The shiftability of some wavelet bases

AbstractAn explicit functional called the shiftability value of wavelet basis, which measures a deviation from translation-invariance, is studied. For functions φ with ‖φ‖ = 1, it lies in (0,1] with the value 1 being best possible. The relation between the shiftability value rV of the father wavelet and the corresponding value rW of the mother wavelet is given. The shiftability value of Meyer's wavelet and the B-spline wavelets is computed. For Meyer's wavelet, we prove that rW = 3rV − 2 and demonstrate how to control the shiftability value by properly designing the wavelet. For the B-spline wavelet of order n, we give the asymptotical relation r(n)W ∼ 3r(n)V − 2, and show r(n)V → 1 as n → ∞. The family of B-spline wavelets contains several well-known wavelets such as Franklin's wavelet, Battle's wavelet, and Lemarié's wavelet, which makes our results potentially useful