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 Incoming links: Meyer-Mallat theorem

Let

W_{j} = V_{j + 1} ⊖ V_{j}

V_{j}

inside

V_{j + 1}

W_{0}

. Dilation then gives the orthogonal decomposition into orthogonal complements

L^{2} (R) = ⨁_{j \in Z} W_{j}

, so its translates and dilates form an orthonormal wavelet basis.

Using the MRA low-pass filter, choose the high-pass symbol

n (ξ) = e^{- i ξ} \overline{m (ξ + π)}

ψ (2 ξ) = e^{- i ξ} \overline{m (ξ + π)} φ (ξ) .

Equivalently, with the refinement coefficients from the scaling refinement equation, put

g_{k} = (- 1)^{k - 1} \overline{h_{1 - k}}

and

ψ (x) = 2 \sum_{k} g_{k} φ (2 x - k)

. This index choice gives precisely the displayed high-pass symbol. Changing all

g_{k}

by one constant unit phase gives the same orthonormal wavelet construction. In particular the often-used

(- 1)^{k} \overline{h_{1 - k}}

convention differs only by a global minus sign. The basis is

ψ_{j, k} (x) = 2^{j /2} ψ (2^{j} x - k)

for

j, k \in Z

Put

W_{j} = V_{j + 1} \cap V_{j}^{⊥}

ψ \in W_{0}

whose integer translates form an orthonormal basis of

W_{0}

. Dilations then give an orthonormal basis of each

W_{j}

, and the density and trivial-intersection axioms imply

L^{2} (R) = ⨁_{j \in Z} W_{j} .

For an explicit filter construction, write the scaling refinement equation

φ (x) = 2 \sum_{k} h_{k} φ (2 x - k)

g_{k} = (- 1)^{k} \overline{h_{1 - k}}

and

ψ (x) = 2 \sum_{k} g_{k} φ (2 x - k)

. A harmless constant sign or integer translation gives an equivalent orthonormal wavelet.

ψ \in L^{2} (R)

such that

{2^{j /2} ψ (2^{j} x - k)}_{j, k \in Z}

V_{j}

with trivial intersection, dense union, dyadic scaling, integer-translation invariance of

V_{0}

, and a generator

ϕ

whose integer translates form an orthonormal basis of

V_{0}

. The Meyer-Mallat theorem says every such analysis has an orthonormal wavelet whose translates span

V_{j + 1} ⊖ V_{j}