Do shifted Chebyshev polynomials form a complete set of independent functions?

Question

Do Chebyshev polynomials form a complete set of independent functions? If yes, what can we say about their shifted versions? E.g. shifted Chebyshev polynomials of the first kind are defined as $$T_{n}^{*}(x)=T_{n}(2x-1)$$ or other possible shitting like e.g. $T_{n}\left(\frac{x^2}{2}-1\right)$.

Answer 1

I assume that by complete you mean a basis for the set of polynomials.

Yes, the Chebyshev polynomials form a complete set of independent functions. That should be clear considering the recurrence relation. For example, for the first kind, that relation is\begin{align} T_0(x) & = 1\\ T_1(x) & = x\\ &\ \ \vdots\\ T_{n+1}(x) & = 2xT_n - T_{n-1}.\end{align} The degree of polynomial $T_n$ is $n$, so any polynomial can be expressed as a linear combination of the $T_n$, and if $c_0 T_0 +\cdots + c_n T_n = 0$, then each $c_i$ is zero. See also the Wikipedia article.

The same holds for the shift $T^*_n(x) = T_n(2x-1)$ because the degree of $T^*_n$ is still $n$ and the argument above applies.

The shift $T^*_n(x^2/2 - 1)$ generates independent polynomials, but they do not form a basis for the polynomials. For example, all the $T^*_n$ in this case are of even degree, so polynomials of odd degree cannot be formed from a linear combination of the $T^*_n$.

Answer 2

Chebyshev polynomials of the first kind $\;T_n(x)\;$ are defined in the interval $\;[-1,1]\;$ and presents one of the popular families of the classic orthogonal polynomials, with the next common features:

• the family is the orthogonal polynomial basis with the weight $\;\dfrac1{\sqrt{1-x^2}};\;$
• are known recurrency relations, differential equations, generating function etc;
• are known expressions via the some other orthogonal polynomials and via hypergeometric function.

Besides, they have some specific features:

• presentations via trigonometric and hyperbolic functions;
• existance of the inverse functions with the closed form in the elementary functions;
• bounded set of values $\;T_n(x)\in[-1,-1],\;$ wherein all extremes have the values $\;\pm1.\;$

Known formulas $$T_3(x)=4x^3-3x=\cos(3\arccos x) = -\sin(3\arcsin x) = \cosh(3\operatorname{arccosh} x)$$ are the base of the "trigonometric" solutions of the cubic equations.

The "bounding" property is the base of the "series economization", which is used for the polynomial approximaion of functions. Idea of method is the presentation of known series via Chebyshev polynomials. Elimination of high-order polynomials leads to the constant approximation errors among the domain $[-1,1].$

Idea of the shifted Chebyshev polynomials is the linear transformation of the domain to $[0,1],$ which is more suitable for the economization technic. Thus, they have all described properties, with the corresponding differencies in the parametrization.

Therefore, the shifted Chebyshev polynomials are the way of the applying of the "bounding" feature of the Chebyshev polynomials of the first kind.