Geodesic
On Banach Limit of Fourier Series and Conjugate Series I
Canadian mathematical bulletin, Tome 11 (1968) no. 2, pp. 255-262

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Let (xn) be a sequence of real numbers. (xn) corresponds to a number Lim xn called the Banach limit of (xn) satisfying the following conditions: (1) Lim (axn + byn) = a Lim xn + b Lim yn (2) If xn ≥ 0 for every n, then Lim xn ≥ 0 (3) Lim xn+1 = Lim xn (4) If xn = 1 for every n, then Lim xn = 1 The existence of such limits is proved by Banach [1]. 
Dayal, S. On Banach Limit of Fourier Series and Conjugate Series I. Canadian mathematical bulletin, Tome 11 (1968) no. 2, pp. 255-262. doi: 10.4153/CMB-1968-030-x
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     author = {Dayal, S.},
     title = {On {Banach} {Limit} of {Fourier} {Series} and {Conjugate} {Series} {I}},
     journal = {Canadian mathematical bulletin},
     pages = {255--262},
     year = {1968},
     volume = {11},
     number = {2},
     doi = {10.4153/CMB-1968-030-x},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1968-030-x/}
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