Construction for a class of interpolation multiscaling functions with dilation factor a≥3a≥3

This paper is devoted to construction interpolatory multiscaling functions with a general dilation factor a≥3a≥3. We first show two-scale matrix symbol associated with interpolatory multiscaling functions is reduced to a special form. Also, a characterization on approximation order for the multiscaling functions is described in terms of elements of this special two-scale matrix symbol. Then, an algorithm is provided for constructing compactly supported interpolating multiscaling functions with dilation factor a=3a=3 and higher approximation order. Finally, the associated several families examples with one-parameter or two-parameters are explicitly presented. The optimal parameter values which make multiscaling functions provide the highest regularity are also computed.