The Relationships Between Chebyshev, Legendre and Jacobi Polynomials: The Generic Superiority of Chebyshev Polynomials and Three Important Exceptions

Abstract

We analyze the asymptotic rates of convergence of Chebyshev, Legendre and Jacobi polynomials. One complication is that there are many reasonable measures of optimality as enumerated here. Another is that there are at least three exceptions to the general principle that Chebyshev polynomials give the fastest rate of convergence from the larger family of Jacobi polynomials. When $f(x)$ is singular at one or both endpoints, all Gegenbauer polynomials (including Legendre and Chebyshev) converge equally fast at the endpoints, but Gegenbauer polynomials converge more rapidly on the interior with increasing order $m$. For functions on the surface of the sphere, associated Legendre functions, which are proportional to Gegenbauer polynomials, are best for the latitudinal dependence. Similarly, for functions on the unit disk, Zernike polynomials, which are Jacobi polynomials in radius, are superior in rate-of-convergence to a Chebyshev–Fourier series. It is true, as was conjectured by Lanczos 60 years ago, that excluding these exceptions, the Chebyshev coefficients $a_{n}$ usually decrease faster than the Legendre coefficients $b_{n}$ by a factor of $\sqrt{n}$. We calculate the proportionality constant for a few examples and restrictive classes of functions. The more precise claim that $b_{n} \sim \sqrt{\pi /2} \sqrt{n} a_{n}$, made by Lanczos and later Fox and Parker, is true only for rather special functions. However, individual terms in the large $n$ asymptotics of Chebyshev and Legendre coefficients usually do display this proportionality.

Appendix: Lacunary Chebyshev Theorem

Definition 1

(m-Lacunary Series) A Chebyshev series of the form

$$\begin{aligned} f(x) = \sum _{n=0}^{\infty } \, a_{n} T_{nm}(x) \end{aligned}$$

(110)

so that each nonzero Chebyshev coefficient is followed by $(m-1)$ zero coefficients is an $m$-lacunary series.

We must append a warning: the term “lacunary”, which is a Latin adjective that simply means “having gaps”, is often applied in mathematics in the narrow sense of a power series or Fourier series whose gaps of zero coefficients increase as a geometric progression or faster so that the sum is a “lacunary function” which cannot be analytically continued beyond a “natural boundary”.

Theorem 7

( $m$ -Lacunary Chebyshev Series) Suppose

$$\begin{aligned} f(x) \equiv g(T_{m}(x)) \end{aligned}$$

(111)

where $m$ is an integer and $g(x)$ is analytic everywhere on $x \in [-1, 1]$ with the Chebyshev series

$$\begin{aligned} g(x) \equiv \sum _{n=0}^{\infty } g_{n} T_{n}(x) \end{aligned}$$

(112)

Then all Chebyshev coefficients of $f(x)$ whose degrees are not multipliers of $m$ are zero. More precisely

$$\begin{aligned} f(x) \equiv \sum _{n=0}^{\infty } g_{n} T_{nm}(x) \end{aligned}$$

(113)

Proof

With the trigonometric change of coordinate, $x=\cos (t), f(\cos (t))= g(\cos (mt))$. With the further change of coordinate $T \equiv m t$, this becomes $g(\cos (T))$ which, because of the analyticity of $g$ has a nice Fourier cosine series in $T$:

$$\begin{aligned} g(\cos (T)) \equiv \sum _{n=0}^{\infty } g_{n} \cos (nT) \end{aligned}$$

(114)

Converting this series back to $t$ gives a Fourier series in which all terms except those whose degrees are multiples of $m$ are missing.

$$\begin{aligned} g(\cos (mt)) \equiv \sum _{n=0}^{\infty } g_{n} \cos (nmT) \end{aligned}$$

(115)

The substitution $T={arccos}(x)$ and recalling that $T_{k}(x)=\cos (k {arccos}(x))$ gives the theorem. $\square$

A simpler version of this theorem was proved in [9]. A special case is the long-known identity $T_{n}(T_{m}(x)) = T_{mn}(x)$.