ASYMPTOTICS OF DAUBECHIES FILTERS, SCALING FUNCTIONS, AND WAVELETS

We study the asymptotic form as p --> infinity of the Daubechies ortho gonal minimum phase filter h(p)[n], scaling function phi(p)(t), and wa velet w(p)(t). Kateb and Lemarie calculated the leading term in the ph ase of the frequency response H-p(omega). The infinite product <(phi)o ver cap>(p)(omega) = Pi H-p(omega/2(k)) leads us to a problem in stati onary phase, for an oscillatory integral with parameter t. The leading terms change form with tau = t/p and we find three regions for phi(p) (tau): (1) An Airy function up to near tau(0): root 42 pi/p Ai(-root 4 2 pi p(2)(tau - tau(0))) + o(p(-1/3)) (2) An oscillating region root 2 /pi pG'(omega(tau))cos [p(G((-1))(omega(tau)) - G(omega(tau))omega(tau )) + pi/4] + o(p(-1/2)) (3) A rapid decay after tau(1): (1/p pi)(1/(ta u - tau(1)))sin[p(G((-1))(pi) - tau pi)] + o(p(-1)) The numbers tau(0) similar or equal to 0.1817 and tau(1) similar or equal to 0.3515 are known constants. The function G and its integral G((-1)) are independe nt of p. Regions 1 and 2 are matched over the interval p(-2/3) much le ss than tau - tau(0) much less than 1. The wavelets have a simpler asy mptotic expression because the Airy wavefront is removed by the highpa ss filter. We also find the asymptotics of the impulse response h(p)[n ] -a different function g(omega) controls the three regions. The diffi culty throughout is to estimate the phase. (C) 1998 Academic Press.