Skip to main content
SpringerLink
Log in

Piecewise Polynomial Approximation Methods in the Theory of Nikol’Skiĭ–Besov Spaces

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

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

References

  1. D. Adams and L. Hedberg, Function Spaces and Potential Theory, Springer, Berlin (1996).

    Book  Google Scholar 

  2. O. Blasco and S. Pott, “Dyadic BMO on the bidisk,” Rev. Math. Iberoam., 21, No. 2, 483–510 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  3. J. Bergh and J. L¨ofstr¨om, Interpolation Spaces. Introduction [Russian translation], Mir, Moscow (1980).

  4. O. V. Besov, “On some families of functional spaces. Imbedding and extension theorems,” Dokl. Akad. Nauk SSSR, 126, 1163–1165 (1959).

    MathSciNet  MATH  Google Scholar 

  5. O. V. Besov, “On conditions of belonging to Lp for derivatives of periodic functions,” Nauch. Dokl. Vyssh. Shk., 1, 12–17 (1959).

    Google Scholar 

  6. O. V. Besov, “Investigation of a class of function spaces in connection with imbedding and extension theorems,” Tr. Mat. Inst. Steklova, 60, 42–81 (1961).

    MathSciNet  MATH  Google Scholar 

  7. O. V. Besov, V. P. Il’in, and S. M. Nikol’skiǐ, Integral Representations of Functions and Embedding Theorems [in Russian], Nauka, Moscow (1975).

  8. M. Sh. Birman and M. Z. Solomyak, “Piecewise-polynomial approximations of functions of the class Wα p ,” Sb. Math., 2, 295–317 (1968).

    Article  Google Scholar 

  9. Ju.A. Brudnyi, “Piecewise polynomial approximation and local approximations,” Dokl. Akad. Nauk SSSR, 201, 16–18 (1971).

    MathSciNet  Google Scholar 

  10. Ju.A. Brudnyi, “Spaces that are definable by means of local approximations,” Tr. Mosk. Mat. Obshch., 24, 69–132 (1971).

    MathSciNet  MATH  Google Scholar 

  11. Ju.A. Brudnyi, “The rearrangement of a smooth function,” Usp. Mat. Nauk, 27, No. 2(164), 165-166 (1972).

  12. Ju.A. Brudnyi, “Spline approximation and functions of bounded variation,” Dokl. Akad. Nauk SSSR, 215, 511–513 (1974).

    MathSciNet  Google Scholar 

  13. Ju.A. Brudnyi, “Approximation spaces,” Geometry of Linear Spaces and Operator Theory, Yaroslavl’ State Univ., Yaroslavl’, 3–30 (1977).

  14. Ju.A. Brudnyi, “Rational approximation and imbedding theorems,” Dokl. Akad. Nauk SSSR, 247, No. 2, 269–272 (1979).

    MathSciNet  Google Scholar 

  15. Ju.A. Brudnyi, “Adaptive approximation of functions with singularities,” Trans. Moscow Math. Soc., 1994, 123–186 (1995).

    MathSciNet  Google Scholar 

  16. Ju.A. Brudnyi and M. I. Ganzburg, “A certain extremal problem for polynomials in n variables,” Izv. Akad. Nauk SSSR Ser. Mat., 37, 344–355 (1973).

  17. Ju.A. Brudnyi and I.E. Gopengauz, “Approximation by piecewise polynomial functions,” Izv. Akad. Nauk SSSR Ser. Mat., 27, 723–743 (1963).

  18. Ju.A. Brudnyi and I.P. Irodova, “Nonlinear spline-approximation and B-spaces,” Proc. Int. Conf. on Approximation Theory, Kiev, 1983, Nauka, Moscow, 71–75 (1987).

  19. Ju.A. Brudnyi and I.P. Irodova, “Nonlinear spline approximation of functions of several variables and B-spaces,” St. Petersburg Math. J., 4, No. 4, 667–694 (1993).

  20. Ju.A. Brudnyi and N. Ja. Krugljak, “A family of approximation spaces,” Studies in the Theory of Functions of Several Real Variables, No. 2, Yaroslavl’ State Univ., Yaroslavl’, 15–42 (1978).

  21. H. G. Buchard, “Splines (with optimal knot) are better,” J. Appl. Anal., 3, 309–319 (1974).

    Article  Google Scholar 

  22. G. J. Butler and F.B. Richards, “An Lp saturation theorem for splines,” Can. J. Math., XXIV, No. 5, 957–966 (1972).

  23. T. G. Bychkova, “Optimal expansions with respect to atoms in dyadic Hardy Hp-spaces and the interpolation of Hp-spaces,” Materials of All-Russsian Reseach Conference Devoted to the 200th Anniversary of the Demidov Yaroslavl’ State University, Yaroslavl’, 50–69 (2003).

  24. N. I. Chernykh, “Approximations by splines with given densities of knots distributions,” Tr. Mat. Inst. Steklova, 138, 174–197 (1975).

    MATH  Google Scholar 

  25. Z. Ciesielski, “Constructive function theory and spline systems,” Stud. Math., 52, 277–302 (1973).

    MathSciNet  Google Scholar 

  26. A. Cohen, R.A. Devore, P.P. Petrushev, and H. Xu, “Nonlinear approximation and the space BV (R2),” Am. J. Math., 121, 587–628 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  27. H. B. Curry and I. J. Schönberg, “On spline distributions and their limits the Pólya distribution functions,” Bull. Am. Math. Soc., 53, No. 11, 1114 (1947).

  28. J. E. Daly and S. Fridli, “Walsh multipliers for dyadic Hardy spaces,” Appl. Anal., 82, No. 7, 689–700 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  29. C. De Boor, “The quasi-interpolant as a tool in elementary polynomial spline theory,” Approximation Theory, Academic Press, N.Y., 266–276 (1973).

  30. C. De Boor, Practical Guide to Splines, Springer, N.Y. (1978).

    Book  MATH  Google Scholar 

  31. C. De Boor and G. J. Fix, “Spline approximation by quasiinterpolants,” J. Approx. Theory, 8, No. 1, 19–45 (1973).

    Article  MATH  Google Scholar 

  32. R. A. Devore, B. Jawerth, and V.A. Popov, “Compression of wavelet decompositions,” Am. J. Math., 114, 737–785 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  33. R. A. Devore, P.P. Petruschev, and X. M. Yu, “Nonlinear wavelet approximation in the space C(Rd),” Comput. Math., 19, 261–283 (1992).

    Google Scholar 

  34. R. A. Devore and V. A. Popov, “Free multivariate splines,” Constr. Approx., 3, 239–248 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  35. R. A. Devore and V. A. Popov, “Interpolation of Besov spaces,” Trans. Am. Math. Soc., 305, 397–414 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  36. R. A. Devore and F. B. Richards, “Saturation and inverse theorems for spline approximation,” Spline Functions and Approximation Theory, Birkhauser, Basel, 73–82 (1972).

  37. R.A. Devore and K. Scherer, “A constructive theory for approximation by splines with an arbitrary sequence of knot sets,” Approximation Theory, Springer, Berlin, 167–183 (1976).

  38. J.R. Dorronsoro, “Characterization of potential spaces,” Proc. Am.Math. Soc., 95, No. 1, 21–31 (1985).

  39. V.K. Dzyadyk, Introduction to the Theory of Uniform Approximation of Functions by Polynomials [in Russian], Nauka, Moscow (1977).

  40. J. Garnett, Bounded Analytic Functions [Russian translation], Mir, Moscow (1984).

  41. J. B. Garnett and P. W. Jones, “BMO from dyadic BMO,” Pacific J. Math., 99, No. 2, 351–371 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  42. S. Geiss, R. Müller, and V. Pillwein, “A remark on extrapolation of rearrangement operators on dyadic H p s , 0 < s < 1,” Stud. Math., 171, No. 2, 197–205 (2005)

  43. K.K. Golovkin, “A generalization of Marcinkiewicz’s interpolation theorem,” Tr. Mat. Inst. Steklova, 102, 5–28 (1967).

    MathSciNet  MATH  Google Scholar 

  44. B. I. Golubov, “On a modified strong dyadic integral and derivative,” Sb. Math., 193, No. 3-4, 507–529 (2002).

  45. B. I. Golubov, Elements of Dyadic Analysis [in Russian], Moskovskiǐ Gosudarstvennyǐ Universitet Pechati, Moscow (2005).

  46. C. S. Herz, “Lipschitz spaces and Bernstein theorem of absolutely convergent Fourier transforms,” J. Math. Mech., 18, No. 4, 283–323 (1968).

    MathSciNet  MATH  Google Scholar 

  47. I.P. Irodova, “On the properties of the scale of spaces B p , (λθ) for 0 < p < 1,” Sov. Math., Dokl., 21, 53–55 (1980).

  48. I.P. Irodova, “Properties of functions given by the rate of decrease of piecewise polynomial approximation,” Studies in the Theory of Functions of Several Real Variables, Yaroslavl’ State Univ., Yaroslavl’, 92–117 (1980).

  49. I.P. Irodova, “On properties of the scale of spaces Bλθ p in the case 0 < p < 1,” Studies in the Theory of Functions of Several Real Variables, Yaroslavl’ State Univ., Yaroslavl’, 57–79 (1982).

  50. I.P. Irodova, A Joint Approximation of a Function and Its Derivatives in Lp ([0, 1]n) via Nonlinear Splines, Yaroslavl’, 1982 [deponated in VINITI, No 1135-82]

  51. I.P. Irodova, “Generalization of the Marchaud inequality,” Studies in the Theory of Functions of Several Real Variables, Yaroslavl’ State Univ., Yaroslavl’, 63–69 (1984).

  52. I.P. Irodova, “Properties of dyadic Besov spaces,” Proc. Int. PFU Conf. on Functions Spaces, Differential Operators, and Problems of Mathematical Education, No. 1, PFU, Moscow, 78–81 (1998).

  53. I.P. Irodova, “Jackson and Bernstein inequalities for dyadic Besov spaces,” Izv. Tul. Gos. Univ. Ser. Mat. Mekh. Inform., 4, No. 1, 83–86 (1998).

    MathSciNet  Google Scholar 

  54. I.P. Irodova, “Dyadic Besov spaces,” St. Petersburg Math. J., 12, No. 3, 379–405 (2001).

    MathSciNet  MATH  Google Scholar 

  55. I.P. Irodova, “Jackson inequality for nondiagonal dyadic Besov spaces,” Optimization of Finite Element Approximation and Splines and Wavelets. Proc. Second Int. Conf. OFEA’2001, St.- Petersburg, 138-139 (2001).

  56. I.P. Irodova, “Constructing algorithms for smooth splines,” Materials of Second Russsian Conference “Mathematics in Contemporary World,” Kaluga, 100–105 (2004).

  57. I.P. Irodova, “On descripition of the continuity modulus in terms of piecewise polynomial approximations,” Izv. Tul. Gos. Univ. Ser. Mat. Mekh. Inform., 11, No. 1, 148–155 (2005).

    MathSciNet  Google Scholar 

  58. I.P. Irodova, “On dyadic Nikol’skiǐ–Besov spaces,” Function Spaces, Approximation Theory, Nonlinear Analysis. Abstr. Conf. Devoted to 100th Birthday of Academician Nikol’skiǐ, Moscow, 115 (2005).

  59. I.P. Irodova, “Piecewise polynomial approximations and the continuity modulus,” Mathematics in Yaroslavl’ University, Yaroslavl’ State Univ., Yaroslavl’, 227–244 (2006).

  60. I.P. Irodova, “On the computation of the K-functional of a pair of B-spaces,” Izv. Tul. Gos. Univ. Ser. Mat. Mekh. Inform., 12, No. 1, 109–123 (2006).

    MathSciNet  Google Scholar 

  61. I.P. Irodova, “Dyadic derivatives and their properties,” Izv. Tul. Gos. Univ. Ser. Nat. Sci., No. 1, 29–36 (2008).

  62. I.P. Irodova, “Dyadic Nikol’skiǐ–Besov spaces and their relationship to classical spaces,” Math. Notes, 83, No. 5-6, 624–634 (2008).

  63. I.P. Irodova, “On Bernstein-type inequalities,” Model. Anal. Inform. Syst., 15, No. 4, 31–41 (2008).

    Google Scholar 

  64. I.P. Irodova, “On Jackson-type inequalities in dyadic BMO-spaces,” Model. Anal. Inform. Syst., 16, No. 3, 29–46 (2009).

    Google Scholar 

  65. I.P. Irodova, “On the computation of K-functionals,” St. Petersburg Math. J., 21, No. 4, 579– 599 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  66. I.P. Irodova and M. V. Nevskiǐ, “Dyadic Besov spaces and other topics of approximation theory,” Mathematics in Yaroslavl’ University, Yaroslavl’ State Univ., Yaroslavl’, 115–131 (2001).

  67. F. John and L. Nirenberg, “On functions of bounded mean oscillation,” Commun. Pure Appl. Math., 14, 415–426 (1961).

    Article  MathSciNet  MATH  Google Scholar 

  68. N. Ja. Krugljak and M. V. Nevskiǐ, “Almost optimality of dyadic approximation algorithms for Lp([0, 1]n) and real interpolation,” Algebra i Analiz, 14, No. 4, 63–90 (2002).

  69. L.D. Kudryavtsev and S.M. Nikol’skiǐ, “Spaces of differentiable functions of several variables and imbedding theorems,” Encyclopaedia Math. Sci., 26, 1–140 (1991).

    Google Scholar 

  70. A. Marchaud, “Sur les derivées et sur les différences des fonctions de variables réebles,” J. Math. Pures Appl., 6, 337–425 (1927).

    MATH  Google Scholar 

  71. V.E. Mayorov, “On best approximations of Wr 1 (Is)-classes in L∞(Is)-spaces,” Mat. Zametki, 19, No. 5, 699–706 (1976).

  72. T. Mei, “BMO is the intersection of two translates of dyadic BMO,” C.R. Math. Acad. Sci. Paris, 336, No. 12, 1003–1006 (2003).

  73. Yu.V. Netrusov, “Nonlinear approximation of functions in Besov–Lorentz spaces in the uniform metric,” J. Math. Sci., 79, No. 5, 1308–1319 (1996).

    Article  MathSciNet  Google Scholar 

  74. M. V. Nevskiǐ, On the Rate of Piecewise Polynomial Approximations in Orlicz Classes [in Russian], Yaroslavl’, 1986 [deponated in VINITI, No 3225-B 86].

  75. S. M. Nikol’skiǐ, Approximation of Functions of Several Variables and Embedding Theorems [in Russian], Nauka, Moscow (1977).

  76. K. I. Oskolkov, “Polygonal approximation of functions of two variables,” Mat. Sb., 107 (149), No. 4, 601–612 (1978).

  77. P. Oswald, “On the degree for nonlinear spline approximation un Besov–Sobolev spaces,” J. Approx. Theory, 61, 131–157 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  78. J. Peetre, “Espaces d’interpolation et théoreme de Soboleff,” Ann. Inst. Fourier, 16, 279–317 (1966).

    Article  MathSciNet  MATH  Google Scholar 

  79. J. Peetre, Thoughts on Besov Spaces, Lund Univ., Lund (1966).

    Google Scholar 

  80. J. Peetre, “Remarques sur les espaces de Besov. Le cas 0 < p < 1,” C.R. Math. Acad. Sci. Paris, 277, 947–949 (1973).

  81. J. Peetre, New Thoughts on Besov Spaces, Duke Univ. Math., Durham, N.C (1976).

  82. J. Peetre and G. Sparr, “Interpolation of normed Abelian groups,” Ann. Mat. Pura Appl. (4), 92, No. 4, 217–262 (1972).

  83. J. Peetre and E. Svensson, “On the Generalized Hardy’s inequality of Mcgehee, Pigno, and Smith and the problem of interpolation between BMO and Besov space,” Math. Scand., 54, 221–241 (1984).

    MathSciNet  MATH  Google Scholar 

  84. B. I. Peleshenko, “Structure theorems for a class of function spaces,” Zap. Nauchn. Semin. LOMI, 39, 191 (1974).

    MATH  Google Scholar 

  85. P.P. Petruschev, “Direct and converse theorems for spline and rational approximation and Besov spaces,” Lecture Notes in Math., 1302, 363–377 (1988).

    Article  Google Scholar 

  86. J. Pirher and L. Ward, “BMO from dyadic BMO on the bidisk,” J. London Math. Soc. (2), 77, No. 2, 524–544 (2008).

  87. N. Popa, “Isomorphism theorems for dyadic Hardy spaces generated by a rearrangement invariant space X on unit interval,” Rev. Roum. Math. Pures Appl., 35, No. 3, 261–281 (1990).

    MATH  Google Scholar 

  88. V. A. Popov, “On the connection between rational and spline approximation,” C.R. Acad. Bulgare Sci., 27, 623–626 (1974).

  89. V.A. Popov and G. Freud, “Lower error bounds in the theory of spline functions,” Stud. Sci. Math. Hungar., 6, 387–391 (1971).

    MathSciNet  Google Scholar 

  90. Rong-Qing Jia, “A Bernstein-type inequality associated with wavelet decomposition,” Constr. Approx., 9, 299–318 (1993).

  91. Y. Sagher and P. Shvartsman, “On the John–Strömberg–Torchinsky characterization of BMO,” Fourier Anal. Appl., 4, No. 4-5, 521–548 (1998).

  92. K. Scherer, “Characterization of generalized Lipschitz classes by best approximation with splines,” SIAM. J. Numer. Anal., 11, No. 2, 283–304 (1974).

    Article  MathSciNet  MATH  Google Scholar 

  93. F. Schipp, “On equivalence of rearrangement of the Haar system in dyadic Hardy and BMO spaces,” Anal. Math., 16, No. 16, 135–141 (1990).

  94. E. Stein, “The characterization of functions arising as potentials,” Bull. Am. Math. Soc., 67, No. 1, 102–104 (1961).

    Article  Google Scholar 

  95. E. Stein, Singular Integrals and Differentiability Properties of Functions [Russian translation], Mir, Moscow (1973).

  96. È. A. Storoženko, “Jackson theorems in Lp-spaces, 0 < p < 1,” Sibirsk. Mat. Zh., 19, No. 4, 630–639 (1978).

  97. R. S. Strichartz, “Multipliers on fractional Sobolev spaces,” J. Math. Mech., 16, 1031–1042 (1967).

    MathSciNet  MATH  Google Scholar 

  98. Yu.N. Subbotin, “Spline approximations and smooth bases in E(0, 2π),” Mat. Zametki, 12, No. 1, 43–52 (1972).

  99. Yu.N. Subbotin and N. I. Chernykh, “Order of the best spline approximations of some classes of functions,” Mat. Zametki, 7, 20–26 (1970).

  100. A. F. Timan, Theory of Approximation of Functions of a Real Variable [in Russian], Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow (1960).

  101. H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, Veb Deutscher Verlag der Wissenschaften, Berlin (1978).

  102. H. Triebel, Theory of Function Spaces [Russian translation], Mir, Moscow (1986).

  103. P. L. Ul’yanov, “Embedding theorems and relations between best approximations (continuity moduli) in different metrics,” Mat. Sb., 81, No. 1, 104–131 (1970).

    MathSciNet  Google Scholar 

  104. D. Yager, “Saturation bei Splines,” Approx. Quadratur. Numer. Math., 16, 129–140 (1970).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Yaroslavl Demidov State University, Yaroslavl, Russia

    I. P. Irodova

Authors
  1. I. P. Irodova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to I. P. Irodova.

Additional information

Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 50, Functional Analysis, 2013.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Irodova, I.P. Piecewise Polynomial Approximation Methods in the Theory of Nikol’Skiĭ–Besov Spaces. J Math Sci 209, 319–480 (2015). https://doi.org/10.1007/s10958-015-2506-2

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-015-2506-2

Keywords

Search

Navigation