Construction of multiscaling functions using the inverse representation theorem of matrix polynomials

Wavelet analysis deals with finding a suitable basis for the class of L^2 functions. Symmetric basis functions are very useful in various applications. In the case of all wavelets other than the famous Haar wavelet, the simultaneous inclusion of compact supportedness, orthogonality and symmetricity is not possible. Theory of multiwavelets assumes significance since it offers orthogonal, compact frames without losing symmetry. We can also construct symmetric, compactly supported and pseudo-biorthogonal bases which are also possible only in the case of multiwavelets. The properties of a multiwavelet directly depends on the corresponding multiscaling function. A multiscaling function is characterized by a unique symbol function, which is a matrix polynomial in complex exponential. A matrix polynomial can be constructed from its spectral data. Our aim is to find the necessary as well as sufficient conditions a spectral data must satisfy so that the corresponding matrix polynomial is the symbol function of a compactly supported, symmetric multiscaling function ɸ(x). We will construct such a multiscaling function ɸ(x) and its dual ɸ(x) such that the functions ɸ(x) and ɸ(x) form a pair of pseudo-biorthogonal multiscaling functions