Alexits György: Approximation theory. Selected papers (Budapest, 1983)

The scientific works of George Alexits

ON SUMMABILITY OF ORTHOGONAL SERIES 205 So the condition (11) is satisfied. (One can easily see that also (5) is satisfied trivially when we choose ßk = k~x log-1 к (log log A:)-2, because in this case the right hand side of (5) becomes infinite.) Put 1 c„ = —=------------------, Vnlogn-log log« then c£l2(X), but Л C"<Pn ^ Л n log n • log log n All the terms of the series 2 cn<Pn(x) being positive, from the divergence to in­finity it follows that 2 cn<Pn(x) is nowhere summable in E by any permanent method T. References [1] S. Banach, Sur la convergence presque partout des fonctionnelles linéaires, I, II, Bull. Sei. Math.. 50 (1926), 27—32 and 36—43. [2] S. Kaczmarz, Sur la convergence et la sommabilité des développements orthogonoux, Studia Math., 1 (1929), 87—121. {3] S. Kaczmarz, Notes on orthogonal series I, Studia Math., 5 (1935), 24—28. [4] A. Kolmogroff et G. Seliverstoff, Sur la convergence des séries de Fourier, C. R. Acad. Sci. Paris, 178 (1925), 303—305. [5] L. Leindler, Über die sehr starke Riesz Summierbarkeit der Orthogonalreihen und Konver­genz lückenhafter Orthogonalreihen, Acta Math. Acad. Sei. Hungar., 13 (1962), 401—414. [6] F. Móricz, The order of magnitude of the Lebesgue functions and summability of function series, Acta Sei. Math. Szeged, 34 (1973), 289—296. [7] A. Plessner, Über Konvergenz von trigonometrischen Reihen, Jourrt. für die r. und angew. Math., 155 (1926), 15—25. [8] G. Sunouchi, On the Riesz summability of Fourier series, Tőhoku Math. Journ., (2) 11 (1959), 319—326. [9] K. Tandori, Über die Cesärosche Summierbarkeit der orthogonalen Reihen, Acta Sei. Math. Szeged. 14 (1951/52), 85—95. [10] K. Tandori, Weitere Bemerkungen über die Konvergenz und Summierbarkeit der Funktionen­reihen, Acta Math. Acad. Sei. Hungar., 28 (1976), 119—127. [11] N. Dunford and J. T. Schwartz, Linear Operators, Part I (New York, 1958). (Received April 21, 1976) MATHEMATICAL INSTITUTE OF THE HUNGARIAN ACADEMY OF SCIENCES 1053 BUDAPEST, REÁLTANODA U. 13—15. JÓZSEF ATTILA UNIVERSITY BOLYAI INSTITUTE 6720 SZEGED, ARADI VÉRTANÚK TERE 1. Acta Mathematica Academiae Scientiarum Hungaricae 29, 1977 270

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