previous overview
$
\def \slechts {\quad \Longleftrightarrow \quad}
$
Let's talk a minute about my article named Cosine
Expansions.
Why has nobody told us that the cosine power series
in that article, as well as in the current one, are actually
Chebyshev
Polynomials of the First kind?
And that the definition which
comes most close to the employment in these two contexts is:
$$
T_n(x) = \left\{ \begin{array}{ll}
\cos(n \arccos(x)) &\qquad \mbox{for} \quad -1 \le x \le +1 \\
\cosh(n\;\mbox{arccosh}(x)) &\qquad \mbox{for} \quad x \ge +1
\end{array} \right.
$$
An immediate corollary is the composite identity (or the "nesting property"):
$$
T_n(T_m(x)) = T_{n.m}(x)
$$
Lemma. The following holds for the trigonometric as well as for the
hyperbolic cosine:
\begin{eqnarray*}
\cos(\alpha+\beta) + \cos(\alpha-\beta) &=& 2\cos(\alpha) \cos(\beta) \\
\cosh(\alpha+\beta) + \cosh(\alpha-\beta) &=& 2\cosh(\alpha) \cosh(\beta)
\end{eqnarray*}
Proof. Add the following equations together.\\For the trigonometric case:
\begin{eqnarray*}
\cos(\alpha+\beta) = \cos(\alpha) \cos(\beta) - \sin(\alpha) \sin(\beta) \\
\cos(\alpha-\beta) = \cos(\alpha) \cos(\beta) + \sin(\alpha) \sin(\beta)
\end{eqnarray*}
For the hyperbolic case:
\begin{eqnarray*}
\cosh(\alpha+\beta) = \cosh(\alpha)\cosh(\beta)+\sinh(\alpha)\sinh(\beta) \\
\cosh(\alpha-\beta) = \cosh(\alpha)\cosh(\beta)-\sinh(\alpha)\sinh(\beta)
\end{eqnarray*}
Theorem. $$ T_{n+1}(x) + T_{n-1}(x) = 2 x T_n(x) $$
Proof. Employ the lemma for $\alpha = n\theta$ and $\beta = \theta$:
\begin{eqnarray*}
\cos((n+1)\theta) + \cos((n-1)\theta) &=& 2\cos(n \theta) \cos(\theta) \\
\cosh((n+1)\theta) + \cosh((n-1)\theta) &=& 2\cosh(n \theta) \cosh(\theta)
\end{eqnarray*}
And substitute $\theta = \mbox{arccos}(x))$ or $\theta = \mbox{arccosh}(x))$
respectively. This gives:
\begin{eqnarray*}
\cos((n+1)\;\mbox{arccos}(x)) + \cos((n-1)\;\mbox{arccos}(x))
&=& 2 x \cos(n\;\mbox{arccos}(x)) \\
\cosh((n+1)\;\mbox{arccosh}(x)) + \cosh((n-1)\;\mbox{arccosh}(x))
&=& 2 x \cosh(n\;\mbox{arccosh}(x)) \\
\end{eqnarray*}
In short:
$$
T_{n+1}(x) + T_{n-1}(x) = 2 x T_n(x) \slechts
T_{n+1}(x) = 2 x T_n(x) - T_{n-1}(x)
$$
The latter is exactly the standard recursion formula for Chebyshev Polynmials.
Its initial values are:
\begin{eqnarray*}
T_0(x) &=& \mbox{cos[h]}(0\;\mbox{arccos[h]}(x)) = 1 \\
T_1(x) &=& \mbox{cos[h]}(1\;\mbox{arccos[h]}(x)) = x
\end{eqnarray*}
Herewith we find, for the Chebyshev Polynomials up to order $5$:
\begin{eqnarray*}
T_2(x) &=& 2 x x - 1 = 2 x^2 - 1 \\
T_3(x) &=& 2 x (2 x^2 - 1) - x = 4 x^3 - 3 x \\
T_4(x) &=& 2 x (4 x^3 - 3 x) - (2 x^2 - 1) = 8 x^4 - 8 x^2 + 1 \\
T_5(x) &=& 2 x (8 x^4 - 8 x^2 + 1) - (4 x^3 - 3 x) = 16 x^5 - 20 x^3 + 5 x
\end{eqnarray*}
Now we could define Han de Bruijn's Polynomials $B_n(y)$ by:
$$
B_n(y) = 2 T_n(y/2 - 1) + 2
$$
Herewith we find:
\begin{eqnarray*}
B_0(y) &=& 2 + 2 = 4 \\
B_1(y) &=& 2 (y/2 - 1) + 2 = y \\
B_2(y) &=& 4 (y/2 - 1)^2 - 2 + 2 = (y - 2)^2 \\
B_3(y) &=& 8 (y/2-1)^3 - 6 (y/2-1) + 2 = (y-2)^3 - 3(y-2) + 2 = y(y-3)^2
\end{eqnarray*}
Thus the DoubleGrid and TripleGrid Coursening iterands are described by the
polynomials $B_2(y)$ and $B_3(y)$ respectively.