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   -> Volume 6, Issue 3

Question: Smoothest Scaling Function
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Scott Gilbert (

PostPosted: Thu Feb 27, 1997 5:56 pm    
Subject: Question: Smoothest Scaling Function
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#21 Question: Smoothest Scaling Function

What is the smoothest scaling function, w(x), solving a
'wavelet equation' of type:

(1) w(x) = c(0)w(2x) + c(1)w(2x-1) + c(2)w(2x-2) +...+ c(m)w(2x-m),

where the coefficients c(i) satisfy the 'usual conditions':

(2) sum c(i) over odd i = 1,

(3) sum c(i) over even i = 1,

and where w is normalized to have, say, integral = 1.

The Schoenberg B-spline, with support [0,m], satisfies (1)-(3),
and I have found no smoother w's, after experimenting
with various c's, for m=3. Are B-splines the
smoothest (non-orthogonal) (father) wavelets?

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