Cesàro summation methods

2020 Mathematics Subject Classification: Primary: 40C05 [MSN][ZBL]

A collection of methods for the summation of series of numbers and functions. Introduced by E. Cesàro [Ce] and denoted by the symbol $(C,k)$.

A series \begin{equation} \label{eq1} \sum_{n=0}^\infty a_n \end{equation} with partial sums $S_n$ is summable by the Cesàro method of order $k$, or $(C,k)$-summable, with sum $S$ if $$ \sigma_n^k = \frac{S_n^k}{A_n^k} \rightarrow S, \quad n \rightarrow \infty, $$ where $S_n^k$ and $A_n^k$ are defined as the coefficients of the expansions $$ \sum_{n=0}^\infty A_n^k x^n = \frac{1}{(1-x)^{k+1}}, \quad \sum_{n=0}^\infty S_n^k x^n = \frac{1}{(1-x)^k} \sum_{n=0}^\infty S_n x^n = \frac{1}{(1-x)^{k+1}}\sum_{n=0}^\infty a_n x^n. $$ Expressions for $\sigma_n^k$ and $A_n^k$ can be given in the form $$ \sigma_n^k = \frac{1}{A_n^k}\sum_{\nu=0}^n A_{n-\nu}^{k-1}S_\nu = \frac{1}{A_n^k}\sum_{\nu=0}^n A_{n-\nu}^{k} a_\nu, $$ $$ A_n^k = \binom{k+n}{n} = \frac{(k+1) \cdots (k+n)}{n!}, \quad k \neq,-1,-2,\ldots $$ The method $(C,k)$ is a matrix summation method with matrix $[a_{n\nu}]$, $$ a_{n\nu} = \begin{cases} \frac{A_{n-\nu}^{k-1}}{A_n^k}, & \nu \leq n, \\ 0, & \nu > n. \end{cases} $$ For $k=0$ the method coincides with ordinary convergence, for $k=1$ it is the method of arithmetic averages. The methods $(C,k)$ are totally regular for $k \geq 0$ and are not regular for $k < 0$. The power of the method increases as $k$ increases: If a series is summable by the method $(C,k)$, then it is summable with the same sum by the method $(C,k')$ for $k' > k > -1$. This property does not hold for $k < -1$. It follows from the summability of the series \ref{eq1} by the method $(C,k)$ that $a_n=o\bigl(n^k\bigr)$. The method $(C,k)$ is equivalent to and compatible with the Hölder summation method $(H,k)$ and the Riesz summation method $(R,n,k)$ for $k>0$. For any $k>-1$ the method $(C,k)$ is weaker than the Abel summation method.

Originally, the methods $(C,k)$ were defined by Cesàro for positive integer values of the parameter $k$, and applied to the multiplication of series. They were later extended to arbitrary values of $k$, including complex values. The methods $(C,k)$ have numerous applications, e.g. to the multiplication of series, in the theory of Fourier series, and to other questions.

References

[Ba]	S.A. Baron, "Introduction to the theory of summability of series", Tartu (1966) (In Russian)
[Ce]	E. Cesàro, Bull. Sci. Math., 14 : 1 (1890) pp. 114–120
[Ha]	G.H. Hardy, "Divergent series", Clarendon Press (1949)
[Zy]	A. Zygmund, "Trigonometric series", 1, Cambridge Univ. Press (1988)