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Framelets (a.k.a. wavelet frames) are of interest in both theory andapplications. Quite often, tight or dual framelets with high vanishing momentsare constructed through the popular oblique extension principle (OEP). ThoughOEP can increase vanishing moments for improved sparsity, it has a seriousshortcoming for scalar framelets: the associated discrete framelet transform isoften not compact and deconvolution is unavoidable. Here we say that a framelettransform is compact if it can be implemented by convolution using onlyfinitely supported filters. On the other hand, in sharp contrast to theextensively studied scalar framelets, multiframelets (a.k.a. vector framelets)derived through OEP from refinable vector functions are much less studied andare far from well understood. Also, most constructed multiframelets often lackbalancing property which reduces sparsity. In this paper, we are particularlyinterested in quasi-tight multiframelets, which are special dual multiframeletsbut behave almost identically as tight multiframelets. From any compactlysupported \emph{refinable vector function having at least two entries}, weprove that we can always construct through OEP a compactly supportedquasi-tight multiframelet such that (1) its associated discrete framelettransform is compact and has the highest possible balancing order; (2) allcompactly supported framelet generators have the highest possible order ofvanishing moments, matching the approximation/accuracy order of its underlyingrefinable vector function. This result demonstrates great advantages of OEP formultiframelets (retaining all the desired properties) over scalar framelets.

Compactly Supported Quasi-tight Multiframelets with High Balancing Orders and Compact Framelet Transforms