The Peano Theorem

3.3.3 The Peano Theorem

One can often, by a combination of theoretical and numerical evidence, rely on asymptotic error estimates. Since there are exceptions, it is interesting that there are two general methods for deriving strict error bounds. We call one of them the norms and distance formula. This is not restricted to polynomial approximation, and it is typically easy to use, but it requires some advanced concepts and often overestimates the error. We therefore postpone the presentation of that method to Sec. 4.5.2.

238 Chapter 3. Series, Operators, and Continued Fractions We shall now give another method, due to Peano. 77 Consider a linear functional

for the approximate computation of another linear functional, for example,

Lf =

xf (x) dx.

Suppose that it is exact when it is applied to any polynomial of degree less than k: In other words, ˜Lf = Lf for all f ∈ P k . The remainder is then itself a linear functional, R = L− ˜L,

with the special property that

Rf = 0 if f ∈ P k .

The next theorem gives a representation for such functionals which provides a universal device for deriving error bounds for approximations of the type that we are concerned with. Let f ∈ C n [a, b]. In order to make the discussion less abstract we confine it to functionals of the following form, 0 ≤ m < n,

Rf = φ (x)f (x) dx

+ (m) j, 0 f (x j +b j, 1 f ′ (x j ) +···+b j,m f (x j ) , (3.3.26)

a j =1

where the function φ is integrable, the points x j lie in the bounded real interval [a, b], and

b j,m

(3.3.27) We define the function 78

Rp =0∀p∈P k .

0 t 1 + sign t

+ = max(t, 0),

2 The function t 0 + is often denoted H (t) and is known as the Heaviside unit step function. 79

The function sign is defined as in Definition 3.1.3, i.e., sign x = 0, if x = 0. Note that j t

+ ∈C −1 , (j ≥ 1). The Peano kernel K(u) of the functional R is defined by the equation

1 K(u) =

, x ∈ [a, b], u ∈ (−∞, ∞). (3.3.29) The subscript in R x indicates that R acts on the variable x (not u).

77 Giuseppe Peano (1858–1932) was an Italian mathematician and logician. 78 We use the neutral notation t here for the variable, to avoid tying up the function too closely with the variables

x and u, which play a special role in the following. 79 Oliver Heaviside (1850–1925), English physicist.

3.3. Difference Operators and Operator Expansions 239 The function K(u) vanishes outside [a, b] because

• if u > b, then u > x; hence (x − u) k −1

= 0 and K(u) = 0.

• if u < a, then x > u. It follows that (x − u) k −1 −1

∈P k ; hence K(u) = 0, by (3.3.29) and (3.3.27).

= (x − u)

If φ(x) is a polynomial, then K(u) becomes a piecewise polynomial; the points x j are the joints of the pieces. In this case K ∈ C k −m−2 ; the order of differentiability may be

lower, if φ has singularities. We are now in a position to prove an important theorem.

Theorem 3.3.8 (Peano’s Remainder Theorem). k . Then, 80 Suppose that Rp = 0 for all p ∈ P k for all f ∈ C [a, b],

Rf

= (k) f (u)K(u) du.

The definition and some basic properties of the Peano kernel K(u) were given above. Proof. By Taylor’s formula,

This follows from putting n = k, z = x − a, t = (u − a)/(x − u) into (3.1.5). We rewrite the last term as ∞ a f (k) (u)(x

du . Then apply the functional R = R x to both sides. Since we can allow the interchange of the functional R with the integral, for the class of functionals that we are working with, this yields

− u) k −1

∞ f (k)

(u)(x k − u) −1

(k)

f (u)R x (x − u)

k −1

Rf =0+R

du =

du.

a (k − 1)! The theorem then follows from (3.3.29).

a (k − 1)!

Corollary 3.3.9.

Suppose that Rp = 0 for all p ∈ P k . Then

R x (x

− a) = k!

K(u) du.

− a) k ) holds for some ξ ∈ (a, b) if and only if K(u) does not change its sign.

R x ((x

If K(u) changes its sign, the best possible error bound reads

|Rf | ≤ sup |f (u) |

(k)

|K(u)| du;

u ∈[a,b]

a formula with f (k) (ξ ) is not generally true in this case.

80 The definition of f (k) (u) for u / ∈ [a, b] is arbitrary.

240 Chapter 3. Series, Operators, and Continued Fractions Proof. First suppose that K(u) does not change sign. Then, by (3.3.30) and the mean value

, ξ ∈ [a, b]. For f (x) = (x − a) k this yields (3.3.31). The “if” part of the corollary follows from the combination of these

theorem of integral calculus, Rf = f ∞ (ξ )

(k)

−∞ K(u)du

formulas for Rf and R(x − a) k . If K(u) changes its sign, the “best possible bound” is approached by a sequence of

functions f chosen so that (the continuous functions) f (k) (u) approach (the discontinuous function) sign K(u). The “only if” part follows.

Example 3.3.7.

The remainder of the trapezoidal rule (one step of length h) reads

Rf =

f (x) dx − (f (h) + f (0)).

0 2 We know that Rp = 0 for all p ∈ P 2 . The Peano kernel is zero for u / ∈ [0, h], while for

u ∈ [0, h]

− u) −u(h − u) =

h h (h

2 h(h

K(u) (x dx

− u) + − < 0.

− u)

2 We also compute Rx 2 h x 2 h ·h 2 h 3 h 3 h 3

((h

0 2 − u) + + 0) =

6− 4=− 12 Since the Peano kernel does not change sign, we conclude that

Example 3.3.8 (Peano Kernels for Difference Operators). Let Rf = 4 3 f (a) , and set x i = a + ih, i = 0 : 3. Note that Rp = 0 for p ∈ P 3 .

Then

Rf = f (x 3 ) − 3f (x 2 ) + 3f (x 1 ) − f (x 0 ),

3 − u) + − 3(x 2 − u) + + 3(x 1 − u) + − (x 0 − u) + ; i.e.,

2 2 2 2K(u) = (x 2

 0 if u > x 3 ,     (x 3 − u) 2

if x 2 ≤u≤x 3 , (x 3 − u) 2 − 3(x 2 − u) 2K(u) = 2

if x 1 ≤u≤x 2 ,    

(x 3 − u) 2 − 3(x 2 − u) 2 + 3(x 1 − u) 2 ≡ (u − x 0 ) 2 if x 0 ≤u≤x 1 , (x 3 − u) 2 − 3(x 2 − u) 2 + 3(x 1 − u) 2 − (x 0 − u) 2 ≡ 0 if u < x 0 .

For the simplification of the last two lines we used that 4 3 u (x 0 − u) 2 ≡ 0. Note that K(u) is a piecewise polynomial in P 3 and that K ′′ (u) is discontinuous at u = x i , i = 0 : 3.

3.3. Difference Operators and Operator Expansions 241 It can be shown (numerically or analytically) that K(u) > 0 in the interval (u 0 ,u 3 ) .

This is no surprise because, by (3.3.4), 4 n f (x) =h n f (n) (ξ ) for any integer n, and, by the above corollary, this could not be generally true if K(u) changes its sign. These calculations

can be generalized to 4 k f (a) for an arbitrary integer k. This example will be generalized in Sec. 4.4.2 to divided differences of nonequidistant data.

In general it is rather laborious to determine a Peano kernel. Sometimes one can show that the kernel is a piecewise polynomial, that it has a symmetry, and that it has a simple form in the intervals near the boundaries. All this can simplify the computation, and might have been used in these examples.

It is usually much easier to compute R((x − a) k ) , and an approximate error estimate is often given by

For example, suppose that x ∈ [a, b], where b − a is of the order of magnitude of a step size parameter h, and that f is analytic in [a, b]. By Taylor’s formula,

= p(x) + k − a) +1 +···, f (k) (a) k !

where p ∈ P k ; hence Rp = 0. Most of the common functionals can be applied term by term. Then

R x (x − a) +1 +···.

(k + 1)!

Assume that, for some c, R x (x

−a) k = O(h +c ) for k = 1, 2, 3, . . . . (This is often the case.) Then (3.3.32) becomes an asymptotic error estimate as h → 0. It was mentioned above

that for formulas derived by operator methods, an asymptotic error estimate is directly available anyway, but if a formula is derived by other means (see Chapter 4) this error estimate is important.

Asymptotic error estimates are frequently used in computing, because they are often much easier to derive and apply than strict error bounds. The question is, however, how to know that “the computation is in the asymptotic regime,” where an asymptotic estimate is practically reliable. Much can be said about this central question of applied mathematics. Let us here just mention that a difference scheme displays well the quantitative properties of a function needed to make the judgment.

If Rp = 0 for p ∈ P k , then a fortiori Rp = 0 for p ∈ P k −i , i = 0 : k. We may thus obtain a Peano kernel for each i, which is temporarily denoted by K k −i (u) . They are

obtained by integration by parts,

K k (u)f (k) (u) du =

K k −1 (u)f (k −1) (u) du (3.3.33)

−2 (u)f

(k −2)

(u) du =...,

where K k −i i = (−D) K k , i = 1, 2, . . . , as long as K k −i is integrable . The lower-order kernels are useful, e.g., if the actual function f is not as smooth as the usual remainder

formula requires.

242 Chapter 3. Series, Operators, and Continued Fractions For the trapezoidal rule we obtained in Example 3.3.7

A second integration by parts can only be performed within the framework of Dirac’s delta functions (distributions); K 0 is not integrable. A reader who is familiar with these generalized functions may enjoy the following formula:

Rf = K 0 (u)f (u)du ≡ − δ(u) 2 +1− δ(u

2 − h)

f (u)du.

This is for one step of the trapezoidal rule, but many functionals can be expressed analo- gously.