Extending Wavelet Regularity Beyond Gevrey Classes

Source abstract

Abstract We present a general construction of smooth orthonormal wavelet $$\psi$$ ψ which, together with its Fourier transform $$\widehat{\psi }$$ ψ ^ belongs to the extended Gevrey class $$\mathcal {E}_{\sigma }(\mathbb {R})$$ E σ ( R ) for $$\sigma > 1$$ σ > 1 , providing an example that lies beyond all classical Gevrey classes. Our approach uses the idea of invariant cycles to extend the initial Lemarié–Meyer support of the low-pass filter $$m_0$$ m 0 from $$[-\frac{2\pi }{3}, \frac{2\pi }{3}]$$ [ - 2 π 3 , 2 π 3 ] to $$[-\frac{4\pi }{5}, \frac{4\pi }{5}]$$ [ - 4 π 5 , 4 π 5 ] . This extension allows precise control of the the decay rate of $$m_0$$ m 0 near $$\frac{2\pi }{3}$$ 2 π 3 , which yields global decay estimates for $$\psi$$ ψ and $$\hat{\psi }$$ ψ ^ . In addition, the decay rates are described using special functions involving the Lambert W function, which plays an important role in our construction.