Orthogonal Vector-Valued Wavelets on $\mathbb {R}_{+}$

Abstract

We have considered the notion of vector-valued multiresolution analysis (VMRA) on positive half line $\mathbb {R}_+$ and studied associated vector-valued wavelets and wavelet packets. Xia and Suter in 1996 generalized the concept of multiresolution analysis (MRA) on $\mathbb {R}$ to vector-valued multiresolution analysis (VMRA) on $\mathbb {R}$ and studied associated vector-valued wavelets. Farkov (Farkov in Orthogonal p-wavelets on $R^{+}$. St. Petersburg University Press, Saint Petersburg, p. 426, 2005) introduced MRA on $\mathbb {R}_+$. In this chapter, we have introduced vector-valued multiresolution analysis (VMRA) on $\mathbb {R}_+$, where the associated subspace $V_0$ of $L^{2}(\mathbb {R}_+,\mathbb {C}^N)$ has an orthonormal basis, a family of translates of a vector-valued function $\varPhi$, i.e, $\{\varPhi (x\ominus l)\}_{l\in \mathbb {Z}_+}$, where $\mathbb {Z}_+$ is the set of nonnegative integers. The necessary and sufficient condition for the existence of associated vector-valued wavelets has been obtained and the construction of vector-valued multiresolution analysis (VMRA) on $\mathbb {R}_{+}$ has been presented.