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
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where and
are defined as the coefficients of the expansions
![]() |
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Expressions for and
can be given in the form
![]() |
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The method is a matrix summation method with matrix
,
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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) |
Cesàro summation methods. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ces%C3%A0ro_summation_methods&oldid=26191