In mathematics, more specifically in mathematical analysis, the Cauchy product is the discrete convolution of two infinite series. It is named after the French mathematician Augustin Louis Cauchy.
Contents
Definitions
The Cauchy product may apply to infinite series or power series. When people apply it to finite sequences or finite series, it is by abuse of language: they actually refer to discrete convolution.
Convergence issues are discussed in the next section.
Cauchy product of two infinite series
Let
Cauchy product of two power series
Consider the following two power series
with complex coefficients
Convergence and Mertens' theorem
Let (an)n≥0 and (bn)n≥0 be real or complex sequences. It was proved by Franz Mertens that, if the series
It is not sufficient for both series to be convergent; if both sequences are conditionally convergent, 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 with
which are only conditionally convergent (the divergence of the series of the absolute values follows from the direct comparison test and the divergence of the harmonic series). The terms of their Cauchy product are given by
for every integer n ≥ 0. Since for every k ∈ {0, 1, ..., n} we have the inequalities k + 1 ≤ n + 1 and n – k + 1 ≤ n + 1, it follows for the square root in the denominator that √(k + 1)(n − k + 1) ≤ n +1, hence, because there are n + 1 summands,
for every integer n ≥ 0. Therefore, cn does not converge to zero as n → ∞, hence the series of the (cn)n≥0 diverges by the term test.
Proof of Mertens' theorem
Assume without loss of generality that the series
with
Then
by rearrangement, hence
Fix ε > 0. Since
(this is the only place where the absolute convergence is used). Since the series of the (an)n≥0 converges, the individual an must converge to 0 by the term test. Hence there exists an integer M such that, for all integers n ≥ M,
Also, since An converges to A as n → ∞, there exists an integer L such that, for all integers n ≥ L,
Then, for all integers n ≥ max{L, M + N}, use the representation (1) for Cn, split the sum in two parts, use the triangle inequality for the absolute value, and finally use the three estimates (2), (3) and (4) to show that
By the definition of convergence of a series, Cn → AB 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
This can be generalised to the case where the two sequences are not convergent but just Cesàro summable:
Theorem
For
Examples
by definition and the binomial formula. Since, formally,
Generalizations
All of the foregoing applies to sequences in
Products of finitely many infinite series
Let
converges and we have:
This statement can be proven by induction over
The induction step goes as follows: Let the claim be true for an
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. A finite sequence can be viewed as a function
Then
More generally, given a unital semigroup S, one can form the semigroup algebra