#### 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

# Dini – Lipschitz Test on the Generalized Haar Systems

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.

- 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).
- 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).
- Monna A. F. Analyse Non-Archimedienne ´ . Berlin, Heidelberg, N. Y., Springer-Veilag, 1970, 118 p.
- 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).
- 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.
- 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).
- Onneweer C. W., Waterman D. Uniform convergence of Fourier Series on groups. Michigan Math. J., 1971, vol. 18, iss. 3, pp. 265–273.
- 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).
- Golubov B. I. About One Class of the Complete Orthogonal Systems. Sib. Math. J., 1968, vol. IX, no. 2, pp. 297–314 (in Russian).
- 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).
- 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).
- Price J. J. Certain groups of orthogonal step functions. Canadian J. Math., 1957, vol. 9, iss. 3, pp. 417–425.
- Chrestenson H. E. A class of generalized Walsh’s functions. Pacific J. Math., 1955, vol. 5, iss. 1, pp. 17–31.
- Walsh J. L. A constructive of normal orthogonal functions. Amer. J. Math., 1923, vol. 49, iss. 1, pp. 5–24.
- Paley R. E. A. C. A remarkable series of orthogonal functions. Proc. London Math. Soc., 1932, vol. 36, pp. 241–264.
- Rademacher H. Enige Satze uber Reihen von allgemeinen Orthogonalfunctionen. Math. Ann., 1922, B. 87, no. 1–2, pp. 112–130.
- Haar A. Zur Theorie der Orthogonalischen Functionsysteme. Math. Ann., 1910, B. 69, pp. 331–371.
- 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).
- 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).

- 1215 reads