Journal
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
Volume 128, Issue 4, Pages 1087-1095Publisher
AMER MATHEMATICAL SOC
DOI: 10.1090/S0002-9939-99-05251-X
Keywords
Fourier transform; scaling function; Sobolev exponent; wavelet
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Let phi(N), N greater than or equal to 1, be Daubechies' scaling function with symbol (1+e(-i xi)/2)(N) Q(N)(xi), and let s(p)(phi(N)), 0 < p less than or equal to 1, be the corresponding L-p Sobolev exponent. In this paper, we make a sharp estimation of s(p)(phi(N)), and we prove that there exists a constant C independent of N such that N - ln\Q(N)(2 pi/3)\/ln 2 - C/N less than or equal to s(p)(phi(N)) less than or equal to N - ln\Q(N)(2 pi/3)\/ln 2. This answers a question of Cohen and Daubeschies (Rev. Mat. Iberoamericana, 12(1996), 527-591) positively.
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