144
II - Convergence: Discrete variables
Example 2 (multiplication of absolutely convergent series). Consider two absolutely summable families (Ui), i E J and (Vj), j E J and the family (UiVj)
of products, indexed by the Cartesian product I x Jj this is again absolutely
summable and
L
UiVj = LUi. LVj
(i,j)ElxJ
iEI
jEJ
as in ordinary algebra. This follows immediately from the corollary above,
since on putting S = L IVjl, the sum of the IUiVjl for given i is equal to SIUil,
the general term of a series which is absolutely convergent by hypothesis. We
shall return to this example in nO 22 a propos power series, but the following
example will immediately give us another application.
Example 3 (convolution product on a discrete group). Consider the set L1(Z)
of functions f : Z --+ C such that
(18.15)
IIflll = L If(n)1 < +00
and, for two such functions, let us define their convolution product f * 9 = h
by the formula
(18.16)
h(n) = L f(n - p)g(p) = f * g(n)
where f*g(n) denotes the value at n ofthe function f*g. The series converges
unconditionally since (15) shows that the numbers If(n)1 are bounded abovein fact, they tend to 0 at infinity -, so up to a constant factor the general term
of (16) is, in modulus, smaller than that of the series (15) for the function g.
If one performs the permutation p f-+ n - p on the terms of (16), which
transforms n - p to p and does not change the sum of the series, one finds
h(n) = L f(p)g(n - p), whence commutativity of the product:
(18.17)
We now show that f * 9 E Ll (Z) and even that
(18.18)
For this, consider the double series with general term f(p)g(q), indexed by
Z x Z. By Example 2, it converges unconditionally and
(18.19)
L If(p)g(q)1 = L If(p)1 L Ig(q)1 = IIflldglll.
Now one can regroup the terms ofthe series L If(p)g(q)1 as one wishes, and,
for example, group them according to the value of the integer p + q = n. One
finds then that
I: If(p)g(q)1 = I: I: If(p)g(n - p)lj
n
p
II - Convergence: Discrete variables
Example 2 (multiplication of absolutely convergent series). Consider two absolutely summable families (Ui), i E J and (Vj), j E J and the family (UiVj)
of products, indexed by the Cartesian product I x Jj this is again absolutely
summable and
L
UiVj = LUi. LVj
(i,j)ElxJ
iEI
jEJ
as in ordinary algebra. This follows immediately from the corollary above,
since on putting S = L IVjl, the sum of the IUiVjl for given i is equal to SIUil,
the general term of a series which is absolutely convergent by hypothesis. We
shall return to this example in nO 22 a propos power series, but the following
example will immediately give us another application.
Example 3 (convolution product on a discrete group). Consider the set L1(Z)
of functions f : Z --+ C such that
(18.15)
IIflll = L If(n)1 < +00
and, for two such functions, let us define their convolution product f * 9 = h
by the formula
(18.16)
h(n) = L f(n - p)g(p) = f * g(n)
where f*g(n) denotes the value at n ofthe function f*g. The series converges
unconditionally since (15) shows that the numbers If(n)1 are bounded abovein fact, they tend to 0 at infinity -, so up to a constant factor the general term
of (16) is, in modulus, smaller than that of the series (15) for the function g.
If one performs the permutation p f-+ n - p on the terms of (16), which
transforms n - p to p and does not change the sum of the series, one finds
h(n) = L f(p)g(n - p), whence commutativity of the product:
(18.17)
We now show that f * 9 E Ll (Z) and even that
(18.18)
For this, consider the double series with general term f(p)g(q), indexed by
Z x Z. By Example 2, it converges unconditionally and
(18.19)
L If(p)g(q)1 = L If(p)1 L Ig(q)1 = IIflldglll.
Now one can regroup the terms ofthe series L If(p)g(q)1 as one wishes, and,
for example, group them according to the value of the integer p + q = n. One
finds then that
I: If(p)g(q)1 = I: I: If(p)g(n - p)lj
n
p