Abel-Tauber Theorem

The Abel – Tauber theorem is the inverse of the Abel power series theorem. The first theorem is like Tauberian theorems. It was proved by A. Tauber in 1897 ( Tauber's theorem ) ^[1] The formulation and proof under more general conditions were then given by J. Littlewood in 1910 ^[2] Then it was proved by R. Schmidt ^[3] , N. Wiener ^{[4 ]} . The simplest evidence was given by J. Karamata ^[5] . Wording and proof under a weaker condition $n|a_{n}|>-K$ ${\ displaystyle n | a_ {n} |> -K}$ ${\ displaystyle n | a_ {n} |> -K}$ given to E. Landau ^[6] .

Wording

Let be $\sum _{0}^{\infty }a_{n}x^{n}$ ${\ displaystyle \ sum _ {0} ^ {\ infty} a_ {n} x ^ {n}}$ converges to $f(x)$ ${\ displaystyle f (x)}$ at $|x|<1$ ${\ displaystyle | x | <1}$ . Let be $\lim _{x\rightarrow {1-0}}f(x)=s$ ${\ displaystyle \ lim _ {x \ rightarrow {1-0}} f (x) = s}$ when $x$ ${\ displaystyle x}$ tends to the left to $one$ ${\ displaystyle 1}$ . Let be $n|a_{n}|<K<\infty$ ${\ displaystyle n | a_ {n} | <K <\ infty}$ . Then $\sum _{0}^{\infty }a_{n}=s$ ${\ displaystyle \ sum _ {0} ^ {\ infty} a_ {n} = s}$ .

Notes

↑ A. Tauber Ein Satz aus der Theorie der undendlichen Reihen // Monatshefte f. Math. 8 (1897), 273-277
↑ Littlewood On the converse of Abel's theorem on power series // Proc. Lond. Math. Soc. (2), 9 (1910), 434–444
↑ R. Schmidt Uber divergente Folgen und lineare Mittelbindungen // Math. Zeitchr., 22 (1925), 89-152
↑ N. Wiener Tauberian Theorems // Annals of Mathematics, 33 (1932), 1-100
↑ J. Karamata Uber die Hardy - Littlewoodschen Umkehrungen des Abelshen Stetigkeitssatzes // Math. Zeitschr .., 32 (1930), 319-320
↑ E. Landau Uber einen Satz des Herrn Littlewood // Rendiconti di Palermo, 35 (1913), 265–276

Literature

Wiener, N. Integral Fourier and some of its applications. - M.: Fizmatlit, 1963 .-- S. 255.

[1] A. Tauber Ein Satz aus der Theorie der undendlichen Reihen // Monatshefte f. Math. 8 (1897), 273-277

[2] Littlewood On the converse of Abel's theorem on power series // Proc. Lond. Math. Soc. (2), 9 (1910), 434–444

[3] R. Schmidt Uber divergente Folgen und lineare Mittelbindungen // Math. Zeitchr., 22 (1925), 89-152

[4] N. Wiener Tauberian Theorems // Annals of Mathematics, 33 (1932), 1-100

[5] J. Karamata Uber die Hardy - Littlewoodschen Umkehrungen des Abelshen Stetigkeitssatzes // Math. Zeitschr .., 32 (1930), 319-320

[6] E. Landau Uber einen Satz des Herrn Littlewood // Rendiconti di Palermo, 35 (1913), 265–276

Abel-Tauber Theorem

Wording

Notes

Literature

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