Weighted Fejér Constants and Fekete Sets

0.18 MB·English

by  Á. P. Horváth

Weighted Fej´er Constants and Fekete Sets
3
1 A´. P. Horva´th ∗
0
2

n
a
J Abstract

8 WegivetheconnectionsamongtheFeketesets,thezerosoforthogonal
2 polynomials, 1(w)-normal point systems, and the nodes of a stable and

mosteconomical interpolatoryprocessviatheFej´ercontants. Finallythe
]
A convergence of a weighted Gru¨nwald interpolation is proved.
C
. 1 Introduction
h
t
a
L. Fej´er introduced the so-called Hermite-Fej´er interpolatory process, and in
m
1934 he gave the definition of normal- and ̺-normal system of nodes for which
[ the Hermite-Fej´er interpolation is a positive interpolatory process. The sur-

1 prising nice convergence properties of Lagrange, Hermite and Hermite-Fej´er
v operators on ̺-normal systems were proved by L. Fej´er, G. Gru¨nwald, etc. On
9 the other hand the experiences in electrostatics ensure a system of nodes: the
6
Fekete set, which has uniform distribution in some sense, so it must be a good
4
set for interpolation. The system of zeros of orthogonal polynomials has very
6
. similar properties, as it it well-known. From another point of view, Egerv´ary
1
and Tur´an asked,that is it possible to find an interpolatoryprocess, and a sys-
0
tem of nodes together, such that the interpolatory polynomial has the minimal
3
1 degree, and the operator has the minimal norm. The above-mentioned point
: systemscanbeasuitablesystemofnodesforaninterpolatoryprocessingeneral
v
i sense and also with respect to the Egerv´ary-Tur´anproblem.
X
The primary aim of this note to revisit the connections among that sets of
r nodes, and interpolatory problems investigated e.g. in [2], [3], [4], [5], [6], [10].
a
In the next section, we summarize and reformulate these results, and complete
them, when the original statement proved only in classical cases. It will be
pointed out, that in these equivalences the so-called Fej´er constants (see(3))
play the key role, that is the characterization of this special system of nodes is
ensured by the Fej´er constants.
As an application of the results of the second section, in the third section
we provea convergencetheorem on Gru¨nwaldinterpolatoryprocess on the real
∗supportedbyHungarianNationalFoundationforScientificResearch,GrantNo. K-100461
Key words: interpolation, Hermite-Fej´er,stableandmosteconomical, Fekete sets, Gru¨nwald
operator
2000 MSClassification: 41A05,41A36
1
line for Freud-type weights. As it turned out, giving the weighted Fekete sets
with respect to a fixed weight is difficult. (However, there are several methods
of giving approximating Fekete sets.) The zeros of orthogonal polynomials are
Fekete sets for some varyingweights. Unfortunately these varying weights tend
to zerolocallyuniformly,so interpolationonFekete setsin this sense givesonly
trivial (convergent) processes. The investigation of these weights at infinity
leads to define a weighted Gru¨nwald operator (see (11)), which has rather nice
convergence properties. Comparing this result with the previous ones of [8],
[13], it turns out that the convergence is valid here for a wider function class.
2 Connections
At first we give the definition of classes of weights in question.
Definition 1 LetΣ Cisaclosedset. w isquasi-admissible onΣ,ifitisnon-
⊂
negative, upper semi-continuous, and if Σ is unbounded, lim z w(z) = 0.
|zz|∈→Σ∞ | |
It is admissible, if cap z Σ:w(z)>0 >0. Let us call an admissible weight
asapproximating on (a{,b)∈ R, ifit has fi}nitemoments, itis twicedifferentiable
⊂
and log 1 ′′ 0 on (a,b), and if a is finite, then lim w(x) = 0, and if
w ≥ x→a+ x−a
b is finite, then lim w(x) =0.
(cid:0) (cid:0) (cid:1)(cid:1) x→b− b−x
Definition 2 [11]III.1 Let w be a quasi-admissible weight on a closed set Σ
C. Then F are called n-th weighted Fekete sets associated with w, if the supre⊂-
n
mum below is attained at the set F = x ,...,x .
n,w 1 n
{ }
d = sup d (z ,...,z )
n,w n,w 1 n
z1,...,zn∈Σ
2
n(n−1)
= sup z z w(z )w z ) (1)
 | i− j| i ( j 
z1,...,zn∈Σ 1≤i<j≤n
Y
 
Usually these points are not unique, but in one dimension by some restric-
tions on the weight, uniqeness can be proved. In the classical, unweighted case
on [ 1,1], the result is proved by Popoviciu (cf. [14] Ch 6.7 p. 139., and the
−
reference therein). In weighted case, after some restrictions on the weight a
representation of Fekete points was given by M. E. H. Ismail ([5] Thms. 2.1,
2.4), wich ensures the unicity of the Fekete sets as well. In the followings the
one-dimensional case will be investigated.
Now let us deal with the weighted Lagrange interpolatory polynomials on
a system of nodes X = x ,k = 1,...,n;n N . Let l (x) = ω(x) ,
{ k,n ∈ } k ω′(xk)(x−xk)
whereω(x)= n (x x )(denotingx =x ,k =1,...,n)thefundamental
k=1 − k k k,n
polynomials of the Lagrange interpolation, and let w(x) = e−Q(x) be an ap-
proximating wQeight. The properties of Lk,w,X(x) = Lk,w(x) = w(x)wl2k((xxk)) will
be investigated. It is clear, that L (x ) = 1, that is the sup-norm of this
k,w k
2
weighted polynomial is at least 1. If this sup-norm is equal to one, then L
k,w
has a maximum at the point x , that is
k
l2(x) w′(x ) 2l′(x )
(L )′(x )=w(x) k k + k k
k,w k
w(x ) w(x ) l (x )
k k k k !
l2(x) ω′′
=w(x) k Q′(x )+ (x ) =0, (2)
w(xk) − k ω′ k !
which ensures that
ω′′ w′
C :=C = (x )+ (x )=0. (3)
k,w k,w,X ω′ k w k
1
This is the case, when X is a Fekete set with respect to w2(n−1), namely
1≤l≤n (x xl)2w2(n2−1)(x)w2(n2−1)(xl)
L = l6=k −
k,w,X Q1≤l≤n (cid:16)(xk xl)2w2(n2−1)(xk)w2(n2−1)(xl(cid:17))
l6=k −
(cid:16) (cid:17)
Q
1≤i<j≤n (xi xj)2w2(n2−1)(xi)w2(n2−1)(xj)
i,j6=l − 1,
×Q1≤i<j≤n (cid:16)(xi xj)2w2(n2−1)(xi)w2(n2−1)(xj)(cid:17) ≤
i,j6=l −
(cid:16) (cid:17)
because in the dQenominator appears dn(n−1) .
1
n,w2(n−1)
It will turnout in the followings,that the behaviorof the constants C as
k,w
an indicator, shows the properties of the point systems, interpolatory systems
andoperators. Emphasizingthe importence ofthese constants, letus callthem
as ”Fej´er constants”.
Following carefully the proof of the above mentioned theorem of Ismail
([5],Thm. 2.1), we get the following
Proposition 1 Let w be an approximating weight on an interval (a,b). Then
dn(n−1) (z ,...,z ) attains its maximum on (a,b) at a unique set F ,
1 1 n 1
n,w2(n−1) n,w2(n−1
for which the following characterization is valid.
F = x ,...,x if and only if C =0, k =1,...,n. (4)
1 1 n k,w
n,w2(n−1) { }
At first we have to note here, that finite moments are not necessary in this
statement. According to Ismail [5], the proof of this theorem is the following:
taking the partial derivatives of logdn(n−1) , it turns out, that
1
n,w2(n−1)
∂ logdn(n−1) (x ,...,x ) = 0 j = 1,...,n, if and only if C = 0, k =
∂xj n,w2(n1−1) 1 n k,w
1,...,n. Computing the Hessian, it can be seen, that H is always positive
−
3
definite, so recalling the boundary condition on w, we get that the maximum-
set is unique, that is it is the unique solution of the equation system: C =
k,w
0, k = 1,...,n. Independently of the previous chain of ideas, an elementary
proof on unicity can be given.
Proposition 2 Let w bean admissible, continuousweight on R suchthatlog 1
w
is convex. Then the associated weighted Fekete sets are unique.
Proof: Contrary, let x n and y n are Fekete points with respect to w
enumeratedinincreasin{gio}rid=e1r,and{leit}zi=1= xi+yi. Thenbecauseoftheordering
i 2
of the points, and the log-convexity of the weight, by the arithmetic-geometric
mean inequality
(x x )+(y y ) x +y x +y
i j i j i i j j
z z w(z )w(z )= − − w w
i j i j
| − | 2 2 2
(cid:12) (cid:12) (cid:18) (cid:19) (cid:18) (cid:19)
(cid:12) (cid:12)
(cid:12) (cid:12)
= |xi−xj|(cid:12)+|yi−yj|w xi+y(cid:12)i w xj +yj
2 2 2
(cid:18) (cid:19) (cid:18) (cid:19)
x x y y w(x ) w(y ) w(x ) w(y ),
i j i j i i j j
≥ | − | | − |
where the ineqquality is aqn equalitypif and ponly if xq= y qfor all indices, wich
i i
establishes the uniqueness.
Forspecialweights,theFeketesetsarethezerosofsomeorthogonalpolyno-
mials(cf. [5],[4]). Beforesettingtheprecisestatementweneedsomedefinitions.
Definition 3 Let w =e−Q be an approximating weight on (a,b). Let
b Q′(t) Q′(x)
A (x)=̺ p2 (t)w(t) − dt, (5)
n n n,w t x
Za −
where p = γ xn+... is the nth orthonormal polynomial with respect to w,
n,w n
and ̺ = γn−1
n γn
Now we can define our weights:
Definition 4 Let w be as in the previous definition.
w(x)̺
w (x)= n (6)
n
A (x)
n
In the following investigations the constant ̺ has not any role, but it will
n
come into the picture inconnection with a convergence theorem in the next
section. Let us see some examples on An(x) ([5]), which in classical cases are
̺n
differentonly in normalizationfromthe weights w ([10]), for whichthe deriva-
1
tives of p -s are orthogonal :
n,w
Example:
4
(1) If w=e−x2,
A (x)
n
=2,
̺
n
that is w = 1w independently of n and x, and here w =w=2w .
n 2 1 n
(2) If w=xαe−x,
A (x) 1
n = ,
̺ x
n
that is w =xα+1e−x =xw independently of n, and here w =w .
n 1 n
(3) If w=(1 x)α(1+x)β,
−
A (x) α+β+1+2n
n
= ,
̺ 1 x2
n
−
thatisw = 1 (1 x)α+1(1+x)β+1,andherew =(α+β+1+2n)w .
n α+β+1+2n − 1 n
(4) If w=e−x4,
A (x)
n =2(x2+̺2 +̺2 ),
̺ n n+1
n
that is w = 1 w.
n 2(x2+̺2+̺2 )
n n+1
Fromanotherpointofviewwnhasalsoanimportance. Denotingbypn√wn =
z ,itsatisfiesthefollowingdifferentialequationwithsomeΦ (cf. [9],Th. 3.6.):
n n
′′
z (x)+Φ (x)z (x)=0 (7)
n n n
InthenextstatementwereformulatetheresultsofIsmail,RutkaandSmarzewski
(cf. [5], [10]).
Proposition 3 Let w be as in the definitions above, and let us assume that
n
w is an approximating weight. Then
n
C =0, k =1,...,n if and only if x the zeros of p (8)
k,wn { k} n,w
The proof of this statement depends on the differential equation of orthog-
onal polynomials. The equation system on C -s means that the differential
k
equation fulfils at the points x , k = 1,...,n. In the classical cases, it is a
k
Sturm-Liouville equation, that is there are polynomials of degree n in the dif-
ferential equation, which is realized at n points. In general cases unicity is
used.
Normal and ̺-normal point systems were introduced on [ 1,1] by L. Fej´er
−
in 1934 ([2]). The weighted analogon of this definition was given in [3]. The
originalaimofthesedefinitionswasassuringthepositivityoftheHermite-Fej´er
interpolatory operator. The limit case, when ̺ = 1 was investigated on the
weighted real line in [4]. Here this last definition is cited only.
5
Definition 5 Let w be an approximating weight on (a,b). A system of nodes
X = x ,k =1,...,n;n N is 1(w)-normal, if there is an L>1 such that
k,n
{ ∈ }
x <La , (9)
k,n n
| |
where a is the M-R-S number, and
n
n l2(x)
w(x) k 1, x R, (10)
w(x ) ≤ ∈
k
k=1
X
where l (x)-s are the fundamental polynomials of the Lagrange interpolation.
k
In this definition the kernel function of the Gru¨nwald operator appears.
Here we will follow the notations of [8] and [13], that is the weighted Gru¨nwald
operator on the nodes x n with respect to an f is
{ k}k=1
n
w(x)Y (f,x)=w(x) l2(x)f(x ) (11)
n k k
k=1
X
Mostly the boundedness of the operator-norm ensures the convergence of the
interpolatory process. The boundedness by one, is a very special criterium.
This is the case for instance, when the reciprocal of the weight function has
non-negative even derivatives, and the Gru¨nwald operator coincides with the
Hermite-Fej´er one. Also on this chain of ideas the Fej´er constants play the
key role. More precisely, with the notations above, the weighted Hermite in-
terpolatorypolinomial (with some weight w) of a differentiable function can be
expressed as (cf. [4])
n (1 C (x x ))l2(x)
w(x)H (f,f′,x)=w(x) − k,w − k k (fw)(x )
n k
w(x )
k
k=1
X
n (x x )l2(x)
+w(x) − k k (fw)′(x ), (12)
k
w(x )
k
k=1
X
and the corresponding weighted Hermite-Fej´er operator is
n (1 C (x x ))l2(x)
w(x)H (f,x)=w(x) − k,w − k k (fw)(x ), (13)
n,w k
w(x )
k
k=1
X
which coincides with the weighted function at the nodes x n , and which
{ k}k=1
has zero derivatives at the nodes. Furthermore by the definition of the Fej´er
constants, H (f,x) is the (unweighted) Hermite interpolatory polynomial of
n,w
1. So when the Fej´er constants are zero
w
1 n l2(x) 1
Y (x):=w(x)Y ,x =w(x) k =w(x)H ( ,x)
n,w n n,w
w w(x ) w
(cid:18) (cid:19) k=1 k
X
6
n (1 C (x x ))l2(x) 1 1 ′
=w(x) − k,w − k k =w(x)H , ,x (14)
n
w(x ) w w
k=1 k (cid:18) (cid:19) !
X
istheHermiteinterpolatorypolynomialof 1 withrespecttothenodes: x n .
w { k}k=1
So the following connections are established.
Proposition 4 Let w be a weight as above.
If a system of nodes x is 1(w)-normal, then C =0, k =1,...,n.
k k,w
{ }
On the other hand, let us suppose further, that 1 2n 0 on x La .
w ≥ | | ≤ n
Now
(cid:0) (cid:1)
if C =0, k =1,...,n then the system of nodes is 1(w)-normal.
k,w
Proof: If x is 1(w)-normal, then w(x) l2k(x) 1, k =1,...,n (see (10)), so
{ k} w(xk) ≤
C =0, k =1,...,n. Accordingto(14),bytheerrorformulaoftheHermite
k,w
interpolation, it is clear, that 1 w(x) n l2k(x) 0, when 1 (2n) 0 on
− k=1 w(xk) ≥ w ≥
x La .
| |≤ n P (cid:0) (cid:1)
The Egerv´ary-Tur´an interpolatory problem (cf. [10], and the references
therein) is to find an interpolatory process of lowest degree, and of smallest
norm. Below we denote by ˆl (x) each polynomial of arbitrary degree for which
k
ˆl (x )=δ , i=1,...,n.
k i ki
Definition 6 Let w be as in Definition 5. The interpolatory system of polyno-
mials ˆl (x), k =1,...,n is w-stable on (a,b) if for all y ,...,y 0
k 1 n
≥
n ˆl (x)
0 w(x) k y maxy , x (a,b). (15)
k k
≤ w(xk) ≤ k ∈
k=1
X
A w-stable interpolatory system on (a,b) is most economical, if
n
deg ˆl (x) (16)
k
kX=1 (cid:16) (cid:17)
is minimal.
Letusremarkthatiftheweightfunctiontendstozeroquicklyatthebound-
ary points of the fundamental interval, then the w-stability of the Gru¨nwald
operatorcoincideswith the 1(w)-normality ofthe nodes. Itis provedfor allthe
classicalweights (cf. [10], Thm. 2.3), that an interpolatory system is w -stable
n
and most economical, if and only if it is the Gru¨nwald operator on the zeros of
p . From the previous investigations, similarly to the classical cases, we can
n,w
state the parallel theorem for general weights.
Let us denote by
n ˆl (x)
I (x):=w(x) k
n,w
w(x )
k
k=1
X
7
Proposition 5 Let w be an approximating weight on an interval (a,b).
If I (x) is w-stable and most economical, then
n
C =0, k =1,...,n. (17)
k,w
Let us assume further that 1 (2n) 0 on x La .
w ≥ | |≤ n
If C =0, k =1,...,n, then
k,w
(cid:0) (cid:1)
I (x)=Y (x) is w-stable and most economical (18)
n n,w
Proof: As it was pointed out eg. in [10], if an interpolatory process I (x) is
n
w-stable and most economical, it must be the Gru¨nwald operator, because by
thepositivityoftheoperator,ˆl (x)haszerosatthepointsx ,i=1,...,n,i=k
k i
6
of even multiplicity, that is n deg ˆl (x) 2n(n 1). It is realized by
k=1 k ≥ −
Yn,w. As it was shown in StPatement 4(cid:16), if Yn(cid:17),w has maxima at xk-s then Ck-s
are zero. The opposite direction is also follows from Statement 4.
Finallyenumeratingthepropertiesdiscussedabove,wecansummarizethese
results as it follows.
(A) C =0, k =1,...,n (A′) C =0, k =1,...,n
k,w k,wn
(B) F = x ,...,x (B′) F = x ,...,x
1 1 n 1 1 n
n,w2(n−1) { } n,wn2(n−1) { }
(C) p (x )=0, k =1,...,n
n,w k
(D) Y (x) is w-stable and most economical
n,w
(D′) Y (x) is w -stable and most economical
n,wn n
(E) x ,...,x is 1(w)-normal (E′) x ,...,x is 1(w )-normal
1 n 1 n n
{ } { }
Through the equivalence of all the above mentioned properties with property
(A) (or (A′)), that is the Fej´er constants are zero, one can get
Corollary:
Let w be an admissible, approximating weight on an interval (a,b). If
1 (2n) 0on(a,b),then(A),(B),(D),(E)areequivalent,andif 1 (2n) 0
w ≥ wn ≥
(cid:0)on(cid:1)(a,b), then (A′),(B′), (C), (D′),(E′) are equivalent. (cid:16) (cid:17)
We have to show an example on the second assumption .
Example: Let
m
Q(x)= d x2k, d 0, k=1,...,m, (19)
k k
≥
k=0
X
(2n)
and let w(x) = e−Q(x). For these special Freud-type weights 1 0
wn ≥
on R for all n N . According to the Leibniz rule it is enough(cid:16)to (cid:17)show that
∈
8
An (j) 1 (2n−j) >0 for j =1,...,2n. Because
̺n w
(cid:16) (cid:17)
(cid:0) (cid:1)
∂j Q¯(t,x) = m 2kd ∂j t2k−1t−−xx2k−1 = m 2kd 2k−2b t2k−2−lxl−j,
∂xj k (cid:16) ∂xj (cid:17) k l
(cid:0) (cid:1) kX=1 k=X⌈j2⌉+1 Xl=j
where b -s arepositive,taking into considerationthat w is anevenweightfunc-
l
tion, (and so p2(w) is also even), one can see that
n
A (j) m 2k−2
n = 2kd b p2(w,t)w(t)t2k−2−ldtxl−j
̺ k l n
(cid:18) n(cid:19) k=X⌈2j⌉+1 Xl=j ZR
is a polynomial of x with nonnegative coefficients, and all the exponents of this
polynomial are even if j is even and are odd if j is odd. By a simple induction
one can see that
1 (j)
=p(j,x)eQ(x),
w(x)
(cid:18) (cid:19)
where p(j,x) is a polynomial having the same properties as the previous one.
Because j and 2n j have the same parity,
−
(2n)
1
(x)=p(x)eQ(x),
w
(cid:18) n(cid:19)
where p(x) is a polynomial with even exponents and positive coefficients, so it
is positive on the real line for all n N.
∈
Finally we have to remark that the assumption 1 (2n) 0 seems to be
w ≥
assymetric, and it is necessary only because of the method of the proof by
(cid:0) (cid:1)
Hermiteinterpolation. Thequestionthatcanitbeweakenedornot,isunsolved
yet.
3 Interpolation
In this section, let w = e−Q be a three times continuously differentiable Freud
weighton R, that is we suppose that Q is even, Q′ >0 on (0, ), and for some
∞
′ ′
A,B 2;A (xQ (x)) B on(0, ),moreoverthereisaconstantcsuchthat
≥ ≤ Q.(x) ≤ ∞
for every x 1, xQ(3)(x) c. By these assumptions there is a d 1 such
| | ≥ Q′′(x) ≤ ≥
that Q′′(x)≥ 1−(B(cid:12)(cid:12)(cid:12)−1)x22−c(B(cid:12)(cid:12)(cid:12)−1), when |x|≥d. Now we can define
Definition 7 With d>1 given above, let
w(x), x 1
w˜(x)= w(x) |x|≤, x d (20)
 Q′(x) | |≥
 twice continuously differentiable,elsewhere

9
Furthermore we assume that log 1 has positive and continuous first and
w˜
second derivatives on (0, ).
∞
′
′
Let us remark at first that Q′(1) Q′(d)+ d Q (x) (d), because Q′
≤ Q′(d) x
(cid:18) (cid:19)
is increasing, and the second member of the right-hand side is positive, when
A 2. That is a suitable connection can be defined between the two parts of
≥
log 1.
w˜
As usually we define
Definition 8
C = f C(R) lim (fw˜)(x)=0
w˜
{ ∈ ||x|→∞
Let Y (f, ) be as in (10), the Gru¨nwaldoperator on the zeros of p . Now
n n,w
·
we have the following
Theorem 1 Let f C Then
w˜
∈
lim (Y (f) f)w˜ =0 (21)
n
n→∞k − k
Comparing this theorem with Cor. 2. of [13], we can see, that we have two
different weights in this theorem, but when A 2, then the function class is
≥
wider here, that is the fuctions can grow more quickly at infinity.
The previousdefinitionofthe weightwas inspiratedby the nextlemma. In-
vestigatingtheweightsw fromthe previoussection,itturnsout,thathowever
n
w tendstozerolocallyuniformlywhenntendstoinfinity,thebehaviorofw -s
n n
arethesameatinfinity. Itmeans,thattheGru¨nwaldoperatoronFeketepoints
with respect to the varying weights w has trivial convergence properties, but
n
it allows to find a non-trivial process, as it is given in the theorem.
The following estimation of A is valid.
n
Lemma 1 Let w be as above, and let A 2. Let L be a constant such that
0
L20an >a2n+[A−1]+1. For every L>L0 ≥
An(x) an2n′, if |x|≤Lan , (22)
̺n ∼( Qx(x), if |x|≥Lan
where the constants in ” ” depend only on L, but they are independent of n.
∼
Proof:
At first we have to note that such an L exists by [7] 5.9. The first line
0
of the inequality is proved by H. N. Mhaskar ([9], Prop. 3.7.). To prove the
second line we have to divide the integral to some parts. Since A is even we
n
can choose x>La .
n
A (x) Q′(t) Q′(x)
n = p2(t)w(t) − dt+ ()dt=I +I
̺n Z|t|≤x2 n t−x Z|t|>x2 · 1 2
10