.