Geodesic
Summation of Unordered Arrays
Funkcionalʹnyj analiz i ego priloženiâ, Tome 52 (2018) no. 1, pp. 43-55.

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An approach to the summation of unordered number and matrix arrays based on ordering them by absolute value (greedy summation) is proposed. Theorems on products of greedy sums are proved. A relationship between the theory of greedy summation and the theory of generalized Dirichlet series is revealed. The notion of asymptotic Dirichlet series is considered.
Keywords: greedy sum, unordered sum, theorem on multiplications of sums, generalized Dirichlet series, asymptotic Dirichlet series, Riesz means, generic zeta-function. 
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E. V. Shchepin. Summation of Unordered Arrays. Funkcionalʹnyj analiz i ego priloženiâ, Tome 52 (2018) no. 1, pp. 43-55. http://geodesic.mathdoc.fr/item/FAA_2018_52_1_a3/

[1] H. Bohr, “Über die Summabilität Dirichlet'scher Reihen”, Gött. Nachr., 1909, 247–262 | Zbl

[2] G. H. Hardy, M. Riesz, The general theory of Dirichlet's series, Cambridge Tracts in Mathematics and Mathematical Physics, 18, Cambridge University Press, Cambridge, 1915 | MR

[3] S. V. Konyagin, V. N. Temlyakov, “Convergence of greedy approximations I. General systems”, Studia Math., 159:1 (2003), 143–160 | DOI | MR | Zbl

[4] E. Landau, “Über die Multiplikation Dirichlet'scher Reihen”, Rendiconti di Palermo, 24 (1907), 81–160 | DOI | Zbl

[5] M. Riesz, “Sur la summation des séries de Dirichlet”, Comptes Rendus, 149 (1909), 18–21

[6] T. J. Stiltijes, “Note sur la multiplication de deux séries”, Nouvelles Annales, ser. 3, 6 (1887), 210–215

[7] R. Grekhem, D. Knut, O. Patashnik, Konkretnaya matematika, Mir, M., 1998 | MR

[8] A. F. Leontev, Ryady eksponent, Nauka, M., 1976 | MR

[9] P. L. Ulyanov, “Ob $A$-skhodimosti ryadov”, DAN SSSR, 368:2 (1999), 160–163 | MR | Zbl

[10] P. L. Ulyanov, “O svoistvakh ryadov”, DAN SSSR, 378:3 (2000), 307–310

[11] E. Shchepin, Greedy sums and Dirichlet series, arXiv: 1110.5285

[12] E. Shchepin, Riesz means and greedy sums, arXiv: 1212.0821

[13] E. Steinitz, “Bedingt konvergente Reihen und konvexe Systeme”, J. Reine Angew. Math., 143 (1913), 128–175 ; ibid, 144 (1914), 1–40 ; ibid, 146 (1916), 1–52 | DOI | MR | Zbl | MR | Zbl | MR