 Second-difference analyzing function

ID: second-difference-analyzing-function

The Haar wavelet provides an explicit one-vanishing moment comparison:

h (u) = ⎩ ⎨ ⎧ 1 - 1 0 0 \leq u < 1/2, 1/2 \leq u < 1, otherwise, \int h = 0, \int u h (u) d u = - \frac{1}{4}, ∥ h ∥_{2} = 1.

It cancels a constant polynomial, but not a nonconstant affine function. To obtain two vanishing moments by an equally transparent construction, take three adjacent unit intervals with heights proportional to

1, - 2, 1

η (u) = \frac{1}{6} ⎩ ⎨ ⎧ 1 - 2 10 0 \leq u < 1, 1 \leq u < 2, 2 \leq u < 3, otherwise .

The L2 norm normalization follows from

1^{2} + (- 2)^{2} + 1^{2} = 6

. Direct integration gives

\int η = \frac{1 - 2 + 1}{6} = 0, \int u η (u) d u = \frac{1/2 - 2 ( 3/2 ) + 5/2}{6} = 0, \int u^{2} η (u) d u = \frac{2}{6} \neq = 0.

Thus

η

c_{η} (f; a, b) = \frac{a}{6} (A_{0} - 2 A_{1} + A_{2}), A_{j} = \int_{j}^{j + 1} f (b + a u) d u .

The weights are those of the second-order central difference. The first moment balances the centres of the three intervals; it is not enough merely to balance their signed areas.

This

η

is a normalized analyzing function, with no orthonormal basis assertion. Indeed, its unit translate has nonzero L2 inner product:

\int_{R} η (u) η (u - 1) d u = \frac{( - 2 ) ( 1 ) + ( 1 ) ( - 2 )}{6} = - \frac{2}{3} .

Hence these translates already fail orthogonality. The Daubechies wavelet construction imposes additional refinement and orthogonality requirements; two vanishing moments and unit L2 norm by themselves do not supply those requirements.

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