Explicit construction of symmetric orthogonal wavelet frames in L2(Rs)

Recently, some researchers propose the concept of orthogonal wavelet frames, which are useful for multiple access communication systems. In this article, we first give two explicit algorithms for constructing paraunitary symmetric matrices (p.s.m. for short), whose entries are symmetric or antisymmetric Laurent polynomials. We also give two algorithms for constructing orthogonal wavelet frames from existing tight or dual wavelet frames in L2(Rs). The constructed orthogonal wavelet frames are also tight or dual ones. Furthermore, based on the constructed p.s.m. and the existing symmetric tight (dual) wavelet frames, we can obtain symmetric orthogonal (s.o. for short) tight (dual) wavelet frames in L2(Rs). From the constructed s.o. wavelet frames in L2(Rs), we can obtain s.o. wavelet frames in L2(Rm) by the projection method, where m≤s. To illustrate our results, we construct s.o. wavelet frames in L2(R) and L2(R2) from the quadratic B-spline B3(x). Especially, in Example 2, we obtain nonseparable tight 2I2-wavelet frames in L2(R2) from a separable tight 2I2-wavelet frame constructed by tensor product.