Electronic Transactions on Numerical Analysis. ETNA Volume 25, pp. 431-438, 2006. Kent State University Copyright 2006, Kent State University. [email protected]
ISSN 1068-9613.
ON THE EIGENSTRUCTURE OF THE BERNSTEIN KERNEL ¡ URI ITAI
Dedicated to Ed Saff on the occasion of his 60th birthday
Abstract. In approximation theory a common technique is to assume duality behavior between the Bernstein operator and the transformation matrix between the standard basis and the Bernstein basis. In this paper we shall produce an example that this assumption is not always correct. In particular, the eigenstructures of the operator and the transformation Matrix are distinguished.
Key words. eigenstructure, Bernstein operator, Bernstein polynomial basis
AMS subject classifications. 42C15, 42C30
§©¨ 1. Introduction. Let ¢¤£¦¥ , be a family of linear independent polynomials.
Let
§
!
)¤*,+
§©¨¨ ¢&£¦¥ §©¨ ('
£#"%$
+/.10
- where ¨ is a continuous function of . The connection between the transformation matrix from the standard basis to a given basis and the induced operator is widely used. The most significant example is when the ma- trix is totally positive (strictly total positive), i.e., all its minors are positive (strictly positive), then the operator is totally positive (strictly totally positive), totally positive and strictly total positive, and vice versa. Therefore, a duality is often assumed between the matrix and the operator. In many cases this duality is valid and natural. In this paper we shall produce an example that shows that this assumption is not always correct. In particular, the eigenstructures of the operator and the transformation matrix are distinct, in focusing on the Bernstein polynomials.
The Bernstein polynomial is [6]
2
§76 £
£
*
.54 £¦¥ § ' +
¨ ¨ 3 839;:
From these polynomials an approximation operator is reduced
§
!
2
<
)¤*
§ £¦¥ § '
: ¨=¨ 1 3 ¨ £#"%$
This operator is totally positive [3] and has many uses in approximation theory, proba- bility, computer graphics, CAGD etc.
These polynomials are linearly independent. Hence, they form a polynomial basis for
> § (the polynomials of order ). The transformation matrix from the monomials basis to the
Bernstein basis is as follows [7]:
6%K
£ §
ML(N CIH
CJH for
P ¨@?A§B =C ¥ £& EDGF
(1.1) F MO(N for .
Received April 28, 2005. Accepted for publication April 8, 2006. Recommended by D. Lubinsky. ¡ Department of Mathematics, Technion, Haifa, Israel ([email protected]). 431 ETNA Kent State University [email protected]
432 U. ITAI
Furthermore, [7]:
§ £
£¦6BC
6%K
4R.
¨ L(N
H H £
C for
F F
¨¦? C ¥ £¤ QD
(1.2) § O(N for .
REMARK 1.1. In this paper we consider the polynomials as a vector space, where the first coefficient of the vector is the leading coefficient of the polynomial. REMARK 1.2. In this paper all the matrices have real eigenvalues arranged in decreasing order. 2. The Bernstein Operator Eigenstructure. In this section we discuss the eigenstruc- ture of the Bernstein operator. The description will be brief, for more details see [1] or [4]. However, the results here are in contrast of those to the next section. The last corollary is new.
Since
<
§ . .
¨@¨ 3
<
§ +
¨@¨¦ M S S
. ¨ T U
one can conclude that ¨ T and are eigenpolynomials with associated eigen-
§YX K
\
.
V( V §W §]6 ^ _X
value . The remaining eigenvalues are §[Z with associated eigenfunction
<
^ b
^ ^c6©K/d
§ 4 §
a (polynomial) ` lower order terms. Hence, the Bernstein operator is
diagonalizable.
THEOREM 2.1 ([4]). For each fixed and for each function ,
i
egf#h
<
d
. 4 0 +
¨@¤ l- ¨ ¨¦m ¨@Y
(2.1) §
i5jTk
< < <
i
§Tn n §
:o:o: where § .
COROLLARY 2.2.
i
i
<
.
d
49p . 4 0 .t4 * *
¨@¤ - ¨ ¨¦m ¨@Y qr 's' :
§
Proof.
i i
<
d
4p . 4 0 4xwmy +
¨¤ - ¨ ¨¦m ¨@Y Iqu ¨¦v
§ §
w z{|z
where y is the direct sum of the unit matrix of order and a null matrix i.e.,
}~ o
~
.
:o:o:
~
~
.
:o:o:
~
:o:o:
wYy : . .
. .
:o:o:
b
$o¥ § K ¥ § . ¥ § C ¥ §
V V OSV NL9
Since V and , we get:
~ ~
} }
~ ~
:/:o:
~ ~
~ ~
:/:o:
~ ~
i
b
i i
<
b
:/:o: V
¥ §
4xw
V : ¨ y M
¥ § § H
. . F
. .
i
:/:o:V
§]¥ § ETNA Kent State University [email protected]
ON THE EIGENSTRUCTURE OF THE BERNSTEIN KERNEL 433
However,
b
.
;
b
¥ § .54
V :
4
¨¦ z[ 1
Which completes the proof.
d d
§ . .
{
COROLLARY 2.3. Let be the matrix,
~
}
~
.
:o:/:
~
~
... . .
:o:/:
~
~
:o:/:
~ : § . .
. .
:o:/:
:o:/:
Then
i
egf#h
<
:
§ i5jTk
Proof. By Theorem 2.1
Cu NLzB: (2.2) £¦¥ for
Choose a polynomial
§
d d d
$ K §
`&¨¦ M :o:/: :
By Theorem 2.1
i
egf#h
<
d
. 4 0
¨¤ 3 - ¨ ¨@Y ¨@Y 1:
(2.3) § i5jTk
Hence,
b
i
e#fgh
<
d do/cd d
0
¨` - ;K &§ $Y:
§
i5jTk
Write the matrix equation
b b
i
egf#h
<
d d dooJd
$ $ $/¥ $ $ KI¥ $ K ¥ $ § §]¥ $ $Y+
¨ ` ¨R
§
i5jTk
and
b b
i
egf#h
<
d d dooJd d dS/od
` Ks ¨R Kr $ K ¥ K ;K KI¥ K K ¥ K ¤§ K ¥ §8 K ¤§ : ¨
§
i5jTk
Since ¢¨ is an arbitrary polynomial:
$/¥ § KI¥ § §]6%K ¥ § + §]¥ § .
S :
Furthermore,
o/
$o¥ K $/¥ §]6%K K ¥ §]6%K §76%K ¥ §]6%K . $o¥ K
B:
This completes the proof. ETNA Kent State University [email protected]
434 U. ITAI
3. The Bernstein Polynomial Transformation Matrix Eigenvectors. The Bernstein > polynomials are linearly independent. Therefore, they constitute a polynomial basis for § . The transformation matrix from the standard basis to the Bernstein basis eigenpolynomials behaves quite differently from the Bernstein operator. First, unlike the Bernstein operator, almost all the Bernstein transformation matrix eigenvalues have algebraic multiplicity two but the Bernstein operator, have geometrical multiplicity one. Hence, the Bernstein transfor- mation matrix is not diagonalizable. 3.1. The Eigenvalues. In this section we describe the eigenstructure of the transforma- tion matrix between the standard basis and the Bernstein basis.
Since ?§ is a triangular matrix, all of its principal diagonal elements are eigenvalues.
6©K
§
9 H
Hence, £ for each is an eigenvalue. Since,
F
* *
' ' +
(3.1)
4
b
§&K the algebraic multiplicity of each eigenvalue is two, except if is odd, then is a simple eigenvalue. Finding the geometrical multiplicity is identical to finding the dimension
of the solution space of the following equation:
6%K
*
§¡ ¢£ ' ¡£
:
(3.2) ?
¢£ § Let be an eigenvector of ? . Then it is a solution of equation (3.2).
REMARK 3.1. In this chapter we look for the eigenvectors, by (3.1) we can assume that
b¦¥ §¤K
£Q¤ . THEOREM 3.2. Let § be a positive lower triangular matrix that for all natural ,
satisfies the following
£¦¥ C £¦¥ £ §]6 £¦¥ §]6 £ª C¥ C £¦¥ £
§ O¨©N§ § 9 § O¨§ N9 :
z
¥
b
§&K
V £& ¦§,£¦¥ £ £a¤ Let £ be the eigenvector associated with the eigenvalue . Then for every
we have «
«
C
4
N£9
£
§]6 £
5:
£ ¬ N Proof. For N£9 the theorem is trivial since the matrix is triangular. If , we consider
two cases.
d .
Case 1. ® ¦z .
« «
Let us write the eigenequation «
£#&K £#&K
£
d
£ ¥ £#&K « £#&KI¥ £#&K £¦¥ £
§ § § :
£
£ £
£
By the condition on the matrix we get £ which completes the proof in this case.
d
.
¦z ¬
« Case 2. . «
In this case we assume« that the theorem is false. Without loss of generality, we can consider
£
.
£ . For the rest of the proof the following lemma is needed.
C
£ d°
. . 4
£9 £ LEMMA 3.3. If then £#%¯ , . ETNA Kent State University [email protected]
ON THE EIGENSTRUCTURE OF THE BERNSTEIN KERNEL 435
° °
Proof. By induction on . If ± then there is nothing to prove. Therefore, we may
)
° £
assume that the lemma is true for every . Write the eigenequation as:
« «
² ²
¯
! ²
£#%¯
£ ¥ £ + £# £#%¯³¥ £#
§ §
£ "%$
or
« « «
² ²
¯³6©K
!²
£# £g©¯³¥ £g £©4 £g%¯ £#%¯³¥ £#%¯
§ ¨ § :
"%$
« «
By the induction hypothesis the left hand side of the equation is strictly greater than zero and
£¦¥ £%4 £g©¯³¥ £g©¯ £#%¯
O¨ O9
¨@§ implies that which completes the proof of the lemma.
°¡d
4
Proof of Theorem 3.2 (continued). When ¤ we get
« « «
² ²
¯
!²
+
£g §,£#%¯³¥ £# ¦V¡£ §]6 £& §,§76 £¦¥ §]6 £ §]6 £ "%$
or
« «
² ²
¯ 6%K
² !
£¦¥ £%4 §]6 £ ¥ §]6 £ §]6 £ £#%¯³¥ £# £g
l¨@§ § : §
« «
"%$
«
C
£
£ ¥ £&4 §]6 £ ¥ §76 £ .
§ § l O´ £
But . By the assumption and by Lemma 3.3 £ . The left
£ hand side is strictly greater than zero. Hence, a contradiction, £ . Since that and the fact
the the matrix is triangular we get «
C
«
4
N£ I:
£
§76 £
4
µ For the case we consider the theorem to be false i.e., £ since the matrix is
triangular we get «
C
«
4
NO I:
£
§]6 £
¢£
S ¬ Hence, is a null vector which cannot be an eigenvector. Hence, £ which completes the proof.
COROLLARY 3.4. Assume that the condition Theorem 3.2 holds. Then each eigenvalue § of the has geometrical multiplicity one. «
Proof. We consider the theorem to be false, i.e., the geometrical multiplicity is greater
§]6 £ a than one. However, Theorem 3.2 ensures that the choice of the orientation £ is not valid. Hence, the geometrical multiplicity is one. The Bernstein transformation matrix is the classical example of the theorem and the corollary.
COROLLARY 3.5. The Bernstein transformation matrix ?§ is not diagonalizable for all . GL .
Proof. Since the algebraic multiplicity of each eigenvalue is two (except, if is odd,
d
. §
¶z ?
for ¨¦ then it is one) and the geometrical multiplicity is one. Hence, is not .
diagonalizable for xL .
<
§ § ? COROLLARY 3.6. Let be the Bernstein operator of dimension and be the
transformation matrix from the standard basis to the Bernstein basis. Then
<
§&· ¸%¹ § + .
S? GL : (3.3) ¬ ETNA Kent State University [email protected]
436 U. ITAI
Proof. The transformation matrix is not diagonalizable and the operator is diagonaliz-
able. Hence,
<
+ . · ¹
S?§ GL : § ¸ ¬
This completes the proof.
THEOREM 3.7. Assume that the condition of Theorem 3.2 holds and
£# C
4R.
Cu ¨ :
sign §,£¦¥
«
6%K
§
£ H
Let be an eigenvector of the matrix associated with the eigenvalue £ , and let the
F
§76 £
.
£
orientation be . Then, «
C
C ©£¦6 §
4R. 4
º ¨ NL I:
(3.4) sign £c»
«
§]6 £
4 .
N ´
Proof. If , there is nothing to prove ( £ ). We use induction for the rest
)
4 3,N£
of the proof. We consider Theorem 3.7 to be true for all , and we will check it )
for N .
6%K
¡£ *u ¢£
§ '
(3.5) :
Therefore,
« «
^
!
6%K ¯ ^
* +
§ ¼'
§ £ £
H ^
(3.6) ¯ ¥
F
¯"©§76 £
or
« « «
^c6%K
!
^ ^ ¯ 6%K 6©K 6%K
d
§ § : §
£ £ £ § § §
H H H
^J¥ ^ £ ¥ £ ^
(3.7) ¯³¥
F F F
¯"©§]6 £ 6©K
By the properties of the matrix § then
« «
^c6%K
!
6%K ¯ 6%K 6©K ^
4
§ § §
§ £ § § £
H H H
¯#^ £¦¥ £ ^
(3.8) ^c¥
F F F
¯"©§]6 £
)
4
O(NL yields
*S4 )B*
' ' OB:
Hence,
« «
^c6%K
!
^ 6©K ¯
½ § :
£ § £¾
H H ^
sign sign ¯³¥
F F ¯"§]6 £
From
£# C
£¦¥ C 4R. +
¨ sign § ETNA Kent State University [email protected]
ON THE EIGENSTRUCTURE OF THE BERNSTEIN KERNEL 437 §
and the definition of we get «
C
C ©£¦6 §
4R. 4 +
º ¨ NL9 (3.9) sign £c» which completes the proof. The choice of the orientation was only for the convenience of the proof and the proof goes through with any real orientation. The classical example of this theorem is the Bernstein transformation matrix. 3.2. The Asymptotic Behavior of the Transformation Matrix. Unlike the operator’s asymptotic behavior which converge and the limit matrix is described in Corollary 2.3. The transformation matrix between the standard (power) basis and the Bernstein basis asymptotic behavior, however, is very different: it diverges. From that we can deduce that the transfor- mation matrix norm is greater than one, for every subordinate (operator) norm. LEMMA 3.8. The transformation matrix from the standard basis to the Bernstein basis
asymptotic behavior diverges. $
Proof. Consider the vector ¿ . Then,
.
K
$
.
¨@?A§7¿c$c ¨¦?§7¿c$c :
Hence,
i i
$
K
$ . $
: ¨@? ¿ ¨¦? ¿ UÀ
§ §
Therefore,
k
i
!
i
e#fgh e#f#h
. .
K
d
$ *
¨@? ¿ L ' ÂÁ:
§
i5jTk i5jTk
i "&K
Using the same computation, we can determine that:
i
egf#h
C
.
¨? ¿c$/ Á NL :
§
i5jTk
i
egf#h
§
? ?
Since is a finite dimensional matrix, there exist entries of § that diverge [2]. This i5jTk
completes the proof.
· · Ã
COROLLARY 3.9. Let à be a matrix norm. Then,
· · . .
?§ ÃO xL :
(3.10) Ã for every
. · §©· .
? ÃL
Proof. Since the greatest eigenvalue is , Ã ([5] p. 150). By Lemma 3.8 the
i
egf#h ?
matrix § diverges. Hence,
i5jTk
i
egf#h
· ·
à ? Ãs Á:
§ i5jTk
By the subordinate of a matrix norm,
i i
egf#h egf#h
· §©· · ·
à ? à L Ã1? Ã5 ¦Á:
§
i5jTk i5jTk
Hence,
· §%· .
à ? ÃO : ETNA Kent State University [email protected]
438 U. ITAI
This completes the proof.
REMARK 3.10. The infimum of all norms is . since it is the spectral radius. REMARK 3.11. All the results of this section are valid for the transformation matrix from the standard polynomial basis to the reduced Bernstein polynomial basis matrix.
REFERENCES
[1] S. COOPER AND S. WALDRON, The eigenstructure of the Bernstein operator, J. Approx. Theory, 105 (2000), pp. 133–165. [2] I. GOHBERG AND S. GOLDBERG, Basic Operator Theory, Birkhauser¨ Boston Inc., Boston, MA, 2001. [3] S. KARLIN, Total Positivity. Vol. I, Stanford University Press, Stanford, Calif, 1968. [4] R. P. KELISKY AND T. J. RIVLIN, Iterates of Bernstein polynomials, Pacific J. Math., 21 (1967), pp. 511– 520. [5] D. KINCAID AND W. CHENEY, Numerical Analysis, Brooks/Cole Publishing Co., Pacific Grove, CA, 1991. [6] G. G. LORENTZ, Bernstein Polynomials, Mathematical Expositions, no. 8, University of Toronto Press, Toronto, 1953. [7] J. R. WINKLER AND D. L. RAGOZIN, A class of Bernstein polynomials that satisfy Descartes’ rule of signs exactly, in The Mathematics of Surfaces, IX (Cambridge, 2000), Springer, London, 2000, pp. 424–437.