The Franklin wavelet is constructed using the multiresolution analysis (MRA) generated from a scaling function [Formula: see text] that is continuous on [Formula: see text], linear on [Formula: see text] and [Formula: see text] for every [Formula: see text]. For [Formula: see text] and [Formula: see text], it is shown that if a function [Formula: see text] is continuous on [Formula: see text], linear on [Formula: see text] and [Formula: see text], for [Formula: see text], and generates MRA with dilation factor [Formula: see text], then [Formula: see text]. Conversely, for [Formula: see text], it is shown that there exists a [Formula: see text], as satisfying the above conditions, that generates MRA with dilation factor [Formula: see text]. The frame MRA (FMRA) is useful in signal processing, since the perfect reconstruction filter banks associated with FMRA can be narrow-band. So it is natural to ask, whether the above results can be extended for the case of FMRA. In this paper, for [Formula: see text], we prove that if [Formula: see text] generates FMRA with dilation factor [Formula: see text], then [Formula: see text]. For [Formula: see text], we prove similar results when [Formula: see text]. In addition, for [Formula: see text] we prove that there exists a function [Formula: see text] as satisfying the above conditions, that generates FMRA. Also, we construct tight wavelet frame and wavelet frame for such scaling functions.
In signal processing, rational [Formula: see text]-wavelets are preferable than the wavelets corresponding to dyadic MRA because it allows more variations in scale factors of signal components. In this paper, for a rational number [Formula: see text] and [Formula: see text], we consider a collection [Formula: see text], the space of all continuous functions in [Formula: see text] that are linear on [Formula: see text] and [Formula: see text] for all [Formula: see text]. For [Formula: see text], under certain conditions, we prove that, if [Formula: see text] generates a [Formula: see text]-MRA, then [Formula: see text]. Also, we show that if [Formula: see text], there exists a function [Formula: see text], satisfying the above conditions, that generates [Formula: see text]-MRA. In addition, we construct orthonormal [Formula: see text]-wavelets corresponding to [Formula: see text]-MRA.
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