6.2 Algebraic Constructions

In view of the results presented in the last section, only the following algebraic problem remains: We have to find trigonometric polynomials that satisfy the identity

and, of course, the condition H( 0) = 1. We shall insist here on real coefficients h _k ; the corresponding scaling functions ? as well as the mother wavelets ? will then be real-valued as well.

According to 5.3.(13) the Fourier transform of ? is given by

Now, on account of what we said in Section 3.5 (see, e.g., Theorem (3.13)), we are interested in our wavelet ? having an order N as high as possible, and according to 3.5.(3) this is equivalent to the requirement that should vanish of an order N as large as possible at ? = 0. As a consequence the generating function H should have a zero of order N ? l at ? = ?,a fact that we express most elegantly by writing

Instead of looking for H we switch for a moment to the function

(1)

that would have to satisfy the linear identity

(2)

For symmetry reasons the function M is a polynomial in cos ?, and M contains the factor

Therefore we may write

(3)

where is a certain polynomial as well. Now we introduce a new variable y by letting y := sin ² . This...