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 rα 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,