By C. W. Groetsch

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Let ip be a refinable function with orthonormal integer shifts. Then X. f m ' o i O ' ^ d ^ = / K ( i + l / 2) l V ( 0 ^(0 diR 1 "2 P roo f. 23) and (1*1^) J T'{0W)d4 ~ J\ [” ^ o(i/2) ^ ( i / 2) + m o(^/2) ^ '( ^ /2)]»tio(^ /2)^ (^ /2) R J m 1. 27). 6. 0 Computational algorithms To use wavelet expansions in practice, we need to compute coefficients of these expansions. One of the advantages of M R A structure is that coefficients of wavelet expansions can be computed recursively. Suppose that we want to expand a function / G with respect to orthonormal wavelet systems generated by scaling functions (p, respectively.

1. Let x* and be centers of / and / , respectively. Prove that the function g{x) := + x*) and its Fourier transform have zero centers and equal radii A / = A^, A ^ = A^. 2. 0795775 holds. Moreover, the equality is attained if and only if / ( x ) = c e 2 -“*5a(® -i> ), where ga{x) := ^ 0, a > 0, a, 6 G R. First we prove an auxiliary statement. Recall that a function f{ x ) is abso lutely continuous if it is differentiable almost everywhere, its derivative is locally X summable, and f{x ) — f{y ) = f f { t ) dt for any x ,y (see Appendix A.

In compliance with the general scheme, we find the Fourier transform for the cor responding wavelet function -0 = -0 ^: V’iO = e” ^ m o (i/2 + l / 2 ) ^ ( i / 2 ) = e’" * « X [ - i ,- i / 2]u[i/2, i )(0 = e - « ( ^ ( ^ /2) - ^ ( 0 ). 2. 5. = 2(p^{2t + 1) — (p^{t + 1/ 2). Uncertainty constants The main feature of wavelet systems is that all elements of such a system are generated by dilations and shifts of one fixed function. A generating wavelet function is a damped oscillation localized in both time and frequency.