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
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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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 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, 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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 A. Šarma and I. Cimbalario, Some linear differential operators, and generalized differences, Mat. Zametki 21 (1977), no. 2, 161172 (Russian). MR 0437989 (55 #10910)
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 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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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
PII:
S 00029939(2014)118843
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.
