### 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 \psi (x). When the dilation parameter *a* and the translation parameter *b* vary continuously, we have the following family of continuous wavelets [19]:

{\psi}_{a,b}(x)={|a|}^{-1/2}\psi \left(\frac{x-b}{a}\right),\phantom{\rule{1em}{0ex}}a,b\in \mathbb{R},a\ne 0.

Chebyshev wavelets {\psi}_{n,m}=\psi (k,n,m,x) have four arguments, n=1,2,\dots ,{2}^{k-1}, *k* can assume any positive integer, *m* is the degree of Chebyshev polynomials of first kind and *x* denotes the time.

{\psi}_{n,m}(x)=\{\begin{array}{cc}\frac{{\alpha}_{m}{2}^{(k-1)/2}}{\sqrt{\pi}}{T}_{m}({2}^{k}x-2n+1),\hfill & \frac{n-1}{{2}^{k-1}}\le x<\frac{n}{{2}^{k-1}};\hfill \\ 0,\hfill & \text{otherwise},\hfill \end{array}

(5)

where

{\alpha}_{m}=\{\begin{array}{cc}\sqrt{2},\hfill & m=0;\hfill \\ 2,\hfill & m=1,2,\dots \hfill \end{array}

and m=0,1,2,\dots ,M-1, n=1,2,\dots ,{2}^{k-1}. Here {T}_{m}(x) are the well-known Chebyshev polynomials of order *m*, which are orthogonal with respect to the weight function \omega (x)=1/\sqrt{1-{x}^{2}} and satisfy the following recursive formula:

\begin{array}{c}{T}_{0}(x)=1,\hfill \\ {T}_{1}(x)=x,\hfill \\ {T}_{m+1}(x)=2x{T}_{m}(x)-{T}_{m-1}(x).\hfill \end{array}

We should note that the set of Chebyshev wavelets is orthogonal with respect to the weight function {\omega}_{n}(x)=\omega ({2}^{k}x-2n+1).

The derivative of Chebyshev polynomials is a linear combination of lower-order Chebyshev polynomials, in fact [20],

\{\begin{array}{cc}{T}_{m}^{\prime}(x)=2m{\sum}_{k=1}^{m-1}{T}_{k}(x),\hfill & m\text{even};\hfill \\ {T}_{m}^{\prime}(x)=2m{\sum}_{k=1}^{m-1}{T}_{k}(x)+m{T}_{0}(x),\hfill & m\text{odd}.\hfill \end{array}

(6)

### 2.2 Function approximation

A function u(x) defined over [0,1) may be expanded as

u(x)=\sum _{n=1}^{\mathrm{\infty}}\sum _{m=0}^{\mathrm{\infty}}{c}_{nm}{\psi}_{nm}(x),

(7)

where {c}_{nm}=(u(x),{\psi}_{nm}(x)), in which (\cdot ,\cdot ) denotes the inner product with the weight function {\omega}_{n}(x). If u(x) in (7) is truncated, then (7) can be written as

u(x)\approx \sum _{n=1}^{{2}^{k-1}}\sum _{m=0}^{M-1}{c}_{nm}{\psi}_{nm}(x)={C}^{T}\Psi (x),

(8)

where *C* and \Psi (x) are {2}^{k-1}M\times 1 matrices given by

\begin{array}{c}C={[{c}_{1},{c}_{2},\dots ,{c}_{{2}^{k-1}}]}^{T},\hfill \\ \Psi (x)={[{\psi}_{1},{\psi}_{2},\dots ,{\psi}_{{2}^{k-1}}]}^{T}\hfill \end{array}

and

\begin{array}{c}{c}_{i}=[{c}_{i0},{c}_{i1},\dots ,{c}_{i,M-1}],\hfill \\ {\psi}_{i}(x)=[{\psi}_{i0}(x),{\psi}_{i1}(x),\dots ,{\psi}_{i,M-1}(x)],\phantom{\rule{1em}{0ex}}i=1,2,3,\dots ,{2}^{k-1}.\hfill \end{array}