The wavelet subsampling theorem arise from the observation that
if we are given the values of a function F in equally spaced points,
F(n T), then we know that (B=π/T):
∑n F(n T) sinc( B (x - n T) )
is B band limited
G(x) = ∑n F(n T) sinc( 2B (x - n T) )
= ∑n F(n T) sinc( B (2x - 2n T) )
is B band limited, G(2n T) = F(n T)
and G vanishes at x = (2n+1) T
H(x) = ∑n F(n T) sinc( B/2 (x - n T) ) = ∑k F(2k T) sinc( B (x/2 - k T) ) +
∑k F( (2K+1) T) sinc( B (x/2 - k T - T/2 ) )
is the sum of two B band limited functions: HE(x) = FE( x/2 ) =
∑k F(2k T) sinc( B (x/2 - k T) ) HO(x) = FO( x/2 - T/2 ) =
∑k F((2k+1) T) sinc( B (x/2 - k T - T/2 ) )
therefore FE and FO
are B/2 band limited
The subsampling theorem can be put in the wavelet framework by letting
B => 2m B.
The scaling function f(x) = sinc(x) satisfies the dilation equation