In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, more specifically in
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
, the Cauchy product is the discrete
convolution
In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions ( and ) that produces a third function (f*g) that expresses how the shape of one is ...
of two
infinite series
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
. It is named after the French mathematician
Augustin-Louis Cauchy
Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He ...
.
Definitions
The Cauchy product may apply to infinite series or power series. When people apply it to finite sequences or finite series, that can be seen merely as a particular case of a product of series with a finite number of non-zero coefficients (see
discrete convolution
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
).
Convergence
Convergence may refer to:
Arts and media Literature
*''Convergence'' (book series), edited by Ruth Nanda Anshen
*Convergence (comics), "Convergence" (comics), two separate story lines published by DC Comics:
**A four-part crossover storyline that ...
issues are discussed in the
next section.
Cauchy product of two infinite series
Let
and
be two
infinite series
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
with complex terms. The Cauchy product of these two infinite series is defined by a discrete convolution as follows:
:
where
.
Cauchy product of two power series
Consider the following two
power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
:
and
with complex coefficients
and
. The Cauchy product of these two power series is defined by a discrete convolution as follows:
:
where
.
Convergence and Mertens' theorem
Let and be real or complex sequences. It was proved by
Franz Mertens that, if the series
converges to and
converges to , and at least one of them
converges absolutely
In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series \textstyle\sum_^\infty a_n is said ...
, then their Cauchy product converges to .
The theorem is still valid in a
Banach algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach ...
(see first line of the following proof).
It is not sufficient for both series to be convergent; if both sequences are
conditionally convergent In mathematics, a series or integral is said to be conditionally convergent if it converges, but it does not converge absolutely.
Definition
More precisely, a series of real numbers \sum_^\infty a_n is said to converge conditionally if
\lim_\,\s ...
, the Cauchy product does not have to converge towards the product of the two series, as the following example shows:
Example
Consider the two
alternating series
In mathematics, an alternating series is an infinite series of the form
\sum_^\infty (-1)^n a_n or \sum_^\infty (-1)^ a_n
with for all . The signs of the general terms alternate between positive and negative. Like any series, an alternatin ...
with
which are only conditionally convergent (the divergence of the series of the absolute values follows from the
direct comparison test
In mathematics, the comparison test, sometimes called the direct comparison test to distinguish it from similar related tests (especially the limit comparison test), provides a way of deducing the convergence or divergence of an infinite series o ...
and the divergence of the
harmonic series). The terms of their Cauchy product are given by
for every integer . Since for every we have the inequalities and , it follows for the square root in the denominator that , hence, because there are summands,
for every integer . Therefore, does not converge to zero as , hence the series of the diverges by the
term test
In mathematics, the ''n''th-term test for divergenceKaczor p.336 is a simple test for the divergence of an infinite series:If \lim_ a_n \neq 0 or if the limit does not exist, then \sum_^\infty a_n diverges.Many authors do not name this test or ...
.
Proof of Mertens' theorem
For simplicity, we will prove it for complex numbers. However, the proof we are about to give is formally identical for an arbitrary
Banach algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach ...
(not even commutativity or associativity is required).
Assume
without loss of generality
''Without loss of generality'' (often abbreviated to WOLOG, WLOG or w.l.o.g.; less commonly stated as ''without any loss of generality'' or ''with no loss of generality'') is a frequently used expression in mathematics. The term is used to indicate ...
that the series
converges absolutely.
Define the
partial sums
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
with
Then
by rearrangement, hence
Fix . Since
by absolute convergence, and since converges to as , there exists an integer such that, for all integers ,
(this is the only place where the absolute convergence is used). Since the series of the converges, the individual must converge to 0 by the
term test
In mathematics, the ''n''th-term test for divergenceKaczor p.336 is a simple test for the divergence of an infinite series:If \lim_ a_n \neq 0 or if the limit does not exist, then \sum_^\infty a_n diverges.Many authors do not name this test or ...
. Hence there exists an integer such that, for all integers ,
Also, since converges to as , there exists an integer such that, for all integers ,
Then, for all integers , use the representation () for , split the sum in two parts, use the
triangle inequality
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.
This statement permits the inclusion of degenerate triangles, but ...
for the
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
, and finally use the three estimates (), () and () to show that
By the
definition of convergence of a series, as required.
Cesàro's theorem
In cases where the two sequences are convergent but not absolutely convergent, the Cauchy product is still
Cesàro summable. Specifically:
If
,
are real sequences with
and
then
This can be generalised to the case where the two sequences are not convergent but just Cesàro summable:
Theorem
For
and
, suppose the sequence
is
summable with sum ''A'' and
is
summable with sum ''B''. Then their Cauchy product is
summable with sum ''AB''.
Examples
* For some
, let
and
. Then
by definition and the
binomial formula
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial into a sum involving terms of the form , where the ...
. Since,
formally,
and
, we have shown that
. Since the limit of the Cauchy product of two
absolutely convergent
In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series \textstyle\sum_^\infty a_n is said ...
series is equal to the product of the limits of those series, we have proven the formula
for all
.
* As a second example, let
for all
. Then
for all
so the Cauchy product
does not converge.
Generalizations
All of the foregoing applies to sequences in
(
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s). The Cauchy product can be defined for series in the
spaces (
Euclidean spaces
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean s ...
) where multiplication is the
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
. In this case, we have the result that if two series converge absolutely then their Cauchy product converges absolutely to the inner product of the limits.
Products of finitely many infinite series
Let
such that
(actually the following is also true for
but the statement becomes trivial in that case) and let
be infinite series with complex coefficients, from which all except the
th one converge absolutely, and the
th one converges. Then the limit
exists and we have:
Proof
Because
the statement can be proven by induction over
: The case for
is identical to the claim about the Cauchy product. This is our induction base.
The induction step goes as follows: Let the claim be true for an
such that
, and let
be infinite series with complex coefficients, from which all except the
th one converge absolutely, and the
-th one converges. We first apply the induction hypothesis to the series
. We obtain that the series
converges, and hence, by the triangle inequality and the sandwich criterion, the series
converges, and hence the series
converges absolutely. Therefore, by the induction hypothesis, by what Mertens proved, and by renaming of variables, we have:
Therefore, the formula also holds for
.
Relation to convolution of functions
A finite sequence can be viewed as an infinite sequence with only finitely many nonzero terms, or in other words as a function
with finite support. For any complex-valued functions ''f'', ''g'' on
with finite support, one can take their
convolution
In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions ( and ) that produces a third function (f*g) that expresses how the shape of one is ...
:
Then
is the same thing as the Cauchy product of
and
.
More generally, given a
monoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
Monoids ...
''S'', one can form the
semigroup algebra
In abstract algebra, a monoid ring is a ring constructed from a ring and a monoid, just as a group ring is constructed from a ring and a group.
Definition
Let ''R'' be a ring and let ''G'' be a monoid. The monoid ring or monoid algebra of ''G'' ...