2.1 Wavelets and Chebyshev wavelets
In recent years, wavelets have been very successful in many science and engineering fields. They constitute a family of functions constructed from dilation and translation of a single function called the mother wavelet . When the dilation parameter a and the translation parameter b vary continuously, we have the following family of continuous wavelets :
Chebyshev wavelets have four arguments, , k can assume any positive integer, m is the degree of Chebyshev polynomials of first kind and x denotes the time.
and , . Here are the well-known Chebyshev polynomials of order m, which are orthogonal with respect to the weight function and satisfy the following recursive formula:
We should note that the set of Chebyshev wavelets is orthogonal with respect to the weight function .
The derivative of Chebyshev polynomials is a linear combination of lower-order Chebyshev polynomials, in fact ,
2.2 Function approximation
A function defined over may be expanded as
where , in which denotes the inner product with the weight function . If in (7) is truncated, then (7) can be written as
where C and are matrices given by