Wavelets of multiplicity 𝑟

Abstract

A multiresolution approximation ( V m ) m ∈ Z {({V_m})_{m \in {\mathbf {Z}}}} of L 2 ( R ) {L^2}({\mathbf {R}}) is of multiplicity r > 0 r > 0 if there are r functions ϕ 1 , … , ϕ r {\phi _1}, \ldots ,{\phi _r} whose translates form a Riesz basis for V 0 {V_0} . In the general theory we derive necessary and sufficient conditions for the translates of ϕ 1 , … , ϕ r , ψ 1 , … , ψ r {\phi _1}, \ldots ,{\phi _r},\;{\psi _1}, \ldots ,{\psi _r} to form a Riesz basis for V 1 {V_1} . The resulting reconstruction and decomposition sequences lead to the construction of dual bases for V 0 {V_0} and its orthogonal complement W 0 {W_0} in V 1 {V_1} . The general theory is applied in the construction of spline wavelets with multiple knots. Algorithms for the construction of these wavelets for some special cases are given.