(Orthonormal system of
translates) For a function
we form
.
If
has the
property
then we call it "orthonormal system of translates" (OST).

Note
that
and the calculation goes in either direction. Thus the OST may be redefined
with the
requirement

(OST property 0)

Proposition

(Property of transport 1) For any function
we
have
where
is the Fourier transform of
and the integral is a
-th
Fourier coefficient of a 1-periodic function
.

(Property of transport 2) If
has compact support then the function
has the
form
for some numbers
and finite
.

Proof

According to the proposition (
Property of
transport
1
),
Since
has compact support,
for a finite number of
.

Proposition

(OST property 1) The set
is OST
iff

Proof

According to the proposition (
Property of
transport
1
),
The Fourier transformation is 1-1 in
and the collection
is orthonormal in
.
Hence,
if and only if
and since the summation in
is
to
,
the above is true for
.