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By Jack K. Hale

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Example text

If one can choose c{z) = c, a constant, and still verify ^(1) = 1, then c = c(l) = 1 + | ^ ( - 1 ) P > 1. Then, by definition of c, ^ should satisfy ^{z)^%z) = c-${z)^%z) = ^-z)$\-z). Possible choices for such ^ include "^{z) = '^^^\z) = u;z^${-z) or ^{z) = ^^'^\z) = ujz^^*{-z) for some integer N andueC with |a;| = 1. 4 Constraints on samples imposed by QMFs For a concrete construction and parameterization, let ^ be the trigonometric polynomial ^(z) = Ylk=^i^ cikZ^. 36) A:=l on the coefficient side.

However, ^1,2(0 = e~^^*^cos2 7r^(l + 2sin2 7r^) gives rise to a square-integrable scaling function. It is worth pointing out that in the biorthogonal case the FWT follows the same pattern as in the orthogonal case. , [79,99]). 4 Regularity for scaling distributions Consequences of the fact that wavelets form unconditional bases for a large scale of function spaces will be a basic theme of Chapter 6. This fact also goes a long way towards explaining their utility as tools for solving PDEs with initial conditions in suitable function spaces.

11) must be satisfied by the dual scaling pair and this is the condition that one seeks to solve. For the linear spline centered at x = 1, the scaling filter has the form F i ( 0 = \ + \e-'-'^ + \e-'''^ = ( ^ ^ ^ ^ ) ' = e-2'^*« cos'ir^ More generally, the B-spUne of order A^ is defined as the A/'+1-fold convolution of X[o,i) with itself. p-27rze\ ^ + 1 HN{0 = [ ^ The Bezout equation ) = e-(^+i)'^« cos^+i TT^. 11) for H{^) = HN{^) (see [99], p. 272 for plots of corresponding scaling functions and wavelets).

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