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Concept
Cauchy product
Summary
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.
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).
Convergence issues are discussed in the next section.
Let and be two infinite series with complex terms. The Cauchy product of these two infinite series is defined by a discrete convolution as follows:
where .
Consider the following two power series
and
with complex coefficients and . The Cauchy product of these two power series is defined by a discrete convolution as follows:
where .
Let (an)n≥0 and (bn)n≥0 be real or complex sequences. It was proved by Franz Mertens that, if the series converges to A and converges to B, and at least one of them converges absolutely, then their Cauchy product converges to AB. 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). The terms of their Cauchy product are given by
for every integer n ≥ 0. Since for every k ∈ we have the inequalities k + 1 ≤ n + 1 and n – k + 1 ≤ n + 1, it follows for the square root in the denominator that ≤ 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.
For simplicity, we will prove it for complex numbers.
Official source
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