On the Asymptotic Convergence of B-Spline Wavelets to Gabor Functions
M. Unser, A. Aldroubi, M. Eden
IEEE Transactions on Information Theory, vol. 38, no. 2, part II, pp. 864–872, March 1992.
A family of nonorthogonal polynomial spline wavelet transforms is considered. These transforms are fully reversible and can be implemented efficiently. The corresponding wavelet functions have a compact support. It is proven that these B-spline wavelets converge to Gabor functions (modulated Gaussian) pointwise and in all Lp-norms with 1 ≤ p < +∞ as the order of the spline (n) tends to infinity. In fact, the approximation error for the cubic B-spline wavelet (n = 3) is already less then 3%; this function is also near-optimal in terms of its time/frequency localization in the sense that its variance product is within 2% of the limit specified by the uncertainty principle.
Erratum
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Equation (1.5), replace the fraction ((n + 1) ⁄ j) by the bionomial coefficient (n + 1)! ⁄ ((n + 1 - j)! j!).
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AUTHOR="Unser, M. and Aldroubi, A. and Eden, M.",
TITLE="On the Asymptotic Convergence of \mbox{{B}-Spline} Wavelets to
{G}abor Functions",
JOURNAL="{IEEE} Transactions on Information Theory",
YEAR="1992",
volume="38",
number="2",
pages="864--872",
month="March",
note="Part {II}")