Approximately dual pairs of wavelet frames

This paper deals with structural issues concerning wavelet frames and their dual frames. It is known that there exist wavelet frames $\{a^{j/2}\psi( a^j\cdot -kb)\}_{j,k\in \mathbb Z}$ in $L^2(\mathbb R)$ for which no dual frame has wavelet structure. We first generalize this result by proving that there exist wavelet frames for which no approximately dual frame has wavelet structure. Motivated by this we show that by imposing a very mild decay condition on the Fourier transform of the generator $\psi\in L^2(\mathbb R),$ a certain oversampling $\{a^{j/2}\psi( a^j\cdot -kb/N)\}_{j,k\in \mathbb Z}$ indeed has an approximately dual wavelet frame; most importantly, by choosing the parameter $N\in \mathbb N$ sufficiently large we can get as close to perfect reconstruction as desired, which makes the approximate dual frame pairs perform equally well as the classical dual frame pairs in applications.