Constr. Approx. (2006) OF1–OF5 DOI: 10.1007/s00365-005-0615-8
CONSTRUCTIVE APPROXIMATION © 2006 Springer Science+Business Media, Inc.
Addendum to “On the L 1 -Condition Number of the Univariate Bernstein Basis” Tom Lyche and Karl Scherer Abstract. The paper mentioned above is a contribution of the authors to Volume 18 (2002) of this journal, see [1]. Recently, J. Domsta pointed out to us that the proof of Theorem 4.2 in that paper contains an error. The purpose of this addendum is to present a correct argument for it.
1. Introduction The 1-norm condition number of the Bernstein basis of degree n can be defined by n
(1.1)
κn,1 := sup
(c j )=0
n j=0 c j B j 1 n sup (c j )=0 j=0 |c j |
n n
j=0
|c j |
, n j=0 c j B j 1
where (1.2)
Bjn (x) :=
n 1 + x j 1 − x n− j , j 2 2
j = 0, . . . , n,
is the Bernstein basis for polynomials of degree n relative to the interval [−1, 1] and 1 f 1 := −1 | f (x)| d x. In Section 4 of [1] we tried to show that an extremal solution f of the problem sup
(1.3)
g∈n g=0
Cn (g)1 , g1
where Cn (g)1 :=
n
|c j |,
j=1
g=
n
c j Bjn ,
j=0
Date received: April 16, 2004. Date revised: June 24, 2005. Date accepted: October 6, 2005. Communicated by Edward B. Saff. Online publication: April 21, 2006. AMS classification: 41A10. Key words and phrases: Condition numbers, polynomials, Bernstein basis, mixed norms. OF1
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or, equivalently, of the inf problem In ( f ) = inf In (g),
(1.4)
g∈n g=0
must belong to the class
nA :=
(1.5)
n
where
g1 , Cn (g)1
In (g) =
d j Bjn : d j−1 d j < 0, j = 1, . . . , n
j=0
of polynomials of degree ≤ n with alternating coefficients. In this respect we stated the following slightly stronger result (Theorem 4.2 of [1]). Theorem 1.1.
An extremal solution of (1.3) has n distinct roots in (−1, 1).
For its proof the following lemma (Lemma 4.1 in [1]) was used. Lemma 1.2. Suppose f = nj=0 d j Bjn has precisely m ≤ n sign changes in (−1, 1). Then there are integers i 0 , . . . , i m with 0 ≤ i 0 < · · · < i m ≤ n such that dik−1 dik < 0 d j di0 ≥ 0
(1.6)
for
k = 1, . . . , m,
for
j = 0, . . . , i 0 .
Moreover, for any integers j0 , . . . , jm with 0 ≤ j0 < · · · < jm ≤ n there is a unique polynomial g = nj=0 c j B jn with the following properties: (i)
g=
m
c jk B jnk
so that
cj = 0
for
j∈ / { j0 , . . . , jm },
k=0
(1.7)
(ii) f (x)g(x) ≥ 0 for x ∈ [−1, 1], (iii) dik c jk > 0 for k = 0, . . . , m, n (iv) |ck | = 1. k=0
We denote this polynomial by g = R( f ; i 0 , . . . , i m ; j0 , . . . , jm ). 2. Corrected Proofs In Lemma 1.2 it is tacitly assumed that m ≥ 1 which was overlooked in [1]. Therefore, we have still to settle this before we can apply it. This is the content of Lemma 2.1. The extremal function for the second supremum in (1.1) must have at least one sign change. Proof.
Recall that
Bjn (x) ≥ 0,
x ∈ [−1, 1],
n j=0
Bjn (x) = 1,
x ∈ R,
1
and −1
B jn (x) d x =
2 . n+1
Addendum to “On the L 1 -Condition Number of the Univariate Bernstein Basis”
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Suppose f = nj=0 d j B jn is an extremal for the second sup in (1.1) or, equivalently, an extremal for the inf problem (1.4). We first show that the d j ’s must change sign and use this to show that f also must have a sign change in (−1, 1). Suppose all the d j ’s have the same sign. Then 1 n 1 n n n 2 j=0 d j B j (x)| d x j=0 |d j | −1 B j (x) d x −1 | n n In ( f ) = = = . n+1 j=0 |d j | j=0 |d j | But this value cannot be the infimum since for the degree n Chebyshev polynomial Un of the second kind we have shown in Theorem 2.1 in [1] that 21−n n + 2 In (Un ) ≤ . n+1 π We conclude that the d j ’s must change sign at least once. Suppose next that the extremal f does not change sign in (−1, 1) and is nonnegative. For δ > 0 we define Z δ := {x ∈ (−1, 1) : f (x) ≤ δ} and let Z δc := [−1, 1]\Z δ be the complement of Z δ . Pick j0 such that d j0 < 0 and let f δ := f − δ B jn0 . We will show that In ( f δ ) < In ( f ) for δ > 0 sufficiently small, a contradiction to the extremality of f . Now since f (x) > δ on Z δc and 0 ≤ B j0 (x) ≤ 1 on [−1, 1] we find f δ 1 = | f (x) − δ B jn0 (x)| d x + | f (x) − δ B jn0 (x)| d x ≤ =
Z δc
Z δc 1
−1
Zδ
| f (x)| d x − δ
Z δc
| f (x)| d x − δ
≤ f 1 − 2δ
B jn0 (x) d x +
1 −1
| f (x)| d x + δ Zδ
Zδ
Bjn0 (x) d x + 2δ
1 − µ(Z δ ) , n+1
Zδ
Bjn0 (x) d x
B jn0 (x) d x
where µ(Z δ ) is the measure of the set Z δ . Since the polynomial f vanishes only at a finite number of isolated points we can choose δ > 0 so small that µ(Z δ ) < 1/(n + 1), and we have shown that f δ 1 < f 1 . Corrected Proof of Theorem 1.1. f =
n j=0
Suppose there is an extremal polynomial
d j Bjn ,
with
n
|d j | = 1,
j=0
which has m < n sign changes in (−1, 1) and let the integers i 0 , . . . , i m be given by (1.6). By Lemma 2.1 we have m ≥ 1. For the proof we first construct a polynomial g with only
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m + 1 nonzero coefficients and such that In (g) = In ( f ). This g is then used to define a polynomial h such that In (h) < In ( f ) and we have a contradiction. The construction of h from g in [1] appears to be correct so we only consider the construction of g. We define g = R( f ; i 0 , . . . , i m ; i 0 , . . . , i m ), i.e., we choose j0 , . . . , jm = i 0 , . . . , i m in Lemma 1.2. Then we consider for ε small enough 1 In ( f − εg) =
| f (x) − εg(x)| d x n , k=0 |dk − εck |
−1
where (ck ) are the Bernstein basis coefficients of g. In [1] we used the relation (2.1)
| f (x) − εg(x)| = | f (x)| − ε|g(x)|
for
x ∈ [−1, 1],
in order to show 1 (2.2)
In ( f ) ≤ In ( f − εg) =
−1
1 | f (x)| d x − ε −1 |g(x)| d x In ( f ) − ε In (g) n n = . 1−ε k=0 |dk | − ε k=0 |ck |
However, as pointed out by J. Domsta, relation (2.1) is not necessarily true for all x ∈ [−1, 1] so that we have to use a slightly more involved argument. Similar to the proof of Lemma 2.1 we consider for δ > 0 the set Z δ := {x ∈ (−1, 1) : | f (x)| ≤ δ}. Let L be an upper bound for |g(x)| on [−1, 1], set ε := δ/L, and f ε := f − εg. Then since f and g have the same sign on [−1, 1] by Lemma 1.2, f ε 1 = | f (x) − εg(x)| d x + | f (x) − εg(x)| d x ≤ =
Z δc
Z δc 1
−1
Zδ
| f (x)| d x − ε
Z δc
| f (x)| d x − ε
|g(x)| d x +
| f (x)| d x + ε Zδ
1 −1
|g(x)| d x Zδ
|g(x)| d x + 2ε
|g(x)| d x Zδ
≤ In ( f ) − ε In (g) + 2εLµ(Z δ ). Similarly, by (iii) in (1.7) and (1.6) we also have n
|dk − εck | =
k=0
n
|dk | − ε
k=0
n
|ck | = 1 − ε,
k=0
so that for δ = Lε positive and sufficiently small In ( f ) − ε In (g) + 2εLµ(Z δ ) f ε 1 ≤ |d − εc | 1−ε k k=0 k
I n ( f ) ≤ I n ( f ε ) = n
Addendum to “On the L 1 -Condition Number of the Univariate Bernstein Basis”
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which is an inequality similar to (2.2). Rearranging this inequality we see that In (g) ≤ In ( f ) + 2Lµ(Z δ ). But since µ(Z δ ) → 0 as δ → 0 this implies that In (g) = In ( f ). References 1.
T. LYCHE, K. SCHERER (2002): On the L 1 -condition number of the univariate Bernstein basis. Constr. Approx., 18:503–528.
T. Lyche CMA and Institute of Informatics P.O. Box 1053, Blindern 0316 Oslo Norway
[email protected]
K. Scherer Universit¨at Bonn Institut f¨ur Angewandte Mathematik Wegelstr. 6 53115 Bonn Germany
[email protected]
