A sufficient condition and estimates of the frame bounds for generalized translation-invariant frames

Jakob Lemvig, Jordy Timo van Velthoven

Published 2017-04-21Version 1

We present a new sufficient condition for the frame property of generalized translation-invariant systems. The sufficient condition is formulated in the Fourier domain and includes estimates for the upper and lower frame bound. Contrary to previously known conditions of a similar nature, the estimates take the phase of the generating functions in consideration and not only their modulus. By considering the phase of the generating functions, these estimates allow phase cancellations to occur and lead to improvements of the known estimates. Moreover, the possibility of phase cancellations makes these estimates optimal for tight frames. Our results on generalized translation-invariant systems will be proved in the setting of locally compact abelian groups, but even in the euclidean setting and the special case of wavelet and shearlet systems our results are new and recover Tchamitchian's estimate for dyadic wavelets.