LAGRANGE INTERPOLATION - DIVERGENCE

G.B. Baker and T.M. Mills

(Received April 1986, revised November 1987)

1. The interpolation problem

Interpolation deals with the problem of fitting curves or surfaces

through data points.

It is important to distinguish the problem from the statistical regres­

sion problem. This statistical problem is illustrated by the classical

problem of finding the line of best fit to data points. Clearly the experi­

mental scientist does not expect the line to pass through the points but

merely that the line is close to the points. The scientist admits that there

is a certain amount of error associated with the data points and hence is it

not proper to expect the curve to pass through the points.

Therefore, the interpolation problem must deal with the case where there

is no error.

Q. Where does one find measurements with no error?

A. In the Mathematics Department.

There is a great deal of truth in this facetious answer. Interpolation

arises naturally in the use of mathematical tables. Lagrange first encoun­

tered the problem in trying to create interpolation methods for astronomical

tables. Actuarial tables provided a number of interpolation problems for

actuaries: it is no coincidence that J. Steffensen, the author of a

classical text ([28]) , on the subject was an actuary. In statistical

quality control, there are still many tables which are very cumbersome to

use on the shop floor and which should be replaced by formulae which could

be found by interpolation methods.

Numerical integration is another area which requires the use of inter-

(bpolation methods. To evaluate !(/) ■= /, the values of f(x) for

'avarious x are given exactly, and used to define a function p(x) which

interpolates / so that I(p) is used to approximate I(,f) .

Math. Chronicle 17(1988), 1 - 18 .

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Newton used interpolation methods for numerical integration in an

interesting manner. To test his interpolation formula, he let

I(t) = f (1 -xrfdx .

He knew the values of 1(0) , J(l) , J(2) , ... from his study of integration.

Using these values he estimated 1(h) by interpolation '>~J ^ u p m e d the

estimate with the exact value which can be c'^ulated by elementary mensuration.

Interpolation is still used extensively in numerical integration methods.

The classic text by P.J. Davis and P. Rabinowitz [7] gives many relevant

details and references.

The finite element method for solving differential equations uses inter­

polation techniques extensively. The following problem arose in the State

Electricity Commission of Victoria. To study the effect of certain vibrations

on an impeller blade which was fixed to a spinning wheel a blade was detached

and set up on a concrete slab. To construct a mathematical model of the blade,

certain measurements were made and the three dimensional co-ordinates of a

large number of points on the surface of the blade were fed into the computer.

An interpolation method was then used to construct a surface which represented

the blade.

The three problems (use of mathematical tables, numerical integration,

and the finite element method) were chosen to illustrate the uses of interpo­

lation in applied mathematics. Perhaps one could superficially assess the

importance of the study of interpolation by listing mathematicians who have

devoted their energies to this area: J. Wallis, I. Newton, J.L. Lagrange,

P.S. Laplace, L. Zuler, A.M. Legendre, C.F. Gauss, F.N. Bessel, A. Cauchy,

Ch. Hermite, P.L. Chebyshev, S.N. Bernstein, G.D. Birkhoff, L. Fejer,

G. Polya, G. Szego, A. Zygmund, P. Turan, P. Erdos, C. de Boor, P.J. Davis

are on the list.

Having discussed the place of interpolation in mathematics we now turn

to giving a m o d e m mathematical setting for this old problem.

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2. A modern setting for an old problem

Let I be the interval [-1,1] and let M be the matrix of points

M =

22

where

l £ x , > x„ > ... > x 2 -1 .In 2 n nn

If fix') is a real-valued function defined on I then there is a unique

polynomial L (/;x) such that n -1

(i) the degree of Ln _ 1 (f;x) does not exceed n - 1

and (ii) L Cf;x. ) = f(x. ) , i = l , 2 , . . . , n .n -1 ^n inJ * , , »

The sequence {LQ(/;x) , L^if-.x) , L2 (/;x) , ...} is the sequence of Lagrange

interpolation polynomials corresponding to the function f(x) and based on

the system of nodes M . Since Ln j(/;x) and / (x) agree for n

distinct values of x , the approximation theorist is interested in the

question, "Does L^ (/;x) converge to /(x) as n tends to infinity?"

Before we answer this question let us become familiar with the formula

for these interpolation polynomials. We may represent L (/;x) in various

ways, but for our purposes, Lagrange's io?- — 1.* will be the most useful.

We w-i I*-

nL if;x) = I /(x.)Z. (x) 71-1 ft. 1 K

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where l^ix) is the polynomial of degree n - 1 such that ** 6^.

j = 1 , 2 , ... , n . This representation shows that Ln j may be

regarded as a linear operator acting on the function f .

(Here, and elsewhere there is no confusion, x . = x . .)J J«

From the uniqueness of the interpolation polynomial we can deduce that,

if p(x) is a polynomial of degree n - 1 or less, then L (p;x) = p(x) .n-1

In particular we see that

I = 1fc.1

by using the polynomial p(x) = 1 . This is a very useful result because

one of the consequences is

n nL if;x) - fix) = I /(xjZ.(x) - /(x) X L(x) n_1 fc-i * K k = \ K

n

= I (/M -/(x))Z.(x) fc-i * *

and we have a nice representation of the error.

An important quantity is the Lebesgue function

X (x) : = [ U.(x)| n fc-1 *

and related to this is the Lebesgue constant

X : = max{X (x) : -1 S x S 1) n '

which is the norm of the operator

Ln l : C{I) — C(J) .

We shall have more to say about X (x) and X later.

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3. Some early examples of divergence

In 1896 Ch. Meray [18] published an interesting paper which studies

the error

H V j t f ) = _A x ) | : -l a * * iJ .

His results were independently published later by C. Runge [26] .

Their fundamental result is as follows.

For convenience, consider I to be the interval [-5,5] .

We choose the nodes in M to be equidistantly spaced in I , and let

/(x) = l/(l+x2) . Then L (/;x) does not converge to f(x) uniformlyn-1

in I . In fact the sequence {\\Ln (/)|| : n = 1 , 2 , 3 , . . . } is

unbounded. We can see from Figures 1 and 2 that there is a point £

such that \L (f;x) -,f(x)| is large for |x| > £ and small for |x| < £ . w-1

Here we have a situation where the nodes are equidistantly spaced, and

the function being interpolated is infinitely differentiable in I , but

the approximation provided by the interpolation polynomials is very poor.

A depressing find for the working calculator, a fascinating twist for the

pure mathematician. Depending on your fancy, there are more depressing/

fascinating matters to come.

It seems that Meray may have suspected this in 1986 when he wrote -

"Mais je viens d' apercevoir une infinite d 1 autres examples

due meme accident".

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4. Later results concerning divergence

In 1914, G. Faber [l1] considered the Lagrange interpolation poly­

nomials for an arbitrary matrix of nodes M . In short his result proves

that for each M one can find a function f such that L■ j (fix') does not

converge to f(x) uniformly on I = [-1,1] as n increases without bound.

His proof utilizes X^ , the norm of the operator L ̂ .

Recall that

n

Ln - " I n * kn k w ^*1

n

■ I l^c*)! k= i

X = max{X (x) : -1 S x § 1} . n n

Faber proceeds as follows.

Lemma. Given arbitrary distinct points Xj , x2 , ... , x^ in I ,

there exists a polynomial P^ with degree n - 1 or lees suah that

|P(x^)| £ 8/tT i = 1 , 2 , ... , n

and for some a f I , P(e) > In n .

Theorem. (G. Faber) X^ < (£hh)/(8/7) .

Proof. Choose P as in the Lemma. Then

nP(x) = I P{x.)l. (x)

k«l * *

and hence

|P(*)| s I |P(x.)| • |l.(x)| fc« l * *

S 8 / 7 X (x) .

6

X (x) B i £ M i" 8 / 7

xn M * J£<£lL >8 / Y

in n

8 / 7

8 / ¥

Corollary. There exists a function f € C([-l,l]) such that

^ 0 as

Proof. This follows immediately from the Theorem and the Uniform Boundedness

Principle.

The Runge-M6ray example showed how bad these interpolation polynomials

may be for a special matrix of nodes but the Faber theorem shows that the

divergence phenomenon occurs with all M . However, in the Runge-Meray

example at least the bad approximation is restricted to the extremities of

the interval: near the middle the approximation is quite good. In 1918

S.N. Bernstein [2] shattered any comfoTt which this last observation may

bring.

Theorem. (S.N. Bernstein) Let f(x) = |x| , x C I and suppose that the

nodes of interpolation are equally spaced in I . Then for 0 < |ac| < 1 »

Ln-i^f;x ̂ ' 0 as n “*■ °° *

The situation described here is worse than previous results suggest.

Bernstein's proof of this result is short but rather complicated. It would

be interesting to develop a proof of this result which is more straight­

forward.

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5. Chebyshev nodes

If we define the matrix of nodes by

x,k,n

cos t fc = l(l)n , n & 12 n

then we call these nodes "Chebyshev nodes" because they are the zeros of the

Chebyshev polynomial

In this case we denote the matrix M by T .

These nodes do not strike the outsider as natural in any way. Let us

explain the importance of these nodes by describing two properties of this

node system.

First, we know that, for any matrix M ,

Now, for the matrix T , we have (according to S.N. Bernstein [3]) ,

So T is nearly optimal: that is, if Lagrange interpolation is to be used

then the Chebyshev nodes are close to best. For this reason, T has

received a considerable amount of attention in the mathematical literature.

Second, if f ( C(I) then we associate with / its Fourier-Chebyshev

expansion:

T (®) - cos(n arc cosx) .

X < 8 + (4/w) In n .Yl

oo

f(x) ~ 1 * 0 Cf) +0= i

o

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Let Sn - = 2 + tjJi aj(f)Tj(x ) • If one replaces the

integrals a.if) by certain Rieraann sums based on the points 6, = (2fe-l)ir/2» tJ ^

k = 1 , 2 , ... , n then Sn (f;x) becomes Ln l(f;x) . So jC/jx)

is approximately equal to Sn (f;x) . Herein lies an important link

between the study of interpolation polynomials and Fourier series. This

link is fully exploited in the text by A. Zygmund [31] .

Having established the reason for studying Chebyshev nodes we now try

to find if they behave better than equidistant nodes. A general folk-theorem

may be stated as follows:

If you can prove a general divergence theorem for the matrix T then

the theorem is probably true for all matrices M .

In 1935, G. Griinwald [13] gave us the first "bad" news concerning T .

Theorem. (G. Grunwald) Let M = T . There exists a function f € C(I)

such that, for almost all x € I ,

Urn sup \L (f;x) | = » . n+°° n ~1

In the next year J. Marcinkiewicz [17] and G. Grunwald [14] improved

this result with the following.

Theorem. ((7. Grunwald, J. Marcinkiewicz) Let M = T . There exists a

function f € C[I) such that, for all x ( I ,

Urn sup | L Af\x) | = » .oo rl~~ L

In 1937 P. Erdos and P. Turan [9] wrote the first of numerous papers

dealing with interpolation. In this paper they state an interesting diver­

gence result dealing with averages of Lagrange interpolation polynomials.

Considering the case M = T , Erdos and Turin pursued the analogy between

Fourier series and interpolating polynomials. Just as L. Fejer considered

°n (f',x) m (S0(f-,x) + ... + Sn l(/;x))/n ,

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Erdos and Turin considered

0n Cf;aO = (ijCf;*) + ... + £,(/;*))/« •

Concerning these means, they proved the following result.

Theorem. (P. Erdos, P. Turari) There ie a function f € C(J) such that

the sequence (0 (/;0) , « » 1 , 2 , ...} is unbounded.

The 1930's were heady days for those interested in divergence of inter­

polation methods. In subsequent decades Erdbs, Turin, Grtlnwald generated

many more deep results. A considerable amount of effort has been devoted to

the study of and ^n C*) in the case when M = T . Asymptotic expan­

sions, estimates of best constants, and studies of monotonicity of various

related functions have been studied by both pure mathematicians and numerical

analysts.

6. Lagrange interpolation and projections

We have referred to the fact that j maybe regarded as a linear

operator. Specifically, let us write

V, ;

where is the space of polynomials of degree n - 1 or less.

From what we have said earlier, we know that if p € II , thenn - 1

i(p) = p : that is I>n j is an example of a projection of C(I) onto

nn-l

There are many other similar projections which arise in numerical

analysis. Suppose we associate with / its Fourier-Chebyshev expansion

f ~ i «„(/> * j

where

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T^(x) : = cos(fc arc cos x) ,

ak (f) : = f l f t W A W - * 2) dt

If we let

• -

fn= i /(cos 6) cos kQ dQ .

Sn (f;x) - j a Q (f) ♦ J a k (f)Tk (x)

be the nth partial sum then it is true that

s„_, : «[-!.!]) - Hb_1

is a projection.

Other examples of projection operators may be obtained by various

orthogonal expansions of f (e.g. in terms of Legendre polynomials).

A natural question is

"Which projection P : C(I) — ► has minimum norm?"

This question has been discussed nicely in an interesting paper by

M. Golomb (1965) [12] and, as far as we know, the problem is still open.

However, Golomb shows that there is a positive constant such that for each

natural number n , and any projection

p : C(I) -*• nn l

we have

||P|| £ K ^n(n-l) - C . v

A corollary of this is the Corollary of Faber's theoi-em.

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The main point to be made here is that the study of Lagrange interpo­

lation has led us into the general study of projections on function spaces.

It would, by the way, be interesting to get more information about the

constant C above.

7. Some recent results on divergence

Let us now survey some of the most important recent results.

A. A. Privalov (USSR) has been generating a number of very technical

papers concerned with Lagrange interpolation polynomials based on the matrix

of Jacobi nodes. The Jacobi polynomials with parameters o , (5 are afa 8)

sequence of polynomials {P^ (x) : n = 0 , 1 , 2 , 3 , . . . } such that

j 1 PJJ°,’B)(x)P/Ja 'B)(x)(l-x)a ( U x ) Bdx - 6 ^

where a > -1 , 0 > -1. The nth row of the matrix M(a,B) of Jacobi

nodes consists of the n distinct zeros of P^°'^(x) . For more detailsn v Jconsult the text by G. Szego [29] .

In 1976, Privalov [22] proved the following result.

Theorem. (A.A. Privalov) Given a > -1 , 0 > -1 , there is a function

f ( C(I) such that

lim sup IL (/;x)I = ® n>«*> n

a.e. in I .

In an interesting survey paper Privalov [23] takes this result even

further.

Theorem. ( A.A. Privalov) Let a > -1 , 6 > -1 and let M(a,B) be the

Jacobi matrix. Then there is a function f ( C(J) such that

(i) the sequence {t^(/;x) : n = 1 , 2 , . . . } diverges at all points

inside (-1,1) ,

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(ii) for any number e , 0 < e < 1 , the Fourier-Jacobi series of f

converges to f uniformly in [-1+e, 1-e] .

A most startling type of result 1 Not only are the classical divergence

results extended but an equiconvergence problem is solved too.

In 1980, P. Erdos and P. Vertesi [10] , proved the following theorem.

Theorem. (P. Erdos, P. Vertesi) For any matrix M one can find a function

f (. C(J) such that

lim sup |L (fix') | = °° n

for almost all x € I .

If we consider the special matrix

then it is clear that we cannot drop the word "almost" for the statement of

this theorem.

To see that the "lim sup" cannot be replaced by "lim" or "lim inf" we

must refer to a convergence theorem of P. Erdos. However, we are endeavour­

ing to avoid mention of convergence theorems in this paper.

This theorem is the climax of a long history dating back to Moray's

example last century. The proof is very complicated indeed.

These divergence results have been extended in another way by I. Muntean

in two recent papers (I. Muntean [19] and S. Cobzas and I. Muntean [5]) .

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A typical result is the following.

Theorem. (I. Mimtean) Given an arbitrary matrix M let

U - {/ € C(I) : Urn sup ||L f \\ = «} .™ M -+-00 «

Then Uy is an uncountable, G^ , dense subset of C(_I~) .

The examples of Meray, Runge, Griinwald, Marrii.’uicwicz were not isolated

examples of "bad" functions. "Rs*'1" functions are everywhere!

Finally, in reviewing recent advances we mention that a long standing

conjecture of S.N. Bernstein was solved recently. Bernstein's conjecture

deals with the problem of finding n distinct points of [-1,1] which

minimize

Bernstein conjectured that X would be minimised if the local maxima

of the graph of X^(x) were all equal. This has been settled by T. Kilgore

[16] . For an interesting discussion of this very deep problem see the

paper by Carl de Boor [4] , or the recent report of Myron S. Henry (American

Mathematical Monthly 91 (1984), 497-499) .

8. Comments on the literature

We close by giving a brief description of some of the most readable

papers and books on the subject.

D. Elliott [8] recently published a paper whose content is similar to

the present paper. The paper is written in a very easy style and would be an

interesting starting point, for someone seeking general information about the

I. Natanson [20] has written a three volume treatise dealing with

approximation theory and the third volume is devoted exclusively to interpo­

lation. The books provide an excellent introduction to the classical analysis

of approximation theory: they would be ideal for final year undergraduate students.

X^ = max U^(a)| : -1 5 i S 1

= max{X (*) : -1 S * S 1} . n

field.

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T.J. Rivlin [25] has produced an excellent text book dealing exclu­

sively with the Chebyshev polynomials. Interpolation polynomials have an

important place in this text which contains a large number of problems.

Rivlin1s book would be another useful text for final year undergraduates.

P. Turan [30] published a long paper which deals with unsolved problems

in approximation theory before he died. Since then a number of them have

been solved and published in Acta Math. Acad. Sci. Hungar. Still it provides

a fair description of the state of the art of interpolation.

P.J. Davis [6] has written a classic text which deals with interpolation

and approximation.

P. Kergin [15] has done some recent work on polynomial interpolation

of functions of serval variables - a problem which has received little

attention.

R. Askey [l] , P. Nevai [21] have written some interesting papers

dealing with mean convergence of Lagrange interpolation polynomials.

P. Printer [24] has an interesting paper dealing with interpolation in

more abstract spaces. Here she has attempted to generalize Lagrange's

interpolation formula as well as merely generalizing the concept of

interpolation.

Smirnov and Lebedev [27] have written a classic text on interpolation

of functions of a complex variable.

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REFERENCES

1. R. Askey, Mean convergence of orthogonal series and Lagrange interpo­

lation, Acta. Math. Acad. Sci. Hungar. 23 (1972), 71-85 .

2. S.N. Bernstein, Quelques remarques sur I' interpolation, Math. Ann. 79

(1918) , 1-12 .

3. S.N. Bernstein, Sur la limitation des valeurs d' interpolation, Bull.

Acad. Sci. de l'URSS, 8 (1931), 1025-1050 .

4. C. de Boor, Polynomial interpolation, Proc. Int. Cong. Math. Helsinki

(1981), 917-922 .

5. S. Cobzas and I. Muntean, Condensation of singularities and divergence

results in approximation theory, J. Approx. Theory 31 (1981) 138-153 .

6. P.J. Davis, Interpolation and Approximation, Dover Publications, N.Y.

(1975) .

7. P.J. Davis and P. Rabinowitz, Numerical Integration, Academic Press,

N.Y. (1975) .

8. D. Elliott, Lagrange interpolation - decline and fall? Int. J. Math.

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142-155 .

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interpolatory polynomials for arbitrary system of nodes, Acta. Math.

Acad. Sci. Hungar. 36 (1980), 71-89 . Corrections in Acta. Math. Acad.

Sci. Hunger. 38 (1981), 263 .

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lationspolynome stetiger Funktionen, Annales of Math. 37 (1936), 908-918 .

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15. P. Kergin, A natural interpolation of u functions, J. Approx. Th.

29 (1980), 278-293 .

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Act. Sci. Math. Szeged. 8 (1937), 131-135 .

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Ungar Pub. Co., N.Y. (1965) .

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constructed over roots of the Jacobi polynomials on sets of positive

measure of Lebesgue, Sibirsk. Math. 17 (1976), 837-859 . (Russian).

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"Fourier Analysis and Approximation Theory (Budapest)", ed. G. Alexits,

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25. T.J. Rivlin, The Chebyshev Polynomials, Wiley, N.Y. (1974) .

26. C. Runge, Uber die Darstellung willkurlicher Funktionen und die

Interpolation zuischen aquidistanten Ordinaten, Z. Angew. Math. Phys.

46 (1901), 224-243 .

27. V.I. Smirnov and N.A. Lebedev, Functions of a Complex Variable:

Constructive Theory, M.I.T. Press, Cambridge (1968) .

28. J. Steffensen, Interpolation, Chelsea Pub. Co., N.Y. (1950) .

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Providence RI (1939) .

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30. P. Turan, On some open problems of approximation theory, J. Approx.

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31. A. Zygmund, Trigonometric Series, Vols. I , II Cambridge U.P., London

(1968) .

Bendigo College of Advanced Education, P.O. Box 199,Bendigo Vic.,AUSTRALIA 3550 .

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