Examples of divergent Fourier series for classes of functions of bounded \(\Lambda \)-variation. (English. Russian original) Zbl 1192.42009
Math. Notes 86, No. 5, 629-636 (2009); translation from Mat. Zametki 86, No. 5, 664-672 (2009).
Summary: The author has shown earlier that the requirement that a continuous function belong to the class \(HBV([-\pi ,\pi ]^m)\) for \(m \geq 3\) is not sufficient for the convergence of its Fourier series over rectangles. The author gave examples of functions of three and more variables from the Waterman class which are harmonic in the first variable and significantly narrower in the other variables and whose Fourier series are divergent at some point even on cubes. In the present paper, this assertion is strengthened. The main result is that such an example can be constructed even when the class with respect to the first variable is somewhat narrowed. Also the one-dimensional result due to Waterman is refined.
MSC:
| 42B08 | Summability in several variables |
| 42B05 | Fourier series and coefficients in several variables |
Keywords:
divergent Fourier series; function of bounded; \(\Lambda\)-variation; convergence of Fourier series over rectangles and cubes; Waterman class of functions; harmonic variationReferences:
| [1] | D. Waterman, ”On convergence of Fourier series of functions of generalized bounded variation,” Studia Math. 44(1), 107–117 (1972). · Zbl 0207.06901 · doi:10.4064/sm-44-2-107-117 |
| [2] | A. A. Saakyan, ”On the convergence of double Fourier series of functions with bounded harmonic variation,” Izv. Akad. Nauk Arm. SSR, Mat. 21(6), 517–529 (1986) [Sov. J. Contemp.Math. Anal., Arm. Acad. Sci. 21 (6), 1–13 (1986)]. · Zbl 0614.42009 |
| [3] | A. I. Sablin, Functions of Bounded {\(\Lambda\)}-Variation and Fourier Series, Candidate Dissertation in Mathematics and Physics (Moscow State University, Moscow, 1987) [in Russian]. · Zbl 0694.42009 |
| [4] | A. I. Sablin, ”{\(\Lambda\)}-variation and Fourier series,” Izv. Vyssh. Uchebn. Zaved.Mat., No. 10, 66–68 (1987). · Zbl 0657.42005 |
| [5] | A. N. Bakhvalov, ”On convergence and localization of multiple Fourier series for classes of functions of bounded {\(\Lambda\)}-variation,” Vestnik Moskov. Univ. Ser. I Mat. Mekh., No. 3, 6–12 (2008). · Zbl 1212.42003 |
| [6] | A. N. Bakhvalov, ”Continuity in {\(\Lambda\)}-variation of functions of several variables and the convergence of multiple Fourier series. (English. Russian original) [J] Sb. Math. 193, No. 12, 1731–1746 (2002 Mat. Sb. 193 (12), 3–20 (2002) [Russian Acad. Sci. Sb.Math., No. 12, 1731–1746 (2002)]. · Zbl 1058.42002 |
| [7] | A. N. Bakhvalov, ”Representation of nonperiodic functions of bounded {\(\Lambda\)}-variation by the Fourier integral in the multidimensional case,” Izv. Ross. Akad. Nauk Ser. Mat. 67(6), 3–22 (2003) [Russian Acad. Sci. Izv. Math. 67 (6), 1081–1100 (2003)]. · Zbl 1068.42010 · doi:10.4213/im458 |
| [8] | S. Perlman and D. Waterman, ”Some remarks on functions of {\(\Lambda\)}-bounded variation,” Proc. Amer. Math. Soc. 74(1), 113–118 (1979). · Zbl 0415.26007 |
| [9] | N. K. Bari, Trigonometric Series (Fizmatlit, Moscow, 1961) [in Russian]. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
