2 . Gibbs Phenomenon for Wavelet Sampling Expansions

We deal with the maximum Gibbs ripple of the sampling wavelet series of a discontinuous .function f at a point t ~ R, .for all possible values o.['a satisfying f (t) = ee.f (t 0) + (1 cO.f (t + 0). For the Shannon wavelet series, we make a complete description of all ripples, .for any ot in [0,1]. We show that Meyer sampling series exhibit Gibbs Phenomenon.lor ce < 0 .12495 and ct > 0.306853. We also give Meyer sampling formulas with maximum overshoots shorter than Shannon's for several et in [0,1]. 1. I n t r o d u c t i o n It is well known that most multiresolution analyses induce sampling expansions for all continuous functions f in L2(R). Recall that a multiresolution analysis is a nested sequence {Vm C Vm+l, m 6 Z} of closed subspaces in L2(R) and a function ~, called a "scaling" function, such that (i) for any m 6 Z, {2 m/2 ~b(2 m x n), n E Z} is an orthonormal basis of Vm, (ii) f (x) E VO r f(2mx) E Vm, for all m ~ Z, (iii) Um Vm = Z2(R) and Nm Vm = {0}. For more details about multiresolution analysis see [3]. It was shown in [4] and [12] that each function g E Vm has a unique sampling expansion g(x)= g ~ s ( 2 m x n ) , x e R , nc=Z where the convergence is in the L2(R) sense and S E Vo is the sampling function defined by s(• = &• r ~ R , ~_, ~a(n) e -i"~'' n6Z Math Subject Classifications. 42C 15, 94A 12.