Polynomial Approximations
to Finitely
Oscillating Functions
By William J. Kammerer
1. Introduction.
Chandler
Davis [1] established
the following theorem:
//
v<t, Vi, • • • , vn are real numbers such that vo > i\ , t'i < vt, Vt > vs, ■■■ , then there
exists a unique polynomial P of degree n and a set of points >jo, yi, • • • , y* such that
(1)
P(Vi) =Vi
(2)
P'iyd = 0
i = 0, 1, ••• ,n
i = 1, 2, • • • , n - 1
0 = 'Jo < 2/i < • • • < 2/n = 1.
The main result of this paper is an algorithm for the calculation of this polynomial,
which is first motivated by an independent proof for the existence of P.
A function / is said to be finitely oscillating if it has at most a finite number of
relative extrema. The following mode of approximating a continuous finitely oscillating function / in the uniform norm so that the oscillations are preserved, is discussed : first obtain a polynomial P of minimal degree which has the same variation
as/ and then obtain an increasing polynomial Q such that P(Q) agrees with / at
all its relative extrema. Two theorems are given in the last section, to show that
this method of approximation is always possible.
2. Proof of Theorem. Let D and Dv denote the set of all n + 1 tuples
X = (xo, Xi, • • • , xn) such that 0•*» x» < X\ < • • • < x„ = 1 and 0 = x0 < x( <
•••<*»
S 1 respectively. Let (P denote the class of all polynomials p of degree n,
for which there exists an element X e D, such that p(x¡) = «¿ (i = 1, 2, • • • , n).
A polynomial p e (P can be written in the following form:
p(x)
= [Xo] +
[Xo , Xi](x — Xo)
(3)
+
• • • + [Xo, Xi , • • • , X„J(X — Xo) • • • (x — X„_i)
where the bracket function is defined by
[x¿l = Vi
'*'
r,
-,
_
1
[X,+1 , • • • , Xi+k] — [x¡,
[Xi , Xj+i , • • • , Xi+ki —Xi+k — X,
• • • , Xi+t_t]
.
Lemma 2.1 If X e D0 and n ^ 1 then [X] > 0 for n even and [X] < 0 for n odd.
Proof. The proof proceeds by induction. The lemma can be shown to hold for
n = 1 by direct computation. Assuming it true for n = k — 1, one has [x0, xi , • • • ,
xt_i] < 0 ( >0) and [xi, x2, • • • , xk] > 0 ( <0) for k even (odd). The lemma is
therefore true for n = A;by (4).
Received March 11, 1960. This wotk was supported -in part by the Office of Ordnance Re-
search, U. S. Army.
115
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116
WILLIAM J. KÄMMERER
Lemma 2.2. a.) If n is even (odd), the function [X] assumes us minimum (maximum) in D.
b) Let | [X]\ attain its minimum value for X e D, at Y = (y0, j/i,
■• • » 2/n). Then the polynomial p(yk) = Vk(k = 0, 1, • • • , n) is the desired polynomial P.
Proof. By lemma 2.1, [X] is of constant sign in D. The function | [X] | approaches
infinity as X approaches the boundary of D, and is continuous in the compact set
xo + e á Xi, Xi + t 2£ xt, • • • , xn-i + e á x« for arbitrarily
small e > 0.
The following notation is introduced, to simplify the proof of part 6:
a(p, i) = min {x:x e I(p, i)}
i = 0,1, ••• ,n
ß(p> i) — max [x'.x e I(p, i)}
where
r,
x
Í [ftà
I(p,t) « \x:\
- Vi"* Vi > "***]
and
{ LpW á Viif Vi< pmJ
i = 1, % •'•• ,n - 1
I(p, 0) = 0
I(p,n)
Xi_i < x < x»i\
= \
and
]
p e (P.
Let K be a vector in fi at which | [X] | attains its minimum and let p be the polynomial in <Pwhich satisfies p(y,) = vt. If p does not satisfy (2), then there exists
an integer k, such that a(p, k) ?¿ ß(p, k). Let Xt = %{a(p, k) + ß(p, k)\ and consider the following two polynomials
p(x)
= A(x) + [Y}yk(x, Y)
pt(x) = h(x) + lî/u, • • • Vk-i, Jfe» ifc+i, • ♦• , y»]ff*(x,F)
where A(x) is a polynomial of degree n — 1, and
(5)
gk(x, Y)(x - yk) - ft (x - *)•
By construction one has |pi(x*)| < |p(x*)|. Investigation of the possible cases
contradicts the hypothesis that F is a relative extrema of [X].
It should be observed that the above theorem and proof are valid for polynomials of the form £,!La atf(x)k where / is a strictly increasing differentiable
function on [0, 1J. The proof is identical except for notation.
3. An Iterative Procedure.
Step 1. Choose an arbitrary element Xi = (xo1, xi , • • • , x»1) in.fi.
Step 2. With one of the standard interpolating formulas construct the polynomial
p\ e <?,such that pi(xt') — vk , k = 0, 1, • • • , n.
Step 3. Determine the vector X2 = (x02, x? , ■• ■ , x»2} in D such that pi'(xi2) = 0
for i = 1, 2, • •• , n — 1.
We now have a new element X2 e D. To obtain pi, repeat this process beginning
with step 2, using Xt in place of Xi and making the obvious change in subscripts.
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POLYNOMIAL APPROXIMATIONS TO FINITELY OSCILLATING FUNCTIONS
117
Continuing this procedure, we obtain a recursive process for obtaining a sequence
ÍP.}Theorem 3.1. The sequence [px] converges uniformly to P.
Proof. The proof will proceed by a series of lemmas.
Lemma 3.2. If p £ (P, then
[xo, • • • , xk-i, a(p, k), x*+i, • • • , x„] = [xo, • • • , x*_i, ß(p, k), xk+i, • ■■ , x„J
k = 1, 2, • • • , n - 1.
Proof. By (3), p can be written in the following two forms
(G)
p(x)
= h(x) + [xu, • • • , xt_i, a(p, k), Xk+i, ■•■ ,x„\ gk(x, X)
(7)
p(x)
= h(x) + [xo, • • • , x*_i, ß(p, k), Xk+i, •■■ , xn] gk(x, X)
where A(x) is a polynomial of degree n — 1 and gk(x, X) is defined as in (5). To
obtain the desired result, subtract (6) from (7).
Lemma 3.3. Let p be an element of (P and let N(p) = maxi | m¿ — i\ \ where
m, = max | p(x) \for xe [a(p, i), ß(p, i)] and i = 1, 2, • • • , n — 1. Then the
following inequalities hold for the sequence | p,} :
a) // N(Pi) * 0 then | [Xi+1]| < | [AM|
b) | pi(x) - pi+l (x) I ^ I [Xi] - [X<+tjIfor everyx c 10,1|
c) N(p,) g rnaxoâIai | p,(x) - p<+i (x) | .
Proof. Let ¿ be any positive integer and let {A*|, k = 1, 2, • • • , n — 1 be the
polynomials of degree «. such that
hk(xm'+L) = i/„
hk(ßm) = vm
for in = 0, I, • ■• , k
for m = k +!,-••
,«
where /3m = /3(p<, m). Let Z* denote the vector (x0'+I, • • • , xk,+\ ßk+i , ■•• ,$»)■
Then I(hk, m) C 7(/i*+i, m) for m = A;+ 2, • • • , n, for otherwise fe»(x) — hk+\(x)
would be identically equal to zero, instead of having n — 1 simple roots in [0, 1).
As in the proof of lemma 2.2, we have | [Z„]\ è I [Zil [2 • • • Ä I [^»] I with at
least one of the inequalities being strict, since by assumption, N(p¡) t¿ 0. This
proves part a.
For k = 0, 1, • • • , n - 1 one has | A* (x) - Ai+1 (x) | = | [Zk] - [Zk+1\\ ■
| gb (x, Zk) | á | [Zk] — [Zk+i] | . Now apply the triangle law to obtain b.
The proof of part c follows from the existence of an integer k, such that N(pi) =
| Pi(Xk'+l)
— p,+i
(x*'+1)
| .
To complete the proof of theorem 3.1, observe that lemmas 2.1 and 3.3.a imply
that {[Xi]} is a Cauchy sequence. Lemma 3.3.b implies |p,| also forms a Cauchy
sequence in the uniform norm on {0, 1] and therefore convergent to a polynomial
P of degree n. Part c of lemma 3.3 implies that P satisfies conditions 1 and 2.
This iteration can be carried out by the use of standard subroutines available in
most computer libraries, and from all empirical evidence the convergence seems
quite rapid. To illustrate, we shall calculate the third-degree Chebyshev poly-
nomial on [0, 1] by the use of this method, starting with Xt = (0, .25, .5, 1).
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118
WILLIAM J. KAMMERER
Pi(x) = 1 - 22x + 68x2 - 48x3
X2 = (0, .20723911,.73720533,1)
p2(x) = 1 - 18.400541x + 48.970905x2- 32.570364x3
Xt = (0, .24979357,.75256767,1)
p3(x) = 1 - 18.000432x 4- 48.002138x2- 32.001707x3
Xi = (0, .24999339,..74999783,1)
p4(x) = 1 - 17.999999x + 47.999999x2 - 31.999999x3
4. Approximation by Composition.
Theorem 4.1. Let X = (x0, xi, ■• • , x»), F = (î/u, 2/i, • • • , y») be any two elements in D. Then there exists a polynomial Q such that Q'(x) è 0 on [0, 1] and
Q(xi) = y¡, i = 0, 1, ••• , n.
Proof. Define the elements Z\ and Z2 in D as follows:
Zl = Ou\ *i\ • • • , z,,1) = (yo, l(Vi - 2/i) + 2/1, HZA - 2/2) 4- 2/2, • • • , Un)
Z2 = (z0\ z{, ■■■, z,;) = (2/0, K2/1 - 2/0) - Vi, K2/2 - 2/1) - 2/2, ■• • , V»).
Let /, for i — 1,2,3, • • • , 2"_1 be distinct piece wise linear functions, which are linear
on the intervals [j/.-i, y,\ i » 1, 2, • • • , n and such that/,(xi)
= z^1or zk, k = 0,
1, • • • , n. Define e = min¿mnu \ z¡ — ijí\íot i = 1, 2, • • • , n — 1, k «■=1, 2.
Using the fact that the Bernstein polynomials of a continuous increasing function
are increasing and uniformly convergent, there exist increasing polynomials Q,-(x)
on [0, 1] such that |/,(x) - Q,-(x) | < §« for i = 1, 2, • • • , 2"~' (see Lorentz [7]
p. 20-23). The vector Y is contained in the convex hull of the vectors (Qi(x0),
Q<(xi), • • • , (?,(x„)), i = 1, 2, ■• • , 2n~', and therefore there exists a convex linear
combination of the Qi's which will give rise to a desired polynomial Q.
Theorem 4.2. Let f be a continuous finitely oscillating function on [0, 1] and let
e > 0 be given. Then there exist polynomials P(y) and Q(x), such that
a) f(x) and P(y) are equal at their corresponding relative extrema. At the relative
extrema off,
P(Q) = /
and ^^
= 0.
b) The polynomial Q is increasing and \f(x) — P(Q(x))
\ < ton [0, 1].
Proof. Let the partition X = (x0, Xi, • • • , x„) be the points of the relative
extrema of /on [0, 1]. By a previous theorem, there exists a polynomial P(y) and a
partition F = (y0, yt, • ■• , y„) such that P(yi) = f(x{), i = 0, 1, • • • , n and
P'(yi) = 0, i = 1, 2, • • • , n — 1. Let X' be a refinement of the partition X which
satisfies the following condition x, = xa < xl2 < • • • < x.jv,- = xj+i such that
Ni\f(xa)
- f(xii+i) I = ¡fixt) - /(x1+i) I £ AT,«fori - 0, 1, ••• , n -
.?' = 1, 2, • • • , Ni. Define F' to be the refinement
of the partition
1,
F such that
Va £ [Vi, 2/i+i]and f(xa) = P(ya) for i = 0, 1, ■• • , n - 1, j = 1, 2, • • • , Ni. By
theorem 4.1 there exists an increasing polynomial Q(x) on [0, 1], such that Q(x,j) =
¡fit, and therefore by construction | p(Q(x))
— f(x) | < e on [0, 1].
As a concluding remark, let C represent the class of all composite polynomials of
the, form P(Q), where both P and Q are of degree greater than unity and Q' è 0
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POLYNOMIAL APPROXIMATIONS TO FINITELY OSCILLATING FUNCTIONS
119
on [0, 1]. Not every polynomial of degree four or larger can be so written (see
H. Levi {6]). However, since every continuous function on [0, 1] can be uniformly
approximated by a polynomial, (i.e., a finitely oscillating function) one finds that
the completion of C in the uniform norm on [0, 1] is the set of ail continuous func-
tions on [0, 1].
The question of the existence of variation preserving approximations
arose from
the investigation of syntonic functions by Professor P. C. Hammer in [2] and [3].
I would also like to thank the referee for pointing out that the ideas in references
[4J, [5] and [8] bear a relation to the problem treated here.
Georgia Institute
of Technology
Atlanta, Georgia
1. Chandler
Davis, "Extrema
679-680.
2. P. C. Hammer,
"Syntonicity
of a polynomial,"
of functions
Amer. Math. Monthly, v. 04, 1957, p.
and the variation
functional,"
Rend. Cire.
variation,"
Rend. Cire.
Mat. Palermo, Serie II, Tomo VIII, Anno 1959,p. 145-151.
3. P. C. Hammer
& J. C. Hoi.laday,
"Functions
of restricted
Mat. Palermo, Serie II, Tomo VIII, Anno 1959,p. 102-114.
4. R. S. Johnson,
"On monosplines
of least deviation,"
Trans. Amer. Math. »Soc, v. 90,
1960, p. 458-477.
5. A. J. Kempner,
"On the shape of polynomial
curves,"
Tôhoku Math. J., Part I, v. 37,
1933, p. 347-352; Part II, v. 42, 1936, p. 318-330.
6. H. Levi, "Composite
polynomials
with coefficients in an arbitrary
field of characteristic
zero," Amer. J. Math., v. 64, 1942, p. 389-400.
7. G. G. Lorentz,
Bernstein Pohjnomials, University of Toronto Press, 1953.
8. G. R. MacLane,
"Concerning the uniformization
of certain Riemann surfaces allied
lo the inverse-cosine and inverse-gamma surfaces," Trans. Amer. Math. Soc, v. 62, 1947, p.
99-113.
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