Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

For citation:

Shcherbakov V. I. Dini – Lipschitz Test on the Generalized Haar Systems. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2016, vol. 16, iss. 4, pp. 435-448. DOI: 10.18500/1816-9791-2016-16-4-435-448, EDN: XHPYIR

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
Full text:
(downloads: 177)

Dini – Lipschitz Test on the Generalized Haar Systems

Shcherbakov Viktor Innokentievich, Moscow Technical University of Communications and Informatics, Russia

Generalized Haar systems, which are generated (generally speaking, unbounded) by a sequence {pn} ∞n=1 and which is defined on the modification segment [0, 1]∗ , thai is on a segment [0, 1], where {pn} — rational points are calculated two times and which is a geometrical representation of zero-dimensional compact Abelians group are considering in this work. The main result of this work is a setting of the pointwise estimation between of an absolute value of difference between continuous in the given point function and it’s n-s particular Fourier sums and “pointwise” module of continuity of this function (this notion (“pointwise” module of continuity ωn(x, f)) is also defined in this work). Based on this a uniform estimation between an absolute value of difference between a continious on the [0, 1]∗ function and it’s particular Fourier Sums and the module of continuity of this function is established. A sufficient condition of the pointwise and uniformly boundedness of particular Fourier Sums by generalized Haar’s systems for the given continuous function is established too. Based on this estimation we establish a test of convergence of Fourier Series with respect to generalized Haar’s systems analogous Dini – Lipschitz test. The unimprovement of the test, which is obtained in this work, is showed too. For any {pn} ∞n=1 with sup n pn = ∞ a model of the continuous on [0, 1]∗ function, which Fourier Series by generalized Haar’s system, which generated by sequence {pn} ∞n=1 boundly diverges in some fixed point, is constructed. This result may be applied to the zero-dimentions compact Abelian groups. 

  1. Vilenkin N. Ya. On a class of complete orthonormal systems. Izv. Akad. Nauk SSSR. Ser. Mat., 1947, vol. 11, no. 4, pp. 363–400 (in Russian).
  2. Agaev G. N., Vilenkin N. Ya., Dzafarli G. M., Rubinstein A. I. Mul’tiplicativnye sistemi funkciy i garmonicheskiy analiz na nul’mernyh gruppah [Multiplicative Systems of Functions and Harmonic Analysis on Zero-dimensional Groups]. Baku, ELM, 1981, 180 p. (in Russian).
  3. Monna A. F. Analyse Non-Archimedienne ´ . Berlin, Heidelberg, N. Y., Springer-Veilag, 1970, 118 p.
  4. Khrennikov A. Y., Shelkovich V. M. Sovremennyi p-addicheskyi analiz i matematicheskaja phizika. Teoria i prilozhenija [The Moderne p-additional Analysis and Mathematical Phisics. Theory and Applications]. Moscow, Fizmatgiz, 2012, 452 p. (in Russian).
  5. Shcherbakov V. I. Divergence of the Fourier series by generalized Haar systems at points of continuity of a function. Russian Math. (Iz. VUZ), 2016, vol. 60, no. 1, pp. 42–59. DOI: https://doi.org/10.3103/S1066369X16010059.
  6. Shcherbakov V. I. About Pointwise convergence of the Fourier Series with Respect to Multiplicative Systems. Vestn. MSU, Ser. Math., Mech., 1983, iss. 2, pp. 37–42 (in Russian).
  7. Onneweer C. W., Waterman D. Uniform convergence of Fourier Series on groups. Michigan Math. J., 1971, vol. 18, iss. 3, pp. 265–273.
  8. Shcherbakov V. I. Dini – Lipschitz Test and Convergence of Fourier Series which Respect to Multiplicative Systems. Analysis Math., 1984. vol. 10, iss. 1, pp. 133–150 (in Russian).
  9. Golubov B. I. About One Class of the Complete Orthogonal Systems. Sib. Math. J., 1968, vol. IX, no. 2, pp. 297–314 (in Russian).
  10. Goluboфv B. I., Rubinshtein A. I. A class of convergence systems. Mat. Sb. (N.S.), 1966, vol. 71, iss. 1, pp. 96–115 (in Russian).
  11. Lukomskii S. F. Haar series on compact zerodimensional abelian group. Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 2009, vol. 9, iss. 1, pp. 24–29 (in Russian).
  12. Price J. J. Certain groups of orthogonal step functions. Canadian J. Math., 1957, vol. 9, iss. 3, pp. 417–425.
  13. Chrestenson H. E. A class of generalized Walsh’s functions. Pacific J. Math., 1955, vol. 5, iss. 1, pp. 17–31.
  14. Walsh J. L. A constructive of normal orthogonal functions. Amer. J. Math., 1923, vol. 49, iss. 1, pp. 5–24.
  15. Paley R. E. A. C. A remarkable series of orthogonal functions. Proc. London Math. Soc., 1932, vol. 36, pp. 241–264.
  16. Rademacher H. Enige Satze uber Reihen von allgemeinen Orthogonalfunctionen. Math. Ann., 1922, B. 87, no. 1–2, pp. 112–130.
  17. Haar A. Zur Theorie der Orthogonalischen Functionsysteme. Math. Ann., 1910, B. 69, pp. 331–371.
  18. Komissarova N. E. Lebesgue functions for Haar system on compact zero-dimensional group. Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 2012, vol. 13, iss. 3, pp. 30–36 (in Russian).
  19. Shcherbakov V. I. Priznak Dini – Lipshitza po obobshchennym sistemam Haara [Dini-Lipschitz Test on the Generalized Haar’s Systems]. Sovremennye problemy teorii funktsii i ikh prilozheniia : materialy 17-i mezhdunar. Sarat. zimn. shk. [Contemporary Problems of Function Theory and Their Applications : Proc. 17th Intern. Saratov Winter School], Saratov, Nauchnaya kniga, 2014, pp. 307–308 (in Russian).