Chebyshev Expansions

3.2.3 Chebyshev Expansions

The Chebyshev 62 polynomials of the first kind are defined by

(3.2.21) that is, T n (z) = cos(nφ), where z = cos φ. From the well-known trigonometric formula

T n (z) = cos(n arccos z), n ≥ 0,

cos(n + 1)φ + cos(n − 1)φ = 2 cos φ cos nφ

61 This is a rigorous upper bound of the error for this value of r, in spite of simplifications in the formulation of the minimization.

62 Pafnuty Lvovich Chebyshev (1821–1894), Russian mathematician, pioneer in approximation theory and the constructive theory of functions. His name has many different transcriptions, for example, Tschebyscheff. This

may explain why the polynomials that bear his name are denoted T n (x) . He also made important contributions to probability theory and number theory.

3.2. More about Series 199 follows, by induction, the important recurrence relation: T 0 (z) = 1, T 1 (z) = z,

(3.2.22) Using this recurrence relation we obtain T (z)

T n +1 (z) = 2zT n (z) −T n −1 (z), (n ≥ 1).

2 = 2z − 1, T 3 (z) = 4z − 3z, T 4 (z) = 8z − 8z + 1, T (z)

= 32z 2 − 48z + 18z − 1, . . . . Clearly T n (z) is the nth degree polynomial,

5 = 16z − 20z + 5z, T 6 (z)

2 1−z + 4 1−z −···. The Chebyshev polynomials of the second kind,

1 sin(nφ)

U n −1 (z) =

T ′ (z) =

= arccos z, (3.2.23) satisfy the same recurrence relation, with the initial conditions U −1 (z) = 0, U 0 (z) = 1; its

n +1

sin φ

degree is n − 1. (When we write just “Chebyshev polynomial,” we refer to the first kind.) The Chebyshev polynomial T n (x) has n zeros in [−1, 1] given by

the Chebyshev points, and n + 1 extrema

(3.2.25) These results follow directly from the fact that cos(nφ) = 0 for φ = (2k + 1)π/(2n), and

x k ′ = cos

= 0 : n.

that cos(nφ) = ±1 for φ = kπ/n. Note that from (3.2.21) it follows that |T n (x) | ≤ 1 for x ∈ [−1, 1], even though its leading coefficient is as large as 2 n −1 .

Example 3.2.3.

Figure 3.2.1 shows a plot of the Chebyshev polynomial T 20 (x) for x ∈ [−1, 1]. Setting z = 1 in the recurrence relation (3.2.22) and using T 0 ( 1) = T 1 ( 1) = 1, it follows that T n ( 1) = 1, n ≥ 0. From T 0 ′ ( 1) = 0, T 1 ′ ( 1) = 1 and differentiating the recurrence relation we get

T n ′ +1 (z) = 2(zT n ′ (z) +T n (z)) −T n ′ −1 (z), (n ≥ 1).

It follows easily by induction that T n ′ (

1) = n 2 , i.e., outside the interval [−1, 1] the Chebyshev polynomials grow rapidly .

200 Chapter 3. Series, Operators, and Continued Fractions

Figure 3.2.1. Graph of the Chebyshev polynomial T 20 (x), x ∈ [−1, 1]. The Chebyshev polynomials have a unique minimax property. (For a use of this

property, see Example 3.2.4.) Lemma 3.2.4 (Minimax Property).

The Chebyshev polynomials have the following minimax property: Of all nth degree polynomials with leading coefficient

1, the polynomial 2 1−n T n (x) has the smallest magnitude

2 1−n in [−1, 1]. Proof. Suppose there were a polynomial p n (x) , with leading coefficient 1 such that

|p n (x) |<2 1−n for all x ∈ [−1, 1]. Let x ′ k , k = 0 : n, be the abscissae of the extrema of T n (x) . Then we would have

p n (x ′ )< 2 1 1−n 2 T n (x 1 ′ 2 ), . . . , etc., up to x n ′ . From this it follows that the polynomial

changes sign in each of the n intervals (x k ′ ,x ′ k +1 ) , k = 0 : n − 1. This is impossible, since the polynomial is of degree n − 1. This proves the minimax property.

The Chebyshev expansion of a function f (z),

f (z) =

c j T j (z),

j =0

is an important aid in studying functions on the interval [−1, 1]. If one is working with a function f (t), t ∈ [a, b], then one should make the substitution

(3.2.27) which maps the interval [−1, 1] onto [a, b].

2 (a

+ b) + 1

2 (b − a)x,

3.2. More about Series 201 Consider the approximation to the function f (x) = x n on [−1, 1] by a polynomial

of lower degree. From the minimax property of Chebyshev polynomials it follows that the maximum magnitude of the error is minimized by the polynomial

(3.2.28) From the symmetry property T n (

p(x)

=x n −2 1−n T n (x).

−x) = (−1) n T n (x) , it follows that this polynomial has in fact degree n − 2. The error 2 1−n T n (x) assumes its extrema 2 1−n in a sequence of n + 1

points, x i = cos(iπ/n). The sign of the error alternates at these points. Suppose that one has obtained, for example, by Taylor series, a truncated power series approximation to a function f (x). By repeated use of (3.2.28), the series can be replaced by a polynomial of lower degree with a moderately increased bound for the truncation error. This process, called economization of power series often yields a useful polynomial approximation to f (x) with a considerably smaller number of terms than the original power series.

Example 3.2.4.

2 If the series expansion cos x = 1 − x 4 / 2+x /

24 − · · · is truncated after the x 4 -term,

the maximum error is 0.0014 in [−1, 1]. Since T 4 (x)

4 = 8x 2 − 8x + 1, it holds that

24 ≈ x 2 / 24 − 1/192

with an error which does not exceed 1/192 = 0.0052. Thus the approximation

2 cos x = (1 − 1/192) − x 2 ( 1/2 − 1/24) = 0.99479 − 0.45833x has an error whose magnitude does not exceed 0.0052 + 0.0014 < 0.007. This is less than

one-sixth of the error 0.042, which is obtained if the power series is truncated after the x 2 -term. Note that for the economized approximation cos(0) is not approximated by 1. It may not be acceptable that such an exact relation is lost. In this example one could have asked for a polynomial approximation to (1 − cos x)/x 2 instead.

If a Chebyshev expansion converges rapidly, the truncation error is, by and large, determined by the first few neglected terms. As indicated by Figures 3.2.1 and 3.2.5 (see Problem 3.2.3), the error curve is oscillating with slowly varying amplitude in [−1, 1]. In contrast, the truncation error of a power series is proportional to a power of x. Note that f (z) is allowed to have a singularity arbitrarily close to the interval [−1, 1], and the convergence of the Chebyshev expansion will still be exponential, although the exponential rate deteriorates, as R ↓ 1.

Important properties of trigonometric functions and Fourier series can be reformulated in the terminology of Chebyshev polynomials. For example, they satisfy certain orthog- onality relations; see Example 4.5.10. Also, results like (3.2.7), concerning how the rate of decrease of the coefficients or the truncation error of a Fourier series is related to the smoothness properties of its sum, can be translated to Chebyshev expansions. So, even if

f is not analytic, its Chebyshev expansion converges under amazingly general conditions (unlike a power series), but the convergence is much slower than exponential. A typical

202 Chapter 3. Series, Operators, and Continued Fractions result reads as follows: if f ∈ C k [−1, 1], k > 0, there exists a bound for the truncation

error that decreases uniformly like O(n −k log n). Sometimes convergence acceleration can

be successfully applied to such series.

Set w = e iφ = cos φ + i sin φ, where φ and z = cos φ may be complex. Then

2 −1 (z) z − 1, where U n −1 (z) is the Chebyshev polynomials of the second kind; see (3.2.23). It follows that

n z + z 2 −1 =T n (z) +U n

the Chebyshev expansion (3.2.26) formally corresponds to a symmetric Laurent expansion,

% 1 g(w)

0 if j = 0. It can be shown by the parallelogram law that |z + 1| + |z − 1| = |w| + |w| −1 . Hence, if

2 (w +w −1 ) maps the annulus {w : R −1 < |w| < R}, twice onto an ellipse E R , determined by the relation

1, z = 1

(3.2.30) with foci at 1 and −1. The axes are, respectively, R + R −1 and R − R −1 , and hence R is

E R = {z : |z − 1| + |z + 1| ≤ R + R −1 },

the sum of the semiaxes . Note that as R → 1, the ellipse degenerates into the interval [−1, 1]. As R → ∞, it 1

becomes close to the circle |z| < 2 R . It follows from (3.2.29) that this family of confocal √ ellipses are level curves of |w| = |z ± z 2 − 1|. In fact, we can also write

(3.2.31) Theorem 3.2.5 (Bernštein’s Approximation Theorem).

E R = z : 1 ≤ |z + z 2 − 1| ≤ R .

Let f (z) be real-valued for z ∈ [−1, 1], analytic and single-valued for z ∈ E R ,R> 1. Assume that |f (z)| ≤ M for z ∈ E R . Then 63

for x ∈ [−1, 1].

2 (w +w )) . Then g(w) is analytic in the annulus R −1 + ǫ ≤ |w| ≤ R − ǫ, and hence the Laurent expansion (3.2.1) converges

Set as before z = 1

2 (w +w −1 )

, g(w) = f ( 1

there. In particular it converges for |w| = 1, hence the Chebyshev expansion for f (x) converges when x ∈ [−1, 1].

63 A generalization to complex values of x is formulated in Problem 3.2.11.

3.2. More about Series 203 Set r = R − ǫ. By Cauchy’s formula we obtain, for j > 0,

|c j | = 2|a j |= (

Mr −j−1 rdφ = 2Mr −j . 2πi |w|=r

g(w)w

We then obtain, for x ∈ [−1, 1], (

1 − 1/r This holds for any ǫ > 0. We can here let ǫ → 0 and thus replace r by R. The Chebyshev polynomials are perhaps the most important example of a family of

j =0

orthogonal polynomials ; see Sec. 4.5.5. The numerical value of a truncated Chebyshev expansion can be computed by means of Clenshaw’s algorithm; see Theorem 4.5.21.