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This appendix contains outline proofs and derivations for the theorems and formulas given in early part of Chapter: The Scaling Function and Scaling Coefficients, Wavelet and Wavelet Coefficients . They are not intended to be completeor formal, but they should be sufficient to understand the ideas behind why a result is true and to give some insight into its interpretation aswell as to indicate assumptions and restrictions.

Proof 1 The conditions given by [link] and [link] can be derived by integrating both sides of

φ ( x ) = n h ( n ) M φ ( M x - n )

and making the change of variables y = M x

φ ( x ) d x = n h ( n ) M φ ( M x - n ) d x

and noting the integral is independent of translation which gives

= n h ( n ) M φ ( y ) 1 M d y .

With no further requirements other than φ L 1 to allow the sum and integral interchange and φ ( x ) d x 0 , this gives [link] as

n h ( n ) = M

and for M = 2 gives [link] . Note this does not assume orthogonality nor any specific normalization of φ ( t ) and does not even assume M is an integer.

This is the most basic necessary condition for the existence of φ ( t ) and it has the fewest assumptions or restrictions.

Proof 2 The conditions in [link] and [link] are a down-sampled orthogonality of translates by M of the coefficients which results from the orthogonality of translates of the scaling function given by

φ ( x ) φ ( x - m ) d x = E δ ( m )

in [link] . The basic scaling equation [link] is substituted for both functions in [link] giving

n h ( n ) M φ ( M x - n ) k h ( k ) M φ ( M x - M m - k ) d x = E δ ( m )

which, after reordering and a change of variable y = M x , gives

n k h ( n ) h ( k ) φ ( y - n ) φ ( y - M m - k ) d y = E δ ( m ) .

Using the orthogonality in [link] gives our result

n h ( n ) h ( n - M m ) = δ ( m )

in [link] and [link] . This result requires the orthogonality condition [link] , M must be an integer, and any non-zero normalization E may be used.

Proof 3 (Corollary 2) The result that

n h ( 2 n ) = n h ( 2 n + 1 ) = 1 / 2

in [link] or, more generally

n h ( M n ) = n h ( M n + k ) = 1 / M

is obtained by breaking [link] for M = 2 into the sum of the even and odd coefficients.

n h ( n ) = k h ( 2 k ) + k h ( 2 k + 1 ) = K 0 + K 1 = 2 .

Next we use [link] and sum over n to give

n k h ( k + 2 n ) h ( k ) = 1

which we then split into even and odd sums and reorder to give:

n k h ( 2 k + 2 n ) h ( 2 k ) + k h ( 2 k + 1 + 2 n ) h ( 2 k + 1 ) = k n h ( 2 k + 2 n ) h ( 2 k ) + k n h ( 2 k + 1 + 2 n ) h ( 2 k + 1 ) = k K 0 h ( 2 k ) + k K 1 h ( 2 k + 1 ) = K 0 2 + K 1 2 = 1 .

Solving [link] and [link] simultaneously gives K 0 = K 1 = 1 / 2 and our result [link] or [link] for M = 2 .

If the same approach is taken with [link] and [link] for M = 3 , we have

n x ( n ) = n x ( 3 n ) + n x ( 3 n + 1 ) + n x ( 3 n + 2 ) = 3

which, in terms of the partial sums K i , is

n x ( n ) = K 0 + K 1 + K 2 = 3 .

Using the orthogonality condition [link] as was done in [link] and [link] gives

K 0 2 + K 1 2 + K 2 2 = 1 .

Equation [link] and [link] are simultaneously true if and only if K 0 = K 1 = K 2 = 1 / 3 . This process is valid for any integer M and any non-zero normalization.

Proof 3 If the support of φ ( x ) is [ 0 , N - 1 ] , from the basic recursion equation with support of h ( n ) assumed as [ N 1 , N 2 ] we have

φ ( x ) = n = N 1 N 2 h ( n ) 2 φ ( 2 x - n )

where the support of the right hand side of [link] is [ N 1 / 2 , ( N - 1 + N 2 ) / 2 ) . Since the support of both sides of [link] must be the same, the limits on the sum, or, the limits on the indices of the non zero h ( n ) are such that N 1 = 0 and N 2 = N , therefore, the support of h ( n ) is [ 0 , N - 1 ] .

Proof 4 First define the autocorrelation function

a ( t ) = φ ( x ) φ ( x - t ) d x

and the power spectrum

A ( ω ) = a ( t ) e - j ω t d t = φ ( x ) φ ( x - t ) d x e - j ω t d t

which after changing variables, y = x - t , and reordering operations gives

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Source:  OpenStax, Intermodular linking test collection. OpenStax CNX. Sep 09, 2015 Download for free at http://legacy.cnx.org/content/col11841/1.4
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