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Applications of general monotone sequences to strong approximation by Fourier series. (English) Zbl 1297.42008

The authors continue the sequence of theorems generalizing results with monotone decreasing sequences to relax the monotonicity.
This is an interesting paper and twofold useful.
1. Their result shows that one of the reviewer’s theorems, the paper is cited in their Abstract (see below), is not correct. Namely they show that one of the conditions of said theorem is not sufficient in a special case, consequently there is a hidden fault in the proof.
2. They establish an accurate sufficient condition such that their new theorem extends the result of S. M. Mazhar and V. Totik (also cited in the Abstract), which was the aim of my cited note, too.
Abstract: “We consider the strong means of Fourier series generated by infinite nonnegative triangular matrices and prove some estimates of such means in the case of a matrix with rows stating the sequences from the class \(GM(5_\beta)\). Our theorems correspond to the results of L. Leindler [Anal. Math. 29, No. 3, 195–199 (2003; Zbl 1027.41013)] and essentially extend the result of S. M. Mazhar and V. Totik [J. Approximation Theory 60, No. 2, 174–182 (1990; Zbl 0741.42001)].”

MSC:

42A24 Summability and absolute summability of Fourier and trigonometric series
41A25 Rate of convergence, degree of approximation
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References:

[1] Leindler, L., Strong Approximation by Fourier Series (1985), Akadémiai Kiadó: Akadémiai Kiadó Budapest · Zbl 0588.42001
[2] Leindler, L., On the uniform convergence and boundedness of a certain class of sine series, Anal. Math., 27, 279-285 (2001) · Zbl 1002.42002
[3] Leindler, L., A new class of numerical sequences and its applications to sine and cosine series, Anal. Math., 28, 279-286 (2002) · Zbl 1026.42007
[4] Leindler, L., A note on strong approximation of Fourier series, Anal. Math., 29, 195-199 (2003) · Zbl 1027.41013
[5] Leindler, L., A new extension of monotone sequence and its application, J. Inequal. Pure Appl. Math., 7, 1, 7 (2006), Art. 39
[6] Leindler, L., Integrability conditions pertaining to Orlicz space, J. Inequal. Pure Appl. Math., 8, 2, 6 (2007), Art. 38 · Zbl 1214.42006
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[8] Mazhar, S. M.; Totik, V., Approximation of continuous functions by T-means of Fourier series, J. Approx. Theory, 60, 174-182 (1990) · Zbl 0741.42001
[9] Szal, B., A note on the uniform convergence and boundedness a generalized class of sine series, Commentat. Math., 48, 1, 85-94 (2008) · Zbl 1173.42303
[10] Szal, B., On the degree of strong approximation of continuous functions by special matrix, J. Inequal. Pure Appl. Math., 10, 4, 8 (2009), Art. 111 · Zbl 1185.42003
[11] Tikhonov, S., On uniform convergence of trigonometric series, Mat. Zametki. Mat. Zametki, Math. Notes, 81, 2, 268-274 (2007), Translation in · Zbl 1183.42005
[12] Tikhonov, S., Trigonometric series with general monotone coefficients, J. Math. Anal. Appl., 326, 1, 721-735 (2007) · Zbl 1106.42003
[13] Tikhonov, S., Best approximation and moduli of smoothness: computation and equivalence theorems, J. Approx. Theory, 153, 19-39 (2008) · Zbl 1215.42002
[14] Totik, V., Notes on Fourier series: strong approximation, J. Approx. Theory, 43, 105-111 (1985) · Zbl 0558.42001
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