Multivariate tile $\mathrm{B}$-splines

Cover Page

Cite item

Full Text

Open Access Open Access
Restricted Access Access granted
Restricted Access Subscription Access

Abstract

Tile $\mathrm{B}$-splines in $\mathbb R^d$ are defined as autoconvolutionsof indicators of tiles,which are special self-similar compact sets whose integer translates tilethe space $\mathbb R^d$. These functions are not piecewise-polynomial,however, being directgeneralizations of the classical $\mathrm{B}$-splines, they enjoy many oftheir properties and have some advantages. In particular, exactvalues of the Hölder exponents of tile $\mathrm{B}$-splinesare evaluated and are shown, in some cases, to exceed those of the classical $\mathrm{B}$-splines.Orthonormal systems of wavelets based on tile B-splines are constructed,and estimates of their exponential decay are obtained.Efficiency in applications of tile $\mathrm{B}$-splines is demonstrated on an example of subdivision schemesof surfaces. This efficiency is achieved due to high regularity, fast convergence, and small numberof coefficients in the corresponding refinement equation.

About the authors

Tatyana Ivanovna Zaitseva

Lomonosov Moscow State University, Faculty of Mechanics and Mathematics; Moscow Center for Fundamental and Applied Mathematics

without scientific degree, no status

References

  1. К. де Бор, Практическое руководство по сплайнам, Радио и связь, М., 1985, 304 с.
  2. И. Добеши, Десять лекций по вейвлетам, НИЦ “Регулярная и хаотическая динамика”, М.–Ижевск, 2001, 464 с.
  3. И. Я. Новиков, В. Ю. Протасов, М. А. Скопина, Теория всплесков, Физматлит, М., 2005, 613 с.
  4. P. Wojtaszczyk, A mathematical introduction to wavelets, London Math. Soc. Stud. Texts, 37, Cambridge Univ. Press, Cambridge, 1997, xii+261 pp.
  5. A. Yu. Shadrin, “The $L_{infty}$-norm of the $L_2$-spline projector is bounded independently of the knot sequence: a proof of de Boor's conjecture”, Acta Math., 187:1 (2001), 59–137
  6. C. de Boor, R. A. DeVore, A. Ron, “On the construction of multivariate (pre)wavelets”, Constr. Approx., 9:2-3 (1993), 123–166
  7. В. Ю. Протасов, “Фрактальные кривые и всплески”, Изв. РАН. Сер. матем., 70:5 (2006), 123–162
  8. П. А. Терехин, “О наилучшем приближении функций в метрике $L_p$ полиномами по аффинной системе”, Матем. сб., 202:2 (2011), 131–158
  9. C. De Boor, K. Höllig, S. Riemenschneider, Box splines, Appl. Math. Sci., 98, Springer-Verlag, New York, 1993, xviii+200 pp.
  10. M. Charina, C. Conti, K. Jetter, G. Zimmermann, “Scalar multivariate subdivision schemes and box splines”, Comput. Aided Geom. Design, 28:5 (2011), 285–306
  11. D. Van de Ville, T. Blu, M. Unser, “Isotropic polyharmonic B-splines: scaling functions and wavelets”, IEEE Trans. Image Process., 14:11 (2005), 1798–1813
  12. V. G. Zakharov, “Elliptic scaling functions as compactly supported multivariate analogs of the B-splines”, Int. J. Wavelets Multiresolut. Inf. Process., 12:02 (2014), 1450018, 23 pp.
  13. S. Zube, “Number systems, $alpha$-splines and refinement”, J. Comput. Appl. Math., 172:2 (2004), 207–231
  14. J. Lagarias, Yang Wang, “Integral self-affine tiles in $mathbb{R}^n$. II. Lattice tilings”, J. Fourier Anal. Appl., 3:1 (1997), 83–102
  15. K. Gröchenig, A. Haas, “Self-similar lattice tilings”, J. Fourier Anal. Appl., 1:2 (1994), 131–170
  16. K. Gröchenig, W. R. Madych, “Multiresolution analysis, Haar bases, and self-similar tilings of $mathbb{R}^n$”, IEEE Trans. Inform. Theory, 38:2, Part 2 (1992), 556–568
  17. A. S. Cavaretta, W. Dahmen, C. A. Micchelli, Stationary subdivision, Mem. Amer. Math. Soc., 93, no. 453, Amer. Math. Soc., Providence, RI, 1991, vi+186 pp.
  18. E. Catmull, J. Clark, “Recursively generated B-spline surfaces on arbitrary topological meshes”, Comput.-Aided Des., 10:6 (1978), 350–355
  19. T. Zaitseva, TZZZZ/Tile_Bsplines
  20. C. A. Cabrelli, C. Heil, U. M. Molter, Self-similarity and multiwavelets in higher dimensions, Mem. Amer. Math. Soc., 170, no. 807, Amer. Math. Soc., Providence, RI, 2004, viii+82 pp.
  21. A. Krivoshein, V. Protasov, M. Skopina, Multivariate wavelet frames, Ind. Appl. Math., Springer, Singapore, 2016, xiii+248 pp.
  22. G. Strang, G. Fix, “A Fourier analysis of the finite element variational method”, Constructive aspects of functional analysis (Erice, 1971), C.I.M.E. Summer Schools, 57, Cremonese, 1973, 793–840
  23. C. de Boor, R. A. DeVore, A. Ron, “The structure of finitely generated shift-invariant spaces in $L_2(mathbb{R}^d)$”, J. Funct. Anal., 119:1 (1994), 37–78
  24. C. de Boor, R. Devore, A. Ron, “Approximation from shift-invariant subspaces of $L_2(mathbb{R}^d)$”, Trans. Amer. Math. Soc., 341:2 (1994), 787–806
  25. I. Kirat, Ka-Sing Lau, “On the connectedness of self-affine tiles”, J. London Math. Soc. (2), 62:1 (2000), 291–304
  26. C. Bandt, G. Gelbrich, “Classification of self-affine lattice tilings”, J. London Math. Soc. (2), 50:3 (1994), 581–593
  27. G. Gelbrich, “Self-affine lattice reptiles with two pieces in $mathbb{R}^n$”, Math. Nachr., 178 (1996), 129–134
  28. W. J. Gilbert, “Radix representations of quadratic fields”, J. Math. Anal. Appl., 83:1 (1981), 264–274
  29. R. F. Gundy, A. L. Jonsson, “Scaling functions on $mathbb{R}^2$ for dilations of determinant $pm 2$”, Appl. Comput. Harmon. Anal., 29:1 (2010), 49–62
  30. C. Bandt, Combinatorial topology of three-dimensional self-affine tiles
  31. C. Bandt, “Self-similar sets. 5. Integer matrices and fractal tilings of $mathbb{R}^n$”, Proc. Amer. Math. Soc., 112:2 (1991), 549–562
  32. Xiaoye Fu, J.-P. Gabardo, Self-affine scaling sets in $mathbb R^2$, Mem. Amer. Math. Soc., 233, no. 1097, Amer. Math. Soc., Providence, RI, 2015, vi+85 pp.
  33. J. C. Lagarias, Yang Wang, “Haar type orthonormal wavelet bases in $mathbf{R}^2$”, J. Fourier Anal. Appl., 2:1 (1995), 1–14
  34. T. Zaitseva, “Haar wavelets and subdivision algorithms on the plane”, Adv. Syst. Sci. Appl., 17:3 (2017), 49–57
  35. V. G. Zakharov, “Rotation properties of 2D isotropic dilation matrices”, Int. J. Wavelets Multiresolut. Inf. Process., 16:1 (2018), 1850001, 14 pp.
  36. Т. И. Зайцева, В. Ю. Протасов, “Самоподобные 2-аттракторы и тайлы”, Матем. сб., 213:6 (2022), 71–110
  37. Б. В. Шабат, Введение в комплексный анализ, т. 1, 2, 4-е стереотип. изд., Лань, СПб., 2004, 336 с., 464 с.
  38. M. Charina, V. Yu. Protasov, “Regularity of anisotropic refinable functions”, Appl. Comput. Harmon. Anal., 47:3 (2019), 795–821
  39. M. Charina, Th. Mejstrik, “Multiple multivariate subdivision schemes: matrix and operator approaches”, J. Comput. Appl. Math., 349 (2019), 279–291
  40. М. Г. Крейн, М. А. Рутман, “Линейные операторы, оставляющие инвариантным конус в пространстве Банаха”, УМН, 3:1(23) (1948), 3–95
  41. В. Ю. Протасов, “Обобщенный совместный спектральный радиус. Геометрический подход”, Изв. РАН. Сер. матем., 61:5 (1997), 99–136
  42. V. D. Blondel, Yu. Nesterov, “Computationally efficient approximations of the joint spectral radius”, SIAM J. Matrix Anal. Appl., 27:1 (2005), 256–272
  43. V. D. Blondel, S. Gaubert, J. N. Tsitsiklis, “Approximating the spectral radius of sets of matrices in the max-algebra is NP-hard”, IEEE Trans. Automat. Control, 45:9 (2000), 1762–1765
  44. N. Guglielmi, V. Protasov, “Exact computation of joint spectral characteristics of linear operators”, Found. Comput. Math., 13:1 (2013), 37–97
  45. T. Mejstrik, “Algorithm 1011: improved invariant polytope algorithm and applications”, ACM Trans. Math. Software, 46:3 (2020), 29, 26 pp.
  46. P. Oswald, “Designing composite triangular subdivision schemes”, Comput. Aided Geom. Design, 22:7 (2005), 659–679
  47. P. Oswald, P. Schröder, “Composite primal/dual $sqrt{3}$-subdivision schemes”, Comput. Aided Geom. Design, 20:3 (2003), 135–164
  48. Qingtang Jiang, P. Oswald, “Triangular $sqrt 3$-subdivision schemes: the regular case”, J. Comput. Appl. Math., 156:1 (2003), 47–75
  49. Т. И. Зайцева, “Простые тайлы и аттракторы”, Матем. сб., 211:9 (2020), 24–59
  50. M. Charina, C. Conti, T. Sauer, “Regularity of multivariate vector subdivision schemes”, Numer. Algorithms, 39:1-3 (2005), 97–113
  51. N. Dyn, D. Levin, “Subdivision schemes in geometric modelling”, Acta Numer., 11 (2002)
  52. V. Protasov, “The stability of subdivision operator at its fixed point”, SIAM J. Math. Anal., 33:2 (2001), 448–460
  53. C. Conti, K. Jetter, “Concerning order of convergence for subdivision”, Numer. Algorithms, 36:4 (2004), 345–363
  54. A. Cohen, K. Gröchenig, L. F. Villemoes, “Regularity of multivariate refinable functions”, Constr. Approx., 15:2 (1999), 241–255
  55. D. Mekhontsev, IFStile software

Supplementary files

Supplementary Files
Action
1. JATS XML
Download

Copyright (c) 2023 Zaitseva T.I.

Согласие на обработку персональных данных

 

Используя сайт https://journals.rcsi.science, я (далее – «Пользователь» или «Субъект персональных данных») даю согласие на обработку персональных данных на этом сайте (текст Согласия) и на обработку персональных данных с помощью сервиса «Яндекс.Метрика» (текст Согласия).