1.4. MULTILINEAR PARSEVAL-LIKE FORMULAS 5
First, following the ”dual” view of (1.2) as the Parseval-like formula in (1.13),
we derive analogous formulas in arbitrary dimensions. The general result (Theo-
rem 12.4) is cast in a framework of fractional Cartesian products. It is proved by
induction, with key steps provided by a variant of Theorem 3.5.
We illustrate the multilinear formulas of Theorem 12.4 with two archetypal
instances. In the first, we take a simple extension of the (bilinear) dot product in
l2pAq,
(1.20) Δnpx1,..., xnq
ÿ
αPA
x1pαq ¨ ¨ ¨ xnpαq, xi P
l2pAq,
i 1,...,n.
With no additional requirements, integral representations of Δn, which extend the
usual Parseval formula, arise typically as follows. Consider the bounded linear map
(1.21) U :
l2pAq
Ñ
L2pΩA,
PAq,
defined by
(1.22) U pxq
ÿ
αPA
xpαqrα, x P
l2pAq,
where the are Rademacher characters,
(1.23) rαpωq ωpαq, ω P ΩA :“ t´1,
1uA,
α P A.
Then,
Δnpx1, . . . , xnq
ż
pω1 ,...,ωn´1
qPΩA´1 n
ˆ
n´1
ź
i“1
U pxi q
`
ωi
˘
˙
U pxn q
`
ω1 ¨ ¨ ¨ ωn´1
˘
dω1 ¨ ¨ ¨ dωn´1

`
U px1 q ˙ ¨ ¨ ¨ ˙ U pxn q
˘
pω0 q, x1 P
l2pAq,
. . . , xn P
l2pAq,
(1.24)
where the integral above is performed with respect to the pn ´ 1q-fold product of
the Haar measure PA of the compact group ΩA, and is the n-fold convolution of
U px1q, . . . , U pxnq, evaluated at the identity element ω0 P ΩA,
ω0pαq 1, α P A.
Adding the requirement that integrands be uniformly bounded, we obtain (Lemma
12.1)
Δnpx1,..., xnq
ż
pω1 ,...,ωn´1
qPΩA´1 n
ˆ
n´1
ź
i“1
Φnpxiq
`
ωi
˘
˙
Φnpxnq
`
ω1 ¨ ¨ ¨ ωn´1
˘
dω1 ¨ ¨ ¨ dωn´1

`
Φnpx1q ˙ ¨ ¨ ¨ ˙ Φnpxnq
˘
pω0q, x1 P
l2pAq,...,
xn P
l2pAq,
(1.25)
where
(1.26) Φn :
l2pAq
Ñ
L8pΩA,
PAq
is
pl2
Ñ
L2q-continuous,
and
(1.27) }Φnpxq}L8 ď K}x}2, x P
l2pAq,
for an absolute constant K ą 0.
The second instance is the trilinear functional η on
l2pA2q
ˆ
l2pA2q
ˆ
l2pA2q,
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