Let { V k } be a nested sequence of closed subspaces that constitute a multiresolution analysis of L 2( R ). We characterize the family Φ = { φ} where each φ generates this multiresolution analysis such that the two-scale relation of φ is governed by a finite sequence. In particular, we identify the ϑ ϵ Φ that has minimum support. We also characterize the collection Ψ of functions η such that each η generates the orthogonal complementary subspaces W k of V k , ∈ Z . In particular, the minimally supported ψ ϵ Ψ is determined. Hence, the “ B-spline” and “ B-wavelet” pair (ϑ, ψ) provides the most economical and computational efficient “spline” representations and “wavelet” decompositions of L 2 functions from the “spline” spaces V k and “wavelet” spaces W k , k∈ Z . A very general duality principle, which yields the dual bases of both {ϑ(·− j): j∈ Z and {η(·− j): j∈ Z } for any η ϵ Ψ by essentially interchanging the pair of two-scale sequences with the pair of decomposition sequences, is also established. For many filtering applications, it is very important to select a multiresolution for which both ϑ and ψ have linear phases. Hence, “non-symmetric” ϑ and ψ, such as the compactly supported orthogonal ones introduced by Daubechies, are sometimes undesirable for these applications. Conditions on linear-phase φ and ψ are established in this paper. In particular, even-order polynomial B-splines and B-wavelets φ m and ψ m have linear phases, but the odd-order B-wavelet only has generalized linear phases.