Cesàro summation

In mathematical analysis, Cesàro summation assigns values to some sequences without a sum in the usual sense. The Cesàro sum of a sequence is defined as the limit of the arithmetic means of the partial sums of that sequence.

Cesàro summation is named for the Italian analyst Ernesto Cesàro (1859–1906).

The term summation can be misleading, as some statements and proofs regarding Cesàro summation can be said to implicate the Eilenberg–Mazur swindle. For example, it is commonly applied to Grandi's series with the conclusion that its sum is 1/2.

Definitions

A sequence ${a i}$ is called Cesàro summable, if the arithmetic mean of its first $n$ partial sums, ${\frac {1}{n}}\sum _{i=1}^{n}(a_{1}+\cdots +a_{i})$ , for $n\to\infty$ tends to a limit (the Cesàro sum of the sequence).
It is easy to show that any summable sequence is Cesàro summable, and its usual sum equals the Cesàro sum. However, as the first example below demonstrates, Cesàro summability does not imply usual summability.

Examples

The sequence 1, -1, 1, -1, 1, ··· (sequence of Grandi) has as sum sequence (sequence of partial sums): 1, 0, 1, 0, 1, ··· . This (divergent) sum sequence has as Cesàro means: 1/1 = 1, (1+0)/2 = 1/2, (1+0+1)/3 = 2/3, (1+0+1+0)/4 = 2/4, (1+0+1+0+1) = 3/5, (1+0+1+0+1+0) = 3/6, 4/7, 4/8, 5/9, 5/10, ··· , converging to 1/2.
Hence, the non-summable 1, -1, 1, -1, 1, ··· is Cesàro summable, with Cesàro sum 1/2 .

The sequence 1, 2, 3, 4, 5, ··· has as sum sequence: 1, 3, 6, 10, 15, ··· . This sum sequence has as Cesàro means: 1/1 = 1, (1+3)/2 = 4/2, (1+3+6)/3 = 10/3, (1+3+6+10)/4 = 20/4, ··· , diverging to plus infinite.
So the sequence of natural numbers is not Cesàro summable. In fact, any sequence which diverges to (positive or negative) infinity, has a sum sequence with diverging Cesàro means, and hence such a series is never Cesàro summable.

$(C, α)$ summation

In 1890, Ernesto Cesàro stated a broader family of summation methods which have since been called $(C, α)$ for non-negative integers $α$ . The $(C, 0)$ method is just ordinary summation, and $(C, 1)$ is Cesàro summation as described above.

The higher-order methods can be described as follows: given a series $\sum a n$ , define the quantities

{\begin{aligned}A_{n}^{-1}&=a_{n}\\A_{n}^{\alpha }&=\sum _{k=0}^{n}A_{k}^{\alpha -1}\end{aligned}}

(where the upper indices do not denote exponents) and define $E α n$ to be $A α n$ for the series 1 + 0 + 0 + 0 + …. Then the $(C, α)$ sum of $\sum a n$ is denoted by $(C, α)-\sum a n$ and has the value

(C,\alpha ){\text{-}}\sum _{j=0}^{\infty }a_{j}=\lim _{n\to \infty }{\frac {A_{n}^{\alpha }}{E_{n}^{\alpha }}}

if it exists (Shawyer & Watson 1994, pp.16-17). This description represents an $α$ -times iterated application of the initial summation method and can be restated as

(C,\alpha ){\text{-}}\sum _{j=0}^{\infty }a_{j}=\lim _{n\to \infty }\sum _{j=0}^{n}{\frac {\binom {n}{j}}{\binom {n+\alpha }{j}}}a_{j}.

Even more generally, for $α ∈ ℝ \ ℤ−$ , let $A α n$ be implicitly given by the coefficients of the series

\sum _{n=0}^{\infty }A_{n}^{\alpha }x^{n}={\frac {\displaystyle {\sum _{n=0}^{\infty }a_{n}x^{n}}}{(1-x)^{1+\alpha }}},

and $E α n$ as above. In particular, $E α n$ are the binomial coefficients of power $-1 - α$ . Then the $(C, α)$ sum of $\sum a n$ is defined as above.

If $\sum a n$ has a $(C, α)$ sum, then it also has a $(C, β)$ sum for every $β > α$ , and the sums agree; furthermore we have $a n = o (n α)$ if $α > -1$ (see little- $o$ notation).

Cesàro summability of an integral

Let $α \geq 0$ . The integral $\int \infty 0 f (x) d x$ is Cesàro summable $(C, α)$ if

\lim _{\lambda \to \infty }\int _{0}^{\lambda }\left(1-{\frac {x}{\lambda }}\right)^{\alpha }f(x)\,dx

exists and is finite (Titchmarsh 1948, §1.15). The value of this limit, should it exist, is the $(C, α)$ sum of the integral. Analogously to the case of the sum of a series, if $α = 0$ , the result is convergence of the improper integral. In the case $α = 1$ , $(C, 1)$ convergence is equivalent to the existence of the limit

\lim _{\lambda \to \infty }{\frac {1}{\lambda }}\int _{0}^{\lambda }\int _{0}^{x}f(y)\,dy\,dx

which is the limit of means of the partial integrals.

As is the case with series, if an integral is $(C, α)$ summable for some value of $α \geq 0$ , then it is also $(C, β)$ summable for all $β > α$ , and the value of the resulting limit is the same.

References

Shawyer, Bruce; Watson, Bruce (1994), Borel's Methods of Summability: Theory and Applications, Oxford University Press, ISBN 0-19-853585-6
Titchmarsh, E. C. (1986) [1948], Introduction to the theory of Fourier integrals (2nd ed.), New York, NY: Chelsea Publishing, ISBN 978-0-8284-0324-5
Volkov, I. I. (2001) [1994], "Cesàro summation methods", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Zygmund, Antoni (1988) [1968], Trigonometric series (2nd ed.), Cambridge University Press, ISBN 978-0-521-35885-9

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