Consider a periodic function \(f(x), ~ x \in [0,1)\), that is symmetric about \(x = 1/2\), so that \(f(x + 0.5) = f(x - 0.5)\). For example, consider the function \[f(x) = (1/2-x)^2 + \cos(2 \pi x).^2.\]
The form of the function \(f(x)\) does not have to be analytically available. Knowing the function at \(n\) equispaced collocation points \(x_i = i/(n+1)\), \(i = 0, 1, 2, ..., n\), is sufficient. Let us label the function at these collocation points \(f_i = f(x_i)\).
Due to its periodicity, and its symmetry, the discrete cosine series (DCS) is an ideal approximating function for such a data-series. The DCS family consists of members, \(\{1, \cos(2 \pi x), \cos(4 \pi x),... \cos(2 \pi j x), ...\},\) where \(v_j(x) = \cos(2 \pi j x)\) is a typical orthogonal basis function.
The members are orthogonal in the following sense. Let the inner product of two basis function be defined as,\[\langle v_i, v_j\rangle = \frac{2}{n+1} \sum_{i=0}^n v_i(x_i) v_j(x_j).\] Then we have, \[
\langle v_i, v_j\rangle = \begin{cases} 0, & \text{ if } j \neq k \\ 1 & \text{ if } j = k >0 \\ 2 & \text{ if } j = k = 0 \end{cases}.\] This can be verified easily by the following Octave commands:
This snippet yields \[\frac{2}{n+1} V^T V = \begin{bmatrix} 2 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0& 0\\ 0 & 0& 0& 1 & 0\\ 0 & 0 & 0& 0& 1\end{bmatrix}\]
The idea then is to approximate the given function in terms of the basis functions,\[f(x) = \sum_{j=0}^{N-1} a_j v_j(x_i),\] where \(N\) is the number of basis functions used.
From a linear algebra perspective we can think of the vector f and matrix V as, \[\mathbf{f} = \begin{bmatrix} f_0 \\ f_1 \\ \vdots \\ f_n \end{bmatrix}, ~~~ \mathbf{V} = \begin{bmatrix} | & | & ... & | \\ v_0(x_i) & v_1(x_i) & ... & v_{N-1}(x_i) \\ | & | & ... & | \\ \end{bmatrix}. \] The \((n+1) \times N\) matrix V contains the basis vectors evaluated at the collocation points.
We are trying to project f onto the column space of V, f = Va, where the column vector a specifies the linear combination of the matrix columns. In the usual case, the number of collocation points is greater than the number of DCS modes that we want to use in the approximating function.
Thus, we have to solve the problem in a least-squared sense by the usual technique, \[\mathbf{V^T f} = \mathbf{V^T V a}.\] Due to discrete orthogonality, we have already shown that, \[\mathbf{V^T V} = \frac{n+1}{2} \begin{bmatrix} 2 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0& 0\\ 0 & 0& 0& 1 & 0\\ 0 & 0 & 0& 0& 1\end{bmatrix}\]
Therefore, \[\begin{bmatrix} 2 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0& 0\\ 0 & 0& 0& 1 & 0\\ 0 & 0 & 0& 0& 1\end{bmatrix} \mathbf{a} = \frac{2}{n+1} \mathbf{V^T f}.\] In Octave, we can write the following
This yields,
a =
0.5839844
0.1026334
0.5266735
0.0126556
0.0078125
and the plot:
Due to its periodicity, and its symmetry, the discrete cosine series (DCS) is an ideal approximating function for such a data-series. The DCS family consists of members, \(\{1, \cos(2 \pi x), \cos(4 \pi x),... \cos(2 \pi j x), ...\},\) where \(v_j(x) = \cos(2 \pi j x)\) is a typical orthogonal basis function.
The members are orthogonal in the following sense. Let the inner product of two basis function be defined as,\[\langle v_i, v_j\rangle = \frac{2}{n+1} \sum_{i=0}^n v_i(x_i) v_j(x_j).\] Then we have, \[
\langle v_i, v_j\rangle = \begin{cases} 0, & \text{ if } j \neq k \\ 1 & \text{ if } j = k >0 \\ 2 & \text{ if } j = k = 0 \end{cases}.\] This can be verified easily by the following Octave commands:
This snippet yields \[\frac{2}{n+1} V^T V = \begin{bmatrix} 2 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0& 0\\ 0 & 0& 0& 1 & 0\\ 0 & 0 & 0& 0& 1\end{bmatrix}\]
The idea then is to approximate the given function in terms of the basis functions,\[f(x) = \sum_{j=0}^{N-1} a_j v_j(x_i),\] where \(N\) is the number of basis functions used.
From a linear algebra perspective we can think of the vector f and matrix V as, \[\mathbf{f} = \begin{bmatrix} f_0 \\ f_1 \\ \vdots \\ f_n \end{bmatrix}, ~~~ \mathbf{V} = \begin{bmatrix} | & | & ... & | \\ v_0(x_i) & v_1(x_i) & ... & v_{N-1}(x_i) \\ | & | & ... & | \\ \end{bmatrix}. \] The \((n+1) \times N\) matrix V contains the basis vectors evaluated at the collocation points.
We are trying to project f onto the column space of V, f = Va, where the column vector a specifies the linear combination of the matrix columns. In the usual case, the number of collocation points is greater than the number of DCS modes that we want to use in the approximating function.
Thus, we have to solve the problem in a least-squared sense by the usual technique, \[\mathbf{V^T f} = \mathbf{V^T V a}.\] Due to discrete orthogonality, we have already shown that, \[\mathbf{V^T V} = \frac{n+1}{2} \begin{bmatrix} 2 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0& 0\\ 0 & 0& 0& 1 & 0\\ 0 & 0 & 0& 0& 1\end{bmatrix}\]
Therefore, \[\begin{bmatrix} 2 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0& 0\\ 0 & 0& 0& 1 & 0\\ 0 & 0 & 0& 0& 1\end{bmatrix} \mathbf{a} = \frac{2}{n+1} \mathbf{V^T f}.\] In Octave, we can write the following
This yields,
a =
0.5839844
0.1026334
0.5266735
0.0126556
0.0078125
and the plot:
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