Symmetric and antisymmetric tight wavelet frames

For a given set of wavelets Ψ, we provide a general, and yet simple, method to derive a new set of wavelets Ψ′ such that each wavelet in Ψ′ is either symmetric or antisymmetric. The affine system generated by Ψ′ is a tight frame for the space L2(Rd) whenever the affine system generated by Ψ is so. Further, when Ψ is constructed via a multiresolution analysis, Ψ′ can also be derived from a, but possibly different, multiresolution analysis. If moreover the multiresolution analysis for constructing Ψ is generated by a symmetric refinable function, then Ψ′ is obtained from the same multiresolution analysis.