Cesàro summation methods

A collection of methods for the summation of series of numbers and functions. Introduced by E. Cesàro [1] and denoted by the symbol .

A series

(*)

with partial sums is summable by the Cesàro method of order , or -summable, with sum if

where and are defined as the coefficients of the expansions

Expressions for and can be given in the form

The method is a matrix summation method with matrix ,

For the method coincides with ordinary convergence, for it is the method of arithmetic averages. The methods are totally regular for and are not regular for . The power of the method increases as increases: If a series is summable by the method , then it is summable with the same sum by the method for . This property does not hold for . It follows from the summability of the series (*) by the method that . The method is equivalent to and compatible with the summation methods of Hölder and of Riesz (cf. Hölder summation methods; Riesz summation method). For any the method is weaker than Abel's method (cf. Abel summation method).

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

References

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