Valerii T. Shevaldin     (N.N. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences; Ural Federal University, Ekaterinburg, Russian Federation)


The paper deals with generalized linear and parabolic B-splines with the uniform nodes constructed by means only one function \(\varphi(x)\). For such splines in this paper conditions have been found that guarantee satisfaction of two-scale relations.


B-spline, Uniform nodes, Two-scale relations

Full Text:



  1. Alberg J., Nilson E., Walsh J. Theory of splines and their applications. Moscow: Mir, 1972. 318 p. [in Russian]
  2. Shevaldin V.T. Estimations from below of diameters of classes of source-represented functions // Trudy Steklov Math. Institute of RAS, 1989. Vol. 189. P. 185–201. [in Russian]
  3. Rvachev V.A. Finite solutions of functional-differentional equations and their applications // Uspekhy Math. Nauk, 1990. Vol. 45, no. 1. P. 77–103. [in Russian]
  4. Leontiev V.L. Orthogonal finite functions and numerical methods. Ulyanovsk: Ulyanovsk State University, 2003. 181 p. [in Russian]
  5. Kvasov B.I. Methods for the iso-geometric approximation by splines. Moscow: Fizmatlit, 2006. 360 p. [in Russian]
  6. Demyanovich Yu.K. Wavelet basis for \(B_{\varphi}\)–splines on a non-uniform mesh // Math. Modelling, 2006. Vol. 18, no. 10. P. 123–126. [in Russian]
  7. Shevaldin V.T. Three-point scheme for approximation by local splines // Proceedings of International Summer Math. School by the name of S.B. Stechkin on the Theory of Functions. Tula: Tula State University, 2007. P. 151–156. [in Russian]
  8. Chui Ch. Introduction into wavelets. Moscow: Mir, 2001. 412 p. [in Russian]
  9. Subbotin Yu.N., Chernykh N.I. Construction of \(W_2^m(\mathbb{R})\) wavelets and their approximative properties in various metrics // Proc. of Instit. of Math. and Mech. Ural Branch of RAS, 2005. Vol. 11, no. 2. P. 131–167. [in Russian]
  10. Novikov I.Ya., Protasov V.Yu., Skopina M.A. Theory of wavelets. Moscow: Fizmatlit, 2005. 616 p. [in Russian]
  11. Zavyalov Yu.S., Kvasov B.I., Miroshnichenko V.L. Spline–functions methods. Moscow: Nauka, 1980. 355 p. [in Russian]
  12. Shevaldin V.T. Calibration relations for B-L-splines // Modern problems of mathematics: Abstracts of 42-nd Russian Youth Conference. Instit. of Math. and Mech. Ural Branch of RAS: Ekaterinburg, 2011.P. 151–153. [in Russian]


Article Metrics

Metrics Loading ...


  • There are currently no refbacks.