In functional analysis, compactly supported wavelets derived from Legendre polynomials are termed Legendre wavelets or spherical harmonic wavelets. Legendre functions have widespread applications in which spherical coordinate system is appropriate. As with many wavelets there is no nice analytical formula for describing these harmonic spherical wavelets. The low-pass filter associated to Legendre multiresolution analysis is a finite impulse response (FIR) filter.

Wavelets associated to FIR filters are commonly preferred in most applications. An extra appealing feature is that the Legendre filters are linear phase FIR (i.e. multiresolution analysis associated with linear phase filters). These wavelets have been implemented on MATLAB (wavelet toolbox). Although being compactly supported wavelet, legdN are not orthogonal (but for N = 1).

Legendre multiresolution filters

Associated Legendre polynomials are the colatitudinal part of the spherical harmonics which are common to all separations of Laplace's equation in spherical polar coordinates.-order differential equation, n integer:

of a multiresolution analysis (MRA).

A low-pass behaviour is exhibited for the filter H is equal to the degree of the Legendre polynomial. Therefore, the roll-off.

The low-pass filter transfer function is given by

is chosen according to Quadrature mirror filter condition, yielding:

, as expected.

Legendre multiresolution filter coefficients

to the form

are given by:

from which the symmetry:

.

1. is the wavelet order.)

   N.B. The minus signal can be suppressed.

References

Bibliography

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