Only few results are known on continuity properties of the set-valued metric projection in nonlinear uniform approximation. In this paper we investigate this mapping in the case of best uniform approximation by splines of degree m with k free knots. A characterization of those functions at which the metric projection is upper semicontinuous is given. It follows that the metric projection is upper semicontinuous if and only if k &#8804; m, and that it is upper semicontinuous at all "normal" functions. On the other hand, it is shown that the metric projection is never lower semicontinuous

Nürnberger, Günther

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The metric projection for free knot splines

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The Metric Projection for Frre Knot Splinffi
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G. Nürnberger
Nr. 112, 1990
Abstract
Only few results are known on continuity properties of the set-valued metric
projection in nonlinear uniform approximation. Inl this paper we investigate this
mapping in the case of best uniform approximationi by splines of degree m with k
free knots. A characterization of those functions at which the metric projection
is upper semicontinuous is given. It follows that the metric projection is upper
semicontinuous if and only if k $; m, and that it 1s upper semicontinuous at all
"normal" functions. On the other hand, it is show~ that the metric projection is
never lower semicontinuous.
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1. Introduction
There is a vast literat ure on continuity properties of th~ set-valued metric projection onto
linear subspaces (see e.g. the surveys Deutsch [7], [8], Nü~nberger & Sommer [14]' Singer [18],
Vlasov [19] and the references therein). On the other hand, not as many results are known
about this mapping in nonlinear approximation (see e.g. Berens & Finzel [1], Brosowski &
Deutsch [5]' Deutsch [6], Nürnberger [9], Schmidt [15] a.I1-dSinger [18]).
The aim of this paper is to investigate the metric projection onto Sm k, the set of po-. "
lynomial splines of degree m with k free knots. This is the mapping which associates to
each function f E C[a, b], the set PSm,kU) = {sf E S~,k : Ilf - sfll = inf IIf - slI} of
. . ! SESm,k
its best uniform approximations from Sm,k. We give a characterization of those functions in
C[a, b] at which PSm,k is upper semicontinuous. As a con~equence we get that PSm,k is upper
semicontinuous on C[a, b] if and only if k ~ m. Moreover, it follows that PSm k is upper
semicontinuous on the set {f E C[a, b] : PSm,kU) C C[ai b] and PSm,kU) n Sm,k-~ = 0}. On
the other hand, we show that PSm k is never lower semicpntinuous.
The same statements hold for the set-valued mappiJg which associates to each function
f E C[a, b], the nonempty set PSm,kU) n C[a, b] of its cobtinuous best approximations.
In a furt her paper we apply the results to derive uniq~eness theorems (announced in [12])
for Sm,k.
2. Main Results
Let C[a, b] be the space of all continuous real-valued fJnctions fon [a, b] endowed with the
supremum norm Ilfll = sup If(t)l. Moreover, let points' a = Xo < Xl < ... < Xr < xr+l = b
tE[a,b] :
and integers mI, ... , mr E {1, ... ,m + 1} be given, wh,ere m ~ 1 and r ~ 1. We denote
by Sm(~~::::::;';Jthe spa.ce of polynomial splines of degree m with r fixed knots XI, ••• , Xr of
multiplicities mI, ... , mr, and by Sm,k the set of polynOl;nial splines of degree m with k free
(multiple) knots, where k ~ 1 (see e.g. Nürnberger [11]1and Schumaker [17]). Here we use
the convention that a spline has a knot of multiplicity m
l
+ 1 if for this spline no continuity
is required at the knot.
A spline sf E Sm,k is called best uniform approximation of a function f E C[a, b] from
Sm,k, if IIf - s fll = inf IIf - sll. The nonempty set of best uniform approximations of f from
SESm,k
Sm,k is denoted by PSm,kU), and the resulting set-valued mapping PSm,k : C[a, b] -t 2Sm,k is
called the metrie projeetion onto Sm,k.
In the following we investigate continuity properties of this mapping.
Definition 1 The metric projection PSm k : C[a, b] -t 2Sm,k is called upper semieontinuous
(u.s.c.) (respectively lower semieontinuou~ (l.s.c.)) at f ~ C[a, b] if for ea.ch sequence Un) C
C[a, b] with fn -t f and ea.ch c10sed subset A of Sm,k w'ith PSm,kUn) nA i- 0 (respectively
PSm,kUn) C A) for all n, we have PSm,kU) n A i- 0 (r~spectively PSm,kU) cA). PSm,k is
called upper semieontinuous (respectively lower semieontinuous) if it is U.S.C. (respectively
l.s.c.) at every function fE C[a,b].
The first result shows that the upper semicontinuity of the metric projection PSm k at
a given function depends on the multiplicities of the knpts of its best approximations from
Sm,k. I
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Theorem 1 For a lunction I E C[a, b]\Sm,k, the lollow~ng statements are equivalent:
;
(i) PSm,k is upper semicontinuous at I.
(ii) There does not exist a spline s E PSm,kU) n Sm(~~:::::~U such that s is discontinous
r
or m + 2 + ~ mi - .max mi ::; k.
1=1 l=l, ...r
Prool: (ii) => (i). Suppose that (ii) holds. Let a clo~ed set A in Sm,k, I E C[a, b] and
Un) C C[a, b] be given such that In - I and PSm,kUn) h A =j:. 0 for all n. We have to show
that PSm,kU) n A =j:. 0 which implies that PSm,k is uppe~ semicontinuous at I. For all n, we
choose a spline Sn E PSm,kUn) n A. We will show that t~ere exists a spline S E PSm,kU) and
a subsequence (snq) of (sn) such that lim IIs - snqll = Ö. Since A is closed, it follows that
q-+oo
S E A which proves the claim. It is easy to see that (sn) is a bounded sequence. Therefore,
it follows from Bra.ess [4, p. 229] that there exists a spline S E PSm,kU) n Sm(~~:::::~Jsuch
that a subsequence of (sn), again denoted by (sn), converges to S uniformlyon each compact
subset of [a, b]\{ xl ... , xr}. Moreover, the knots of (sn) converge to the knots of s. It follows
r i
from (ii) that S is continuous and m +2+ ~ mi - .max mi > k. For all i E {l, ... , r}, let mi
1=1 l=l, ...r :
be the minimal multiplicity of Xi such that S E Sm(~~:::::~JNow, let an index j E {l, ... , r}
be given. By going to a subsequence, we mayassurne that for all n, the same number of
(multiple) knots of Sn, say Y1,n ::; ... ::; YPi,n, converges to Xj. Then we have Pj 2: mj.
Because, if Pj < mj ::; m, then it follows from Bra.ess [3, :p. 229] that
I
IIs - snll[t(xi-1+xi),t(Xi+Xl+t)] - 0
and that S has a knot of multiplicity Pj at X j which is f' contradiction. Moreover, we have
pj ::;m + 1. Because, if Pj 2: m + 2, then, since (ii) holds,
r r rLPi 2: m + 2 + L mi 2: m + 2 +L mi - .'pax mi > k
i=l i=1 i=l 1-1, ...,r
i~i
which is a contradiction to Sn E Sm,k. We define
Km(z, t) = (t - z)+ ' I(z, t) ~ [a, b] X [a, b]
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and denote by Km[zl, ... , Zl+l, t] the divided difference of:order 1 oft he function z - Km(z, t)
with respect to the points Zl, .. :, ZIH' Then for all n, the spline Sn can be written as
m Pi
Sn(t) = l:ai,nti + I)i,nKm[Y1,n, ... , Yi,n, t] ,
1=0 1=1
For sufficiently large n, we have
3 3
Xj-1 + 4(Xj - Xj-I) ::;Y1,n ::; ... ::; YPi,n ::;Xj + 4(Xj+1 - Xj).
Now, we choose points tl, ... , tm+pi+1 such that
~(Xj-1 + Xj) ::;t1 < ... < tm+1 < Xj-1 + £(Xj - Xj-'-l) < Xj + £(Xj+1 - Xj) < tm+2
< ... < tm+pi+1 ::; ~(Xj + Xj+1)' : .
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It is weil known and easy to vertfy that the determinan t generated by inserting these points
into the m + pj + 1 functions
1, t, ... , tm, Km[xj,.]' ... , Km[xJ, ... , Xj,.]
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is different from zero. Therefore, since (sn) is bounded apd for all t E [a, b]\{xj},
i=l, ... ,pj,
the sequence (ai,n), i = 0, ... , m, and (bi,n), i = 1, ... ,Pj, are bounded. Thus by going to
subsequences, we mayassume that these sequences convfrge.
Moreover, since the spline S is continuous, we have lim bm+l n = 0, if PJ' = m + 1. Thisn-+eo '
implies that '
IIs - Snll[Hxj_l+xj),~(Xj+XHt}] ~ O.
Since this holds for every index j E {1, ... , r}, it foilows :that IIs - snll ~ O.
(i) => (ii). Suppose that (ii) fails. We will show thatl PSm k is not upper semicontinuous
at I. We first assurne that there exists a spline s E PSm lU) ~hich is discontinuous at some
knot Xj. Then it foilows from Schumaker [16] (see'also ~raess [4, p.230]) that there exists a
sequence (sn) c PSm,kU) with the foilowing properties. For all n, the spline Sn has a simple
knot at x j - an and a knot of multiplicity m at x j + ßnJ where an > 0, ßn > 0 and an ~ 0,
ßn~ O. i
Moreover, for all n, I
Sn(t) = s(t)
and
Sn(t) = s(t) tE [a, b]\{xj}.
• I
We set for all n, Sn = Sn +~ and In = I +~. Since 1- J has alternating extreme points, for
all n, Sn (j. PSm kU), Moreover, since Sn E PSm kU), it foilows that Sn E PSm kUn). The set
A = {sn: n E'IN} is closed, since no subsequ~nce of (sn) converges uniform'ty. Now, since
In ~ I, PSm,kUn) n A i= 0 for all n, but PSm,kU) n A = 0, the metric projection PSm,k is not
upper semicontinuous at I.
Finally, suppose that there exists a spline
SEPSmku)nsm(Xl, ,xr)1 CC[a,b], m1, ,mrY
i
such that m + 2 + t mi - .max mi ~ k. Let Xj be a ~not with mj = .max mi ~ m. We
set t=1 t=l, ...,r . I t=l, ...,r
Yi,n = Xj, t = 2, ... , m
l
j + 1,
and choose points
Yl,n < Xj < Ymj+2,n < ... <Ym+2,n
such that
Yi,n ~ Xi i = 1, ... , rn + 2.
Let Bn be the normalized B-spline of degree m associat~d with the knots
Yl,n ~ ... ~ Ym+2,n.
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By multiplying B. with an appropriate lodor lor an n, Lmayassume that
B.(xj) = ~(f(Xj) - 8(xL).
For all n, we set sn = s+Bn. Then for sufficiently large n, Sn E PSm k(J). As above, we set for
all n, sn = Sn +~, fn = f + ~ and A = {sn: nEIN}. SiI}ce no sub~equence of (sn) converges
uniformly, the set Ais closed. Analogously as above, we Ihave fn -+ f and PSm,k(Jn) n A i= 0
for all n, but PSm,k(J) n A = 0. Therefore, PSm,k is not upper semicontinuous at f. This
proves Theorem 1. .' I .
As a first consequence of Theorem 1, we obtain a characterization of the upper semicon-
tinuity of PSm k' I, I
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Corollary 1 The metrie projeetion PSm,k is upper semieontinuous on C[a, b] if and only if
k ~ m.
Proof: It is easy to verify that Ps k is upper semicontinuous on Sm k. Suppose that
m, I '
k ~ m and let f E C[a, b]\Sm,k be given. Then all splines s E PSm,k(J) are continuous
and the inequality in Theorem 1 is obviously not satisfi~ed for s. Therefore, it follows from
Theorem 1 that PSm,k is upper semicontinuous at f. :
Now, suppose that k > m. Then there exists a spline s E Sm,k which is not continuous. It
is clear that we can construct a function f E C[a, b]\Sm,k such that f - s has m + 2k + 2
alternating extreme points on some knot-interval of s. T~en by Schumaker [16], s E PSm,k(J)
and by Theorem 1, PSm,k is not upper semicontinuous. This proves Corollary 1.
The second conclusion of Theorem 1 shows that PSm 1 is upper semicontinuous on a large
subset of C[a, b], namely at all "normal" functions. 'I
Corollary 2 The metrie projeetion PSm k is upper semi~ontinuous on, ,
{f E C[a, b] : PSm,k(J) C C[a, b] and Pfm'k n Sm,k-1 = 0}.
Proof: Let a function fE C[a, b] be given such that lfSm,k(J) C C[a, b] and
PSm,k(J) n Sm,k-1 = 0. This means that for all s E PSm,k(J) n Sm(~;::::,~;;J,we have mi ~ m,
i = 1, ... , r, and t mi = k. Therefore, the inequality inl Theorem 1 cannot be satisfied and
z=l
PSm,k is upper semicontinuous at f. This proves Corolla,ry 2.
While by Corollary 1, the metric projection PSm k is upper semicontinuous if and only if
k ~ m, we now show that PSm,k is never lower semicontiiuous.
Theorem 2 The metrie projeetion PSm,k : C[a, b] -+ 2Sr,kiS not lower semieontinuous.
Proof: We construct a function f E C[a, b] and a sequence (Jn) in C[a, b] such that
fn -+ f, PSm,k(Jn) = {so} for all n and {so} ~PSni,k(J), rhich shows that PSm,k is not lower
semicontinuous. For doing this, we choose arbitrary poi:qts
!
a = Xo < Xl < ... < Xk < Xk+1 = b
and a spline So E Sm,k\Sm,k-1 which has active knots at XiI,' .. , Xk such that so(t) = (t-Xk)m,
t E [Xk-1, Xk], and so(t) = 0, t E [Xk, b]. Moreover, we define f E C[Xk, Xk+1] such that
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I(Xk) = -1, I ("+,'-t,) = 1, l(xHÜ = -1 and I islnnear elsewhere on [Xk,Xk+1]. We
mayextend I to a functlon in C[a, b] such that 111- soll = 1, I - So is piecewise linear and
I - So has sufficiently many (which will be specified later) alternating extreme points on each
knot-interval [Xi, Xi+l], i = 0, ... , k - 1. We now define ~ sequence Un) in C[a, b] as folIows.
For all n, we set
.' . 1
In(t) = I(t) t E [a, Xk] U [Xk + n' b],
In(t) = -1 tE [Xk, Xk + ln]'
In linear on (Xk+2~'Xk*"1 ~).
Then it folIows that In -+ I. .
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Now, let Yl ::; ... ::; Y2k be the knots of So counting eaeh knot twice. Moreover, we choose
arbitrary points Y-m < ... < Y-l < Yo = a and b = Y2k+I < Y2k+2 < ... < Y2k+m+l' We have
the freedom to define I on [a, Xk] such that for all n, In - So has at least j + 1 alternating
extreme points in each knot-interval (Yi, Yi+m+j) C (Y-rrl, Y2k+m+l), j 2: 1.
Note, that by constructioh the interval (Y2k-b Y2k+m+lj C (Y-m, Y2k+m+l), j 2: 1, contains
three alternating extreme points of In - So for all n, but IlonlYtwoalternating extreme points
of I - So.
Moreover, by construction I - So has the same number of alternating extreme points on
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[a, b] as 1- sn,and therefore, I - So has at least m + 21:- + 2 alternating extreme points on
(Y-m, Y2k+m+Ü. Therefore, it folIows from Schumaker [16] and Braess [3] that So E PSm,kU),
Moreover, since In - So has sufficiently many alternatibg extreme points in each interval
(Yi, Yi+m+j), it follows from Nürnberger[9] that So is a (s~rongly) unique best approximation
of In from Sm,k for all n. We now show that {so} :j; PSm,kU), For all c > 0 we define
Se E Sm,k - Sm,k-l by
and
se(t) = set)
se(t) = (t - Xk)m ,
I .
t E [a, Xk-l], .
tE [Xk .•...l, Xk + cl,
se(t) = (t - Xk)m + ae(t - (Xk + c))m I tE [Xk + C, b].
Where 3 I
Ci(Xk+l - Xk))j
ae = - (i(Xk+l - Xk) - 4)m'
Then it folIows that !
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se(t) > 0 ,
and
Since I is linear on
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se(t) < 0, tE (Xk + -(Xk+l - Xk), b].
[Xk. "+,'-t,], there exis" a s';icie~tlY small € > 0 such that
II(t) - se(t)/ ::; 1
Moreover, since IIsell -+ 0 for c -+ 0, for sufficiently small c > 0,
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,
This shows that So =I SE E PSm,k(J) and proves Theorem I 2.
We note that the proofs of the above results show tllat the same statements hold, if we
consider the mapping' Ps k : C[a, b] -+ 2Sm,knC[a,b], defi~ed by Ps ' k(J) = PSm k(J) nC[a, b]m, 1 m,....."
for all 1 E C[a, b], instead of PSm k' It was shown by Seh~maker [16] that PSm k(J) =I 0 for all
1E C[a, b]. In [12] we incorreetl~ announeed the result !that PSm,k is upper ~emieontinuous
(eompare the statemen t in Corollary 1 for PSm,k)' '
We finally eonsider a furt her eontinuity property. A ~ontinuous mapping
F : C[a, b] -+ Sm,k is ealled continuous selection for PSm,k if F(J) E PSm,k(J) for all 1 E
C[a, b]. I
In the fixed knot ease, it was proved by Nürnberger, & Sommer [13] that there exists a
eontinuous selection for the melrie projeetion PSm("'t',::fk) if and only if k ~ m+ 1 (for further
eontinuity results see Berens & Nürnberger [2]' Nürnberger & Sommer [14], and Nürnberger
[11]). On the other hand, the problem of the existenee hf eontinuous selections for PSm,k is
unsolved at present.' I
..
..
which implies that
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111 - sEil = 1= 111 - soJI.
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References I
[1] H. Berens and M. Finzel, A continuous selection ofth~ metric projection in matrix spaces,
in "Numerical Methods of Approximation Theory, "i0l. 8", 1. Collatz, G. Meinardus and
G. Nürnberger (eds.), ISNM 81, Birkhäuser, Basel, i1987, 21-29.
[2] H. Berens and G. Nürnberger, Nonuniqueness and selections in spline approximation,
Constr. Approx. 6 (1990), 181-193.
[3] D. Braess, Chebyshev approximation by spline funciions with free knots, Numer. Math.
17 (1974), 357-366.
[10]
[4] D. Braess, Nonlinear Approximation Theory, Springer, Berlin, 1986.
[5] B. Brosowski and F. Deutsch, Radial continuity c1>fset-valued metric projections, J.
Approx. Theory 11 (1974), 236-253. I
[6] F. Deutsch, Existence of best approximation, J. Approx. Theory 28 (1980), 132-154.
I
[7] F. Deutsch, A survey of metric selections, in "Fixed ~oints and Nonexpansive Mappings",
R.C. Sine (ed.), Contemporary Mathematics 18, Providence, 1983, 49-7l.
[8] F. Deutsch, An exposition of recent results on metril selections, in "Numerical Methods
I
of Apporximation Theory, Vol. 8", L. Collatz, G. Meinardus and G. Nürnberger (eds.),
ISNM 81, Birkhäuser, Basel, 1987, 67-80. !
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[9] G. Nürnberger, Continuous selections for the metri~ projection and alternation, J. Ap-
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G. Nürnberger, Strongly unique spline approximati0n with free knots, Constr. Approx.
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[12]
G. Nürnberger, Approximation by Spline Functions, Springer, Berlin, 1989.
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projection for spline functions, J. Approx. Theory 2~ (1978), 320-330.
[14]
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G. Nürnberger and M. Sommer, Continuous select~ns in Chebyshev approximation, in
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ISNM 72, Birkhäuser, Basel, 1985, 248-263. I
E. Schmidt, On the continuity of the set-valued me~ric projection, J. Approx. Theory 7
(1973), 36-40. I
1.L. Schumaker, Uniform approximation by Chebyshev spline functions 11: free knots,
SIAM J. Numer. Anal. 5 (1968), 647-656. I
I
L.L. Schumaker, Spline Functions: Basic Theory, Wiley-Interscience, New York, 1981.
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Philadelphia, 1974.
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[19] L.P. Vlasov, Approximative properties of sets in normed linear spaces, Russian Math.
Surveys 28 (1973), 1-66.
G. Nürnberger
Fakultät für Mathematik und Informatik
Universität Mannheim
D-6800 Mannheim Germany
•
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The metric projection for free knot splines

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