A Characterization of a Modulus of Smoothness in Multidimensional Setting

Laura Angeloni

Bollettino dell'Unione Matematica Italiana (2011)

  • Volume: 4, Issue: 1, page 79-108
  • ISSN: 0392-4041

Abstract

top
A classical result of approximation theory states that lim δ → 0 ⁡ ω ⁢ ( f , δ ) = 0 , where ω is the modulus of smoothness of f defined by means of the variation functional, if and only if f is absolutely continuous. Such theorem is crucial in order to obtain results about convergence and order of approximation for linear and non-linear integral operators in BV-spaces. It was an open problem to extend the above result to the setting of φ -variation in the multidimensional frame. In this paper, working with a concept of multidimensional W-variation introduced in [3], we prove that an analogous characterization holds for the multidimensional φ -modulus of smoothness.

How to cite

top

Angeloni, Laura. "A Characterization of a Modulus of Smoothness in Multidimensional Setting." Bollettino dell'Unione Matematica Italiana 4.1 (2011): 79-108. <http://eudml.org/doc/290753>.

@article{Angeloni2011,
abstract = {A classical result of approximation theory states that $\lim_\{\delta \to 0\} \omega(f, \delta) = 0$, where $\omega$ is the modulus of smoothness of $f$ defined by means of the variation functional, if and only if $f$ is absolutely continuous. Such theorem is crucial in order to obtain results about convergence and order of approximation for linear and non-linear integral operators in BV-spaces. It was an open problem to extend the above result to the setting of $\varphi$-variation in the multidimensional frame. In this paper, working with a concept of multidimensional W-variation introduced in [3], we prove that an analogous characterization holds for the multidimensional $\varphi$-modulus of smoothness.},
author = {Angeloni, Laura},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {2},
number = {1},
pages = {79-108},
publisher = {Unione Matematica Italiana},
title = {A Characterization of a Modulus of Smoothness in Multidimensional Setting},
url = {http://eudml.org/doc/290753},
volume = {4},
year = {2011},
}

TY - JOUR
AU - Angeloni, Laura
TI - A Characterization of a Modulus of Smoothness in Multidimensional Setting
JO - Bollettino dell'Unione Matematica Italiana
DA - 2011/2//
PB - Unione Matematica Italiana
VL - 4
IS - 1
SP - 79
EP - 108
AB - A classical result of approximation theory states that $\lim_{\delta \to 0} \omega(f, \delta) = 0$, where $\omega$ is the modulus of smoothness of $f$ defined by means of the variation functional, if and only if $f$ is absolutely continuous. Such theorem is crucial in order to obtain results about convergence and order of approximation for linear and non-linear integral operators in BV-spaces. It was an open problem to extend the above result to the setting of $\varphi$-variation in the multidimensional frame. In this paper, working with a concept of multidimensional W-variation introduced in [3], we prove that an analogous characterization holds for the multidimensional $\varphi$-modulus of smoothness.
LA - eng
UR - http://eudml.org/doc/290753
ER -

References

top
  1. ANGELONI, L. - VINTI, G., Convergence in Variation and Rate of Approximation for Nonlinear Integral Operators of Convolution Type, Results Math., 49 (1-2) (2006), 1-23. Erratum: 57 (2010), 387-391. Zbl1110.41006MR2651122DOI10.1007/s00025-010-0019-3
  2. ANGELONI, L. - VINTI, G., Approximation by means of nonlinear integral operators in the space of functions with bounded φ -variation, Differential Integral Equations, 20 (3) (2007), 339-360. Erratum: 23 (7-8) (2010), 795-799. Zbl1212.26016MR2654270
  3. ANGELONI, L. - VINTI, G., Convergence and rate of approximation for linear integral operators in B ⁢ V φ -spacces in multidimensional setting, Journal of Mathematical Analysis and Applications, 349 (2009), 317-334. Zbl1154.26017MR2456191DOI10.1016/j.jmaa.2008.08.029
  4. ANGELONI, L. - VINTI, G., Approximation with respect to Goffman-Serrin variation by means of non-convolution type integral operators, Numerical Functional Analysis and Optimization, 31 (2010), 519-548. Zbl1200.41015MR2682828DOI10.1080/01630563.2010.490549
  5. BARDARO, C., Alcuni teoremi di convergenza per l'integrale multiplo del Calcolo delle Variazioni, Atti Sem. Mat. Fis. Univ. Modena, 31 (1982), 302-324. 
  6. BARDARO, C. - BUTZER, P. L. - STENS, R. L. - VINTI, G., Convergence in variation and rates of approximation for Bernstein-type polynomials and singular convolution integrals, Analysis, 23 (2003), 299-340. Zbl1049.41015MR2052372DOI10.1524/anly.2003.23.4.299
  7. BARDARO, C. - MUSIELAK, J. - VINTI, G., Nonlinear Integral Operators and Applications, De Gruyter Series in Nonlinear Analysis and Applications (New York, Berlin, 9, 2003). MR1994699DOI10.1515/9783110199277
  8. BARDARO, C. - SCIAMANNINI, S. - VINTI, G., Convergence in B ⁢ V φ ⁢ ( ℝ ) by nonlinear Mellin-Type convolution operators, Funct. Approx. Comment. Math., 29 (2001), 17-28. Zbl1075.41013MR2135595
  9. BARDARO, C. - VINTI, G., On convergence of moment operators with respect to φ -variation, Appl. Anal. (1991), 247-256. Zbl0702.42009MR1103861DOI10.1080/00036819108840029
  10. BARDARO, C. - VINTI, G., On the order of B ⁢ V φ -approximation of convolution integrals over the line group, Comment. Math., Tomus Specialis in Honorem Iuliani Musielak (2004), 47-63. Zbl1068.47034MR2111754
  11. BURKILL, J. C., Functions of intervals, Proc. London Math. Soc., 22 (1923), 275-310. Zbl49.0177.02MR1575708DOI10.1112/plms/s2-22.1.275
  12. BUTZER, P. L. - NESSEL, R. J., Fourier Analysis and Approximation, I, Academic Press (New York-London, 1971). Zbl0217.42603MR510857
  13. CESARI, L., Sulle funzioni a variazione limitata, Ann. Scuola Norm. Sup. Pisa, 5 (1936), 299-313. Zbl62.0247.03MR1556778
  14. CHISTYAKOV, V. V. - GALKIN, O. E., Mappings of Bounded Φ -Variation with Arbitrary Function Φ , J. Dynam. Control Systems, 4 (2) (1998), 217-247. Zbl0940.26010MR1626529DOI10.1023/A:1022889902536
  15. DE GIORGI, E., Su una teoria generale della misura (r-1)-dimensionale in uno spazio ad r dimensioni, Ann. Mat. Pura Appl., 36 (4) (1954), 191-213. Zbl0055.28504MR62214DOI10.1007/BF02412838
  16. GIUSTI, E., Minimal Surfaces and Functions of Bounded Variation, Monographs in Mathematics, vol. 80, Birkhäuser Verlag, Basel, 1984. Zbl0545.49018MR775682DOI10.1007/978-1-4684-9486-0
  17. HERDA, H. H., Modular spaces of generalized variation, Studia Math., 30 (1968), 21-42. Zbl0159.18103MR231187DOI10.4064/sm-30-1-21-42
  18. JORDAN, C., Sur la serie de Fourier, C. R. Acad. Sci. Paris, 92 (1881), 228-230. 
  19. LOVE, E. R. - YOUNG, L. C., Sur une classe de fonctionnelles linéaires, Fund. Math., 28 (1937), 243-257. 
  20. MALIGRANDA, L. - ORLICZ, W., On some properties of functions of generalized variation, Mh. Math., 104 (1987), 53-65. Zbl0623.26009MR903775DOI10.1007/BF01540525
  21. MANTELLINI, I. - VINTI, G., Φ -variation and nonlinear integral operators, Atti Sem. Mat. Fis. Univ. Modena, Suppl. Vol. 46, a special issue of the International Conference in Honour of Prof. Calogero Vinti (1998), 847-862. MR1645758
  22. MATUSZEWSKA, W. - ORLICZ, W., On Property B1 for Functions of Bounded φ -Variation, Bull. Polish Acad. Sci. Math., 35 (1-2) (1987), 57-69. MR894138
  23. MUSIELAK, J., Orlicz Spaces and Modular Spaces, Springer-Verlag, Lecture Notes in Math., 1034, 1983. Zbl0557.46020MR724434DOI10.1007/BFb0072210
  24. MUSIELAK, J., Nonlinear approximation in some modular function spaces I, Math. Japon., 38 (1993), 83-90. Zbl0779.46017MR1204187
  25. MUSIELAK, J. - ORLICZ, W., On generalized variations (I), Studia Math., 18 (1959), 11-41. Zbl0088.26901MR104771DOI10.4064/sm-18-1-11-41
  26. RAMAZANOV, A. R. K., On approximation of functions in terms of Φ -variation, Anal. Math., 20 (1994), 263-281. Zbl0821.41009MR1301164DOI10.1007/BF01904057
  27. RAO, M. M. - REN, Z. D., Theory of Orlicz Spaces, Monograph Textbooks Pure Appl. Math., Marcel Dekker Inc. (New York, 1991). MR1113700
  28. SCIAMANNINI, S. - VINTI, G., Convergence and rate of approximation in B ⁢ V φ ⁢ ( ℝ ) for a class of integral operators, Approx. Theory Appl., 17 (2001), 17-35. Zbl1075.41013MR1910706DOI10.1023/A:1022811632587
  29. SCIAMANNINI, S. - VINTI, G., Convergence results in B ⁢ V φ ⁢ ( ℝ ) for a class of nonlinear Volterra-Hammerstein integral operators and applications, J. Concrete Appl. Anal., 1 (4) (2003), 287-306. Zbl1078.47026MR2132911
  30. SZELMECZKA, J., On convergence of singular integrals in the generalized variation metric, Funct. Approx. Comment. Math., 15 (1986), 53-58. Zbl0613.42014MR880134
  31. TONELLI, L., Su alcuni concetti dell'analisi moderna, Ann. Scuola Norm. Super. Pisa, 11 (2) (1942), 107-118. MR15463
  32. VINTI, C., Perimetro-variazione, Ann. Scuola Norm. Sup. Pisa, 18 (3) (1964), 201-231. MR168726
  33. VINTI, C., L'integrale di Fubini-Tonelli nel senso di Weierstrass, I - Caso parametrico, Ann. Scuola Norm. Sup. Pisa, 22 (1968), 229-263. MR231254
  34. WIENER, N., The quadratic variation of a function and its Fourier coefficients, Massachusetts J. of Math., 3 (1924), 72-94. Zbl50.0203.01
  35. YOUNG, L. C., An inequality of the Hölder type, connected with Stieltjes integration, Acta Math., 67 (1936), 251-282. MR1555421DOI10.1007/BF02401743
  36. YOUNG, L. C., Sur une généralisation de la notion de variation de puissance pieme bornée au sens de M. Wiener, et sur la convergence des séries de Fourier, C. R. Acad. Sci. Paris, 204 (1937), 470-472. Zbl63.0182.03

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.