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Difference between revisions of "Chebyshev polynomials of the first kind"
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using the composition identity and the fact that <math>T_m(1) = \cos(m \arccos 1) = \cos 0 = 1</math> for all <math>m</math>.
 
using the composition identity and the fact that <math>T_m(1) = \cos(m \arccos 1) = \cos 0 = 1</math> for all <math>m</math>.
  
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Particular cases include that <math>1</math>, being a root of <math>T_1(x) - 1 = x - 1</math>, is a root of <math>T_n(x) - 1</math> for all <math>n</math>, and <math>-1</math>, being a root of <math>T_2(x) - 1 = 2x^2 - 2</math>, is a root of <math>T_n(x) - 1</math> for all even <math>n</math>. All other roots of <math>T_n(x) - 1</math> correspond to roots of unity which fall into [[Complex conjugate|conjugate pairs]] with the same real part.
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Particular cases include that <math>1</math>, being a root of <math>T_1(x) - 1 = x - 1</math>, is a root of <math>T_n(x) - 1</math> for all positive <math>n</math>, and <math>-1</math>, being a root of <math>T_2(x) - 1 = 2x^2 - 2</math>, is a root of <math>T_n(x) - 1</math> for all positive even <math>n</math>. All other roots of <math>T_n(x) - 1</math> correspond to roots of unity which fall into [[Complex conjugate|conjugate pairs]] with the same real part. One might therefore suspect that all remaining roots of <math>T_n(x) - 1</math> are double roots. In fact, there exist polynomials <math>f_n(x)</math> and <math>g_n(x)</math> such that <cmath>T_{2n+1} = (x - 1)(f_n(x))^2</cmath> and <cmath>T_{2n+2} = 2(x - 1)(x + 1)(g_n(x))^2</cmath> for all nonnegative integers <math>n</math>.

Revision as of 18:40, 2 March 2022

The Chebyshev polynomials of the first kind are defined recursively by \[T_0(x) = 1,\] \[T_1(x) = x,\] \[T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x),\] or equivalently by \[T_n(x) = \cos (n \arccos x).\]

Proof of equivalence of the two definitions

In the proof below, $T_n(x)$ will refer to the recursive definition.

For the $n = 0$ base case, \[\cos(0 \arccos x) = \cos 0 = 1 = T_0(x);\] for the $n = 1$ base case, \[\cos(1 \arccos (x)) = \cos(\arccos (x)) = x = T_1(x).\]

Now for the inductive step, let $y = \arccos x$, so that $x = \cos y$. We then assume that $\cos ((n-1)y) = T_{n-1}(x)$ and $\cos ny = T_n(x)$, and we wish to prove that $\cos ((n+1)y) = T_{n+1}(x)$.

From the cosine sum and difference identities we have \[\cos ((n+1)y) = \cos (ny+y) = \cos ny \cos y - \sin ny \sin y\] and \[\cos ((n-1)y )= \cos (ny-y) = \cos ny \cos y + \sin ny \sin y.\] The sum of these equations is \[\cos ((n+1)y) + \cos ((n-1)y) = 2 \cos ny \cos y;\] rearranging, \[\cos ((n+1)y) = 2 \cos y \cos ny - \cos ((n-1)y).\] Substituting our assumptions yields \[\cos ((n+1)y) = 2xT_n(x) - T_{n-1}(x) = T_{n+1}(x),\] as desired.

Composition identity

For nonnegative integers $m$ and $n$, the identity $T_{mn} = T_m(T_n(x))$ holds.

First proof

By the trigonometric definition, $T_m(T_n(x)) = \cos(m(\arccos(\cos(n(\arccos(x))))))$.

As before, let $\arccos x = y$. We have $\arccos(\cos(ny)) = 2k\pi \pm ny$ for some integer $k$. Multiplying by $m$ and distributing gives $2mk\pi \pm mny$; taking the cosine gives $\cos (2mk\pi \pm mny) = \cos mny = \cos ( mn (\arccos x)) = T_{mn}(x)$.

For now this proof only applies where the trigonometric definition is defined; that is, for $x \in [-1,1]$. However, $T_{mn}(x)$ is a degree-$mn$ polynomial, and so is $T_m(T_n(x))$, so the fact that $T_{mn}(x) = T_m(T_n(x))$ for some $mn + 1$ distinct $x \in [-1,1]$ is sufficient to guarantee that the two polynomials are equal over all real numbers.

Second proof (Induction)

First we prove a lemma: that $T_{k+n}(x) = 2T_n(x)T_k(x) - T_{k-n}(x)$ for all $n \leq k$. To prove this lemma, we fix $k$ and induct on $n$.

For all $k$, we have \[T_{k}(x) = 2T_{k}(x) - T_k(x) = 2T_0(x)T_{k}(x) - T_{k}(x),\] and for all $k \geq 1$, \[T_{k+1}(x) = 2xT_k(x) - T_{k-1}(x) = 2T_1(x)T_k(x) - T_{k-1}(x),\] proving the lemma for $n = 0$ and $n = 1$ respectively.

Suppose the lemma holds for $n - 1$ and $n$; that is, $T_{k+n}(x) = 2T_{n}(x)T_k(x) - T_{k-n}(x)$ and $T_{k+n-1}(x) = 2T_{n-1}(x)T_k(x) - T_{k-n+1}(x)$. Then multiplying the first equation by $2x$ and subtracting the second equation gives \[2xT_{k+n}(x) - T_{k+n-1}(x) = 2(2xT_{n}(x) - T_{n-1}(x))T_k(x) - (2xT_{k-n}(x) - T_{k-n+1}(x)),\] which simplifies to \[T_{k+n+1}(x) = 2T_{n+1}(x)T_k(x) - T_{k-n-1}(x)\] using the original recursive definition, as long as $k-n-1 \geq 0$. Thus, the lemma holds for $n + 1$ (as long as $n + 1 \leq k$), completing the inductive step.

To prove the claim, we now fix $n$ and induct on $m$.

For all $n$, we have \[T_0(T_n(x)) = 1 = T_0(x)\] and \[T_1(T_n(x)) = T_n(x),\] proving the claim for $m = 0$ and $m = 1$ respectively.

Suppose the claim holds for $m - 1$ and $m$; that is, $T_{m-1}(T_n(x)) = T_{(m-1)n}(x)$ and $T_m(T_n(x)) = T_{mn}(x)$. We may also assume $m \geq 2$, since the smaller cases have already been proven, in order to ensure that $n \leq mn$. Then by the lemma (with $k = mn$) and the original recursive definition, \begin{align*} T_{(m+1)n}(x) &= T_{mn+n}(x) \\ &= 2T_n(x)T_{mn}(x) - T_{mn - n}(x) \\ &= 2T_n(x)T_{mn}(x) - T_{(m-1)n}(x) \\ &= 2T_n(x)T_m(T_n(x)) - T_{m-1}(T_n(x)) \\ &= T_{m+1}(T_n(x)), \\ \end{align*} completing the inductive step.

Roots

Since $T_n(\cos y) = \cos ny$, and the values of $ny$ for which $\cos ny = 0$ are $\frac{2k+1}{2}\pi$ for integers $k$, the roots of $T_n(x)$ are of the form \[\cos \left( \frac{2k+1}{2n}\pi \right).\]

These roots are also called Chebyshev nodes.

Connection to roots of unity

Because cosine is $1$ only at integer multiples of $2\pi$, the roots of the polynomial $T_n(x) - 1$ follow a simpler formula: $\cos \frac{2k\pi}{n}$ for integers $k$. The $n$th roots of unity have arguments of $\frac{2k\pi}{n}$ and magnitude $1$, so the roots of $T_n(x) - 1$ are the real parts of the $n$th roots of unity. This lends intuition to several patterns.

All roots of $T_n(x) - 1$ are also roots of $T_{mn}(x) - 1$, since all $n$th roots of unity are also $mn$th roots of unity. This can also be shown algebraically as follows: Suppose $T_n(x) - 1 = 0$. Then \[T_{mn}(x) - 1 = T_m(T_n(x)) - 1 = T_m(1) - 1 = 1 - 1 = 0,\] using the composition identity and the fact that $T_m(1) = \cos(m \arccos 1) = \cos 0 = 1$ for all $m$.

Particular cases include that $1$, being a root of $T_1(x) - 1 = x - 1$, is a root of $T_n(x) - 1$ for all positive $n$, and $-1$, being a root of $T_2(x) - 1 = 2x^2 - 2$, is a root of $T_n(x) - 1$ for all positive even $n$. All other roots of $T_n(x) - 1$ correspond to roots of unity which fall into conjugate pairs with the same real part. One might therefore suspect that all remaining roots of $T_n(x) - 1$ are double roots. In fact, there exist polynomials $f_n(x)$ and $g_n(x)$ such that \[T_{2n+1} = (x - 1)(f_n(x))^2\] and \[T_{2n+2} = 2(x - 1)(x + 1)(g_n(x))^2\] for all nonnegative integers $n$.

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