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On \(\Lambda^2\)-strong convergence of numerical sequences and Fourier series. (English) Zbl 1299.40017
The structure and approaches from F. Móricz’s paper [Acta Math. Hung. 54, No. 3–4, 319–327 (1989; Zbl 0708.42004)] are transferred to the so-called \(\Lambda^2\)-strong convergence rather than to the \(\Lambda\)-strong convergence in [Móricz, loc. cit.]. What makes a difference is that the defining sequence \(\Lambda\) is taken not only nondecreasing as in [Móricz, loc. cit.] but also convex. Correspondingly, some of the proofs need more efforts and sometimes examples are given to show that the obtained results are applicable to a wider range of objects. Open problems posed in [Móricz, loc. cit.] as well as those possible similar related to the \(\Lambda^2\)-strong convergence are not discussed at all.

MSC:
40F05 Absolute and strong summability
40D15 Convergence factors and summability factors
42A20 Convergence and absolute convergence of Fourier and trigonometric series
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References:
[1] C. P. Kahane, Series de Fourier absolument convergentes, Springer (Berlin-Heidelberg-New York, 1970).
[2] Móricz, F., On λ-strong convergence of numerical sequences and Fourier series, Acta Math. Hungar., 54, 319-327, (1989) · Zbl 0708.42004
[3] Tanović-Miller, N., On the strong convergence of trigonometric and Fourier series, Acta Math. Hungar., 42, 35-43, (1983) · Zbl 0543.42002
[4] A. Zygmund, Trigonometric Series, Vol. 1, University Press (Cambridge, 1959). · Zbl 0085.05601
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