Geodesic
Localization Problem of the Absolute Riesz and Absolute Nörlund Summabilities of Fourier Series
Canadian journal of mathematics, Tome 22 (1970) no. 3, pp. 615-625

Voir la notice de l'article provenant de la source Cambridge University Press

1.1. Let Σ an be an infinite series and sn its nth partial sum. Let (pn) be a sequence of positive numbers such that If the sequence (1) is of bounded variation, that is, Σ |tn – tn –1| < ∞, then the series Σ an is said to be absolutely (R, pn, 1) summable or |R, pn, 1| summable.Let ƒ be an integrable function with period 2π and let its Fourier series be (2) Dikshit [4] (cf. Bhatt [1] and Matsumoto [7]) has proved the following theorems.THEOREM I. Suppose that (i) the sequence (pn/Pn) is monotone decreasing, (ii) mn > 0, (iii) the sequence (mnpn/Pn) decreases monotonically to zero, and (iv) the series Σ mnPn/Pn) is divergent. 
Izumi, Masako; Izumi, Shin-Ichi. Localization Problem of the Absolute Riesz and Absolute Nörlund Summabilities of Fourier Series. Canadian journal of mathematics, Tome 22 (1970) no. 3, pp. 615-625. doi: 10.4153/CJM-1970-068-6
@article{10_4153_CJM_1970_068_6,
     author = {Izumi, Masako and Izumi, Shin-Ichi},
     title = {Localization {Problem} of the {Absolute} {Riesz} and {Absolute} {N\"orlund} {Summabilities} of {Fourier} {Series}},
     journal = {Canadian journal of mathematics},
     pages = {615--625},
     year = {1970},
     volume = {22},
     number = {3},
     doi = {10.4153/CJM-1970-068-6},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1970-068-6/}
}
TY  - JOUR
AU  - Izumi, Masako
AU  - Izumi, Shin-Ichi
TI  - Localization Problem of the Absolute Riesz and Absolute Nörlund Summabilities of Fourier Series
JO  - Canadian journal of mathematics
PY  - 1970
SP  - 615
EP  - 625
VL  - 22
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1970-068-6/
DO  - 10.4153/CJM-1970-068-6
ID  - 10_4153_CJM_1970_068_6
ER  - 
%0 Journal Article
%A Izumi, Masako
%A Izumi, Shin-Ichi
%T Localization Problem of the Absolute Riesz and Absolute Nörlund Summabilities of Fourier Series
%J Canadian journal of mathematics
%D 1970
%P 615-625
%V 22
%N 3
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1970-068-6/
%R 10.4153/CJM-1970-068-6
%F 10_4153_CJM_1970_068_6

[1] 1. Bhatt, S. N., An aspect of local property of \R, log n, 1| summability of Fourier series, Töhoku Math. J. (2) 11 (1959), 13–19. Google Scholar

[2] 2. Bhatt, S. N., An aspect of local property of absolute Nörlund summability of a Fourier series, Proc. Nat. Inst. Sri. India Part A 28 (1962), 787–794. Google Scholar

[3] 3. Daniel, E. C., The non-local character of summability \N, pn\ of a Fourier series with factors, Math. Z. 99 (1967), 392–399. Google Scholar

[4] 4. Dikshit, G. D., Localization relating to the summability \R, \n, 1| of Fourier series, Indian J. Math. 17 (1965), 31–39. Google Scholar

[5] 5. Izumi, M. and Izumi, S., The absolute Nörlund summability factors of Fourier series (to appear in Indian J. Math.). Google Scholar

[6] 6. Jurkat, W. and Peyerimhoff, A., Localisation bei absoluter Cesaro-Summierbarheit von Potenzreihen und trigonometrischen Reihen. II, Math. Z. 64 (1956), 151–158. Google Scholar

[7] 7. Matsumoto, K., Local property of the summability \R, X, 1|, Töhoku Math. J. (2) 8 (1956), 354–364. Google Scholar

[8] 8. Zygmund, A., Trigonometric series, 2nd éd., Vol. I (Cambridge Univ. Press, New York, 1959). Google Scholar

Cité par Sources :