Summary
We consider a class of functionsF(z) for which a formal expansion in power ofz gives a Stieltjes series with divergent moments. We show that, introducing a proper cut-off for the moments, we can sum such a series in the framework of the Padé-approximant method, through a connection between the limit on the order of the approximants and the limit on the cut-off. Due to this connection, for which we give some general rules, it is possible to obtain rapidly converging approximations toF(z), starting from the knowledge of the first terms of the formal series.
Riassunto
Ricorrendo al metodo delle approssimanti di Padé, si dimostra che le serie di potenze a termini divergenti derivanti dallo sviluppo formale, in potenze diz, della classe di funzioni\(F(z) = \int\limits_0^{ + \infty } {d\psi (t)/(1 + zt){\text{ (}}\psi {\text{(t)}}} \) funzione limitata non decrescente che assume infiniti valori nell’intervallo [0,+∞)), possono essere sommate in modo corretto introducendo un taglio nei termini della serie e correlando opportunamente il limite sul taglio con il limite sull’ordine delle approssimanti. Questa correlazione, di cui si danno alcune prescrizioni generali, permette di ottenere approssimazioni rapidamente convergenti aF(z), a partire dalla conoscenza dei primi termini dello sviluppo formale.
Реэюме
Мы рассматриваем класс функцийF(z), для которых формальное раэложение по степеням г дает ряд Стильтьеса с расходяшимися моментами. Мы покаэываем, что, вводя надлежашее обреэание для моментов, мы можем просуммировать ряд в рамках метода Падз приближений, благодаря свяэи между пределом для порядка приближений и пределом для величины обреэания. С помошью зтой свяэи, для которой мы приводим некоторые обшие правила, можно получить быстро сходяшиеся приближения к функцииF(z), исходя иэ энания первых членов формального ряда.
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Fogli, G.L., Pellicoro, M.F. & Villani, M. A summation method for a class of series with divergent terms. Nuov Cim A 6, 79–97 (1971). https://doi.org/10.1007/BF02723435
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DOI: https://doi.org/10.1007/BF02723435