What is

〈sinπ(t−n)π(t−n),sinπ(t−k)π(t−k)〉=?
t
n
t
n
t
k
t
k
?

(2)
This

inner product
can be hard to calculate in the time domain, so let's use

Plancharel Theorem
〈·,·〉=12π∫−ππe−(iωn)eiωkdω
·
·
1
2
ω
ω
n
ω
k

(3)

if
n=k
n
k

〈
sinc
n
,
sinc
k
〉=12π∫−ππe−(iωn)eiωkdω=1
sinc
n
sinc
k
1
2
ω
ω
n
ω
k
1

(4)
if

n≠k
n
k
〈
sinc
n
,
sinc
k
〉=12π∫−ππe−(iωn)eiωndω=12π∫−ππeiω(k−n)dω=12πsinπ(k−n)i(k−n)=0
sinc
n
sinc
k
1
2
ω
ω
n
ω
n
1
2
ω
ω
k
n
1
2
k
n
k
n
0

(5)
In

Equation 5 we
used the fact that the integral of sinusoid over a complete
interval is 0 to simplify our equation.

So,

〈sinπ(t−n)π(t−n),sinπ(t−k)π(t−k)〉={1 if n=k0 if n≠k
t
n
t
n
t
k
t
k
1
n
k
0
n
k

(6)
Therefore

sinπ(t−n)π(t−n)
n∈Z
t
n
t
n
n
is an

orthonormal basis (ONB) for the space of

−π
π
bandlimited functions.

Sampling is the same as calculating ONB
coefficients, which is inner products with sincs