A generalized modulus of smoothness
Authors:
Borislav R. Draganov and Kamen G. Ivanov
Journal:
Proc. Amer. Math. Soc. 142 (2014), 15771590
MSC (2010):
Primary 41A25; Secondary 41A15, 41A27
Published electronically:
February 6, 2014
MathSciNet review:
3168465
Fulltext PDF
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We construct moduli of smoothness that generalize the wellknown classical moduli and possess similar properties. They are related to a linear differential operator just as the classical moduli are related to the ordinary derivative. The generalized moduli are used to characterize the approximation error of the corresponding splines in , .
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 Ronald A. DeVore and George G. Lorentz, Constructive approximation, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 303, SpringerVerlag, Berlin, 1993. MR 1261635 (95f:41001)
 [2]
 Z. Ditzian, Polynomial approximation and twenty years later, Surv. Approx. Theory 3 (2007), 106151. MR 2342231 (2008f:41010)
 [3]
 Z. Ditzian and K. G. Ivanov, Strong converse inequalities, J. Anal. Math. 61 (1993), 61111. MR 1253439 (94m:41038), http://dx.doi.org/10.1007/BF02788839
 [4]
 Z. Ditzian and V. Totik, Moduli of smoothness, Springer Series in Computational Mathematics, vol. 9, SpringerVerlag, New York, 1987. MR 914149 (89h:41002)
 [5]
 Borislav R. Draganov and Kamen G. Ivanov, A new characterization of weighted Peetre functionals, Constr. Approx. 21 (2005), no. 1, 113148. MR 2105393 (2005i:41028), http://dx.doi.org/10.1007/s0036500405745
 [6]
 Borislav R. Draganov and Kamen G. Ivanov, A new characterization of weighted Peetre functionals. II, Serdica Math. J. 33 (2007), no. 1, 59124. MR 2313796 (2009d:46041)
 [7]
 Borislav R. Draganov and Kamen G. Ivanov, Equivalence between functionals based on continuous linear transforms, Serdica Math. J. 33 (2007), no. 4, 475494. MR 2418191 (2010a:46057)
 [8]
 K. G. Ivanov, Some characterizations of the best algebraic approximation in , C. R. Acad. Bulgare Sci. 34 (1981), no. 9, 12291232. MR 649151 (83i:41021)
 [9]
 K. G. Ivanov, A characterization of weighted Peetre functionals, J. Approx. Theory 56 (1989), no. 2, 185211. MR 982846 (90h:41038), http://dx.doi.org/10.1016/00219045(89)901093
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 J. W. Jerome and L. L. Schumaker, On the distance to a class of generalized splines, Linear operators and approximation, II (Proc. Conf., Math. Res. Inst., Oberwolfach, 1974), Internat. Ser. Numer. Math., Vol. 25, Birkhäuser, Basel, 1974, pp. 503517. MR 0393948 (52 #14755)
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 Tom Lyche and Larry L. Schumaker, spline wavelets, Wavelets: theory, algorithms, and applications (Taormina, 1993) Wavelet Anal. Appl., vol. 5, Academic Press, San Diego, CA, 1994, pp. 197212. MR 1321430 (96c:42069)
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 Larry L. Schumaker, Lower bounds for spline approximation, Approximation theory (Papers, VIth Semester, Stefan Banach Internat. Math. Center, Warsaw, 1975), Banach Center Publ., vol. 4, PWN, Warsaw, 1979, pp. 213223. MR 553767 (81h:41011)
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 Larry L. Schumaker, Spline functions: basic theory, Pure and Applied Mathematics, A WileyInterscience Publication, John Wiley & Sons Inc., New York, 1981. MR 606200 (82j:41001)
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 Karl Scherer and Larry L. Schumaker, A dual basis for splines and applications, J. Approx. Theory 29 (1980), no. 2, 151169. MR 595599 (82i:41015), http://dx.doi.org/10.1016/00219045(80)901136
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 A. Šarma and I. Cimbalario, Some linear differential operators, and generalized differences, Mat. Zametki 21 (1977), no. 2, 161172 (Russian). MR 0437989 (55 #10910)
 [16]
 V. T. Ševaldin, Extremal interpolation with smallest value of the norm of a linear differential operator, Mat. Zametki 27 (1980), no. 5, 721740, 830 (Russian). MR 578257 (81k:41002)
 [17]
 V. T. Shevaldin, Some problems of extremal interpolation in the mean for linear differential operators, Orthogonal series and approximations of functions, Trudy Mat. Inst. Steklov. 164 (1983), 203240 (Russian). MR 752926 (85m:41038)
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 R. S. Varga, Functional analysis and approximation theory in numerical analysis, Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 3, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1971. MR 0310504 (46 #9602)
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Additional Information
Borislav R. Draganov
Affiliation:
Department of Mathematics and Informatics, University of Sofia, 5 James Bourchier Boulevard, 1164 Sofia, Bulgaria – and – Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, bl. 8 Acad. G. Bonchev Str., 1113 Sofia, Bulgaria
Email:
bdraganov@fmi.unisofia.bg
Kamen G. Ivanov
Affiliation:
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, bl. 8 Acad. G. Bonchev Str., 1113 Sofia, Bulgaria
Email:
kamen@math.bas.bg
DOI:
http://dx.doi.org/10.1090/S000299392014118843
Keywords:
Modulus of smoothness,
$K$functional,
rate of convergence,
$L$spline,
linear operator
Received by editor(s):
February 11, 2012
Received by editor(s) in revised form:
May 7, 2012, and May 30, 2012
Published electronically:
February 6, 2014
Additional Notes:
Both authors were supported by grant DDVU 02/30 of the Fund for Scientific Research of the Bulgarian Ministry of Education and Science.
Communicated by:
Walter Van Assche
Article copyright:
© Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
