Cesàro mean
Encyclopedia
In mathematics
, the Cesàro means (also called Cesàro averages) of a sequence
(an) are the terms of the sequence (cn), where
is the arithmetic mean
of the first n elements of (an).
This concept is named after Ernesto Cesàro
(1859 - 1906).
A basic result
states that if
then also
That is, the operation of taking Cesàro means preserves convergent sequence
s and their limits. This is the basis for taking Cesàro means as a summability method in the theory of divergent series
.
If the sequence of the Cesàro means is convergent, the series is said to be Cesàro summable. There are certainly many examples for which the sequence of Cesàro means converges, but the original sequence does not: for example with
we have an oscillating sequence, but the means have limit:
. (See also Grandi's series.)
Cesàro means are often applied to Fourier series
,
since the means (applied to the trigonometric polynomial
s making up the symmetric partial sums) are more powerful in summing such series than pointwise convergence
. The kernel that corresponds is the Fejér kernel, replacing the Dirichlet kernel; it is positive, while the Dirichlet kernel takes both positive and negative values. This accounts for the superior properties of Cesàro means for summing Fourier series, according to the general theory of approximate identities
.
A generalization of the Cesàro mean is the Stolz-Cesàro theorem.
The Riesz mean
was introduced by M. Riesz as a more powerful but substantially similar summability method.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Cesàro means (also called Cesàro averages) of a sequence
Sequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
(an) are the terms of the sequence (cn), where
is the arithmetic mean
Arithmetic mean
In mathematics and statistics, the arithmetic mean, often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample space...
of the first n elements of (an).
This concept is named after Ernesto Cesàro
Ernesto Cesàro
Ernesto Cesàro was an Italian mathematician who worked in the field of differential geometry.Cesàro was born in Naples. He is known also for his 'averaging' method for the summation of divergent series, known as the Cesàro mean.-Books by E. Cesaro:* * Ernesto Cesàro (March 12, 1859 – September...
(1859 - 1906).
A basic result
states that if
then also
That is, the operation of taking Cesàro means preserves convergent sequence
Limit of a sequence
The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...
s and their limits. This is the basis for taking Cesàro means as a summability method in the theory of divergent series
Divergent series
In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a limit....
.
If the sequence of the Cesàro means is convergent, the series is said to be Cesàro summable. There are certainly many examples for which the sequence of Cesàro means converges, but the original sequence does not: for example with
we have an oscillating sequence, but the means have limit:
. (See also Grandi's series.)Cesàro means are often applied to Fourier series
Fourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
,
since the means (applied to the trigonometric polynomial
Trigonometric polynomial
In the mathematical subfields of numerical analysis and mathematical analysis, a trigonometric polynomial is a finite linear combination of functions sin and cos with n a natural number. The coefficients may be taken as real numbers, for real-valued functions...
s making up the symmetric partial sums) are more powerful in summing such series than pointwise convergence
Pointwise convergence
In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function.-Definition:...
. The kernel that corresponds is the Fejér kernel, replacing the Dirichlet kernel; it is positive, while the Dirichlet kernel takes both positive and negative values. This accounts for the superior properties of Cesàro means for summing Fourier series, according to the general theory of approximate identities
Approximate identity
In functional analysis and ring theory, an approximate identity is a net in a Banach algebra or ring that acts as a substitute for an identity element....
.
A generalization of the Cesàro mean is the Stolz-Cesàro theorem.
The Riesz mean
Riesz mean
In mathematics, the Riesz mean is a certain mean of the terms in a series. They were introduced by Marcel Riesz in 1911 as an improvement over the Cesàro mean...
was introduced by M. Riesz as a more powerful but substantially similar summability method.