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.
[3][4] When people apply it to finite sequences[5] 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).
Convergence issues are discussed in the next section.
be two infinite series with complex terms.
The Cauchy product of these two infinite series is defined by a discrete convolution as follows: Consider the following two power series with complex coefficients
The Cauchy product of these two power series is defined by a discrete convolution as follows: Let (an)n≥0 and (bn)n≥0 be real or complex sequences.
It was proved by Franz Mertens that, if the series
[6] The theorem is still valid in a Banach algebra (see first line of the following proof).
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: 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).
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 simplicity, we will prove it for complex numbers.
However, the proof we are about to give is formally identical for an arbitrary Banach algebra (not even commutativity or associativity is required).
Assume without loss of generality that the series
By the definition of convergence of a series, Cn → AB as required.
This can be generalised to the case where the two sequences are not convergent but just Cesàro summable: For
The Cauchy product can be defined for series in the
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.
be infinite series with complex coefficients, from which all except the
is identical to the claim about the Cauchy product.
The induction step goes as follows: Let the claim be true for an
be infinite series with complex coefficients, from which all except the
We first apply the induction hypothesis to the series
converges, and hence, by the triangle inequality and the sandwich criterion, the series
Therefore, by the induction hypothesis, by what Mertens proved, and by renaming of variables, we have:
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, one can take their convolution:
More generally, given a monoid S, one can form the semigroup algebra
is a generalization of the Cauchy product to higher dimension.