PERTURBATION OF WAVELET FRAMES AND RIESZ BASES I

Suppose that $\psi{\;}\in{\;}L^2(\mathbb{R})$ generates a wavelet frame (resp. Riesz basis) with bounds A and B. If $\phi{\;}\in{\;}L^2(\mathbb{R})$ satisfies $$\mid$\^{\psi}(\xi)\;\^{\phi}(\xi)$\mid${\;}<{\;}{\lambda}\frac{$\mid$\xi$\mid$^{\alpha}}{(1+$\mid$\xi$\mid$)^{\gamma}}$ for some positive constants $\alpha,{\;}\gamma,{\;}\lambda$ such that $1{\;}<1{\;}+{\;}\alpha{\;}<{\;}\gamma{\;}and{\;}{\lambda}^2M{\;}<{\;}A$, then $\phi$ also generates a wavelet frame (resp. Riesz basis) with bounds $A(1{\;}-{\;}{\lambda}\sqrt{M/A})^2{\;}and{\;}B(1{\;}+{\;}{\lambda}\sqrt{M/A})^2$, where M is a constant depending only on $\alpha,{\;}\gamma$ the dilation step a, and the translation step b.