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 and be two infinite series with complex terms. The Cauchy product of these two infinite series is defined by a discrete convolution as follows:
Cauchy product of two power series
Consider the following two power series with complex coefficients and. The Cauchy product of these two power series is defined by a discrete convolution as follows:
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, then their Cauchy product converges to. 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 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.
Proof of Mertens' theorem
Assume without loss of generality that the series converges absolutely. Define the partial sums with Then by rearrangement, hence Fix. Since by absolute convergence, and since converges to as, there exists an integer such that, for all integers, . Since the series of the converges, the individual must converge to 0 by the term test. 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 for the absolute value, 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.
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 . The Cauchy product can be defined for series in the spaces where multiplication is the inner product. 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.
Let such that and let be infinite series with complex coefficients, from which all except the th one converge absolutely, and the th one converges. Then the series converges and we have: This statement can be proven by induction over : The case for is identical tothe 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.
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 with finite support. For any complex-valued functions f, g on with finite support, one can take their convolution: Then is the same thing as the Cauchy product of and. More generally, given a unital semigroup S, one can form the semigroup algebra of S, with, as usual, the multiplication given by convolution. If one takes, for example,, then the multiplication on is a sort of a generalization of the Cauchy product to higher dimension.
