This paper provides new sufficient and necessary conditions for the frame property of generalized translation-invariant systems. The conditions are formulated in the Fourier domain and consists of estimates involving the upper and lower frame bound. Contrary to known conditions of a similar nature, the estimates take the phase of the generating functions in consideration and not only their modulus. The possibility of phase cancellations makes these estimates optimal for tight frames. The results on generalized translation-invariant systems will be proved in the setting of locally compact abelian groups, but even for euclidean space and the special case of (composite) wavelet systems the results are new.
@cite_41: In this paper we extend the investigation of quasi-affine systems, which were originally introduced by Ron and Shen [J. Funct. Anal. 148 (1997), 408-447] for integer, expansive dilations, to the class of rational, expansive dilations. We show that an affine system is a frame if, and only if, the corresponding family of quasi-affine systems are frames with uniform frame bounds. We also prove a similar equivalence result between pairs of dual affine frames and dual quasi-affine frames. Finally, we uncover some fundamental differences between the integer and rational settings by exhibiting an example of a quasi-affine frame such that its affine counterpart is not a frame.
@cite_27: 
@cite_40: Abstract : Shift invariance fiberization techniques are applied for the study of the synthesis and analysis operators of affine (wavelet) systems. In this approach, one has first to circumvent the fact that affine systems are not shift invariant. The results obtained include characterizations of the Bessel property, the Riesz basis property and the frame property of such sets in terms of the behaviour of simpler operators. Various estimates of the lower and upper frame (Riesz) bounds are included, too.
@cite_46: The notion of quasi-affine frame was recently introduced by Ron and Shen [9] in order to achieve shift-invariance of the discrete wavelet transform. In this paper, we establish a duality-preservation theorem for quasi-affine frames. Furthermore, the preservation of frame bounds when changing an affine frame to a quasi-affine frame is shown to hold without the decay assumptions in [9]. Our consideration leads naturally to the study of certain sesquilinear operators which are defined by two affine systems. The translation-invariance of such operators is characterized in terms of certain intrinsic properties of the two affine systems.
@cite_17: Abstract : Discrete affine systems are obtained by applying dilations to a given shift-invariant system. The complicated structure of the affine system is due, first and foremost, to the fact that it is not invariant under shifts. Affine frames carry the additional difficulty that they are 'global' in nature: it is the entire interaction between the various dilation levels that determines whether the system is a frame, and not the behaviour of the system within one dilation level. We completely unravel the structure of the affine system with the aid of two new notions: the affine product, and a quasi-affine system. This leads to a characterization of affine frames; the induced characterization of tight affine frames is in terms of exact orthogonality relations that the wavelets should satisfy on the Fourier domain. Several results, such as a general oversampling theorem follow from these characterizations. Most importantly, the affine product can be factored during a multiresolution analysis construction, and this leads to a complete characterization of all tight frames that can be constructed by such methods. Moreover, this characterization suggests very simple sufficient conditions for constructing tight frames from multiresolution. Of particular importance are the facts that the underlying scaling function does not need to satisfy any a priori conditions, and that the freedom offered by redundancy can be fully exploited in these constructions. (AN)
An alternative route for deriving necessary and sufficient conditions for wavelet frames with integer, expansive dilations goes through quasi-affine systems @cite_17 @cite_27 @cite_46 . This link is known to generalize to rational, expansive dilations @cite_41 , although one has to consider a family of quasi-affine systems to capture the frame property of the given wavelet system. Since quasi-affine systems are shift-invariant, sufficient and necessary conditions for rational wavelet systems are readily available. We stress that such estimates differ slightly from the ones presented in this paper. The estimates presented in @cite_40 for wavelet systems with integer, expansive dilations utilize the quasi-affine route, but they ignore the phase of the wavelet generator and are therefore not optimal for tight frames.