The Wavelet Scaling Equation 2.2 2003/01/17 2005/10/05 15:35:19.970 GMT-5 Phil Schniter schniter@ee.eng.ohio-state.edu Charlet Reedstrom charlet@rice.edu Phil Schniter schniter@ee.eng.ohio-state.edu scaling equation wavelet The difference in detail between V k and V k − 1 will be described using W k , the orthogonal complement of V k in V k − 1 : V k − 1 V k W k At times it will be convenient to write W k V k ⊥ . This concept is illustrated in the set-theoretic diagram, .

Suppose now that, for each k , we construct an orthonormal basis for W k and denote it by ψ k , n t n . It turns out that, because every V k has a basis constructed from shifts and stretches of a mother scaling function (i.e., φ k , n t 2 k 2 φ 2 k t n , every W k has a basis that can be constructed from shifts and stretches of a "mother wavelet" ψ t ℒ 2 : ψ k , n t 2 k 2 ψ 2 k t n . The Haar system will soon provide us with a concrete example . Let's focus, for the moment, on the specific case k 1 . Since W1 V0 , there must exist g n n such that: ψ 1 , 0 t n g n φ 0 , n t ⇔ 1 2 ψ 1 2 t n g n φ t n Wavelet Scaling Equation ψ t 2 n g n φ 2 t n To be a valid scaling-function/wavelet pair, φ t and ψ t must obey the wavelet scaling equation for some coefficient set g n .