Multiresolution properties of the wavelet Galerkin operator

This paper extends recent results by the author [J. Math. Phys. 31, 1898 (1990), J. Math. Phys. 32, 57 (1991)] that show a scaling parameter sequence h yields an orthonormal wavelet basis for L2(R) if and only if an associated operator Sh has eigenvalue 1 with multiplicity 1. The operator transforms a sequence a by Sh(a)(k)=2Σm,n∼(h(m))h(n)a(2k+m−n). A correspondence is derived between Sh and Galerkin projection operators related to the multiresolution analysis defined by the orthonormal wavelet basis. The spectrum of Sh is characterized in terms of the Fourier modulus of the (unique) scaling function φ that satisfies φ(x)=2Σnh(n)φ(2x−n). This characterization yields several results including a direct, alternate proof that the eigenvalue 1 of Sh has multiplicity 1.