Table Of ContentISNM
INTERNATIONAL SERIES OF NUMERICAL MATHEMATICS
INTERNATIONALE SCHRIFTENREIHE ZUR NUMERISCHEN MATHEMATIK
SERIE INTERNATIONALE D'ANALYSE NUMERIQUE
Editors:
eh. Blanc, Lausanne; A. Ghizzetti, Roma; P. Henrici, Zürich; A. Ostrowski, Montagnola;
J. Todd, Pasadena; A. van Wijngaarden, Amsterdam
VOL.30
Numerische Methoden
der Approximationstheorie
Band 3
Vortragsauszüge
der Tagung über numerische Methoden der Approximationstheorie
vom 25. bis 31. Mai 1975
im Mathematischen Forschungsinstitut Oberwolfach (Schwarzwald)
Herausgegeben von
L. COLLATZ, Hamburg, H. WERNER, Münster, und G. MEINARDUS, Erlangen
SPRINGER BASEL AG
CIP-Kurztitelaufnahme der Deutschen Bibliothek
Numerische Methoden der Approximationstheorie
Bd. 3. Vortragsauszüge der Tagung über numerische Methoden der Approximationstheorie:
vom 25.-31. Mai 1975 im Math. Forschungsinst. Oberwolfach (Schwarzwald)/hrsg.
von 1. Collatz. ..
(International series of numerical mathematics; Vol. 30)
NE: Collatz, Lothar [Hrsg.]; Tagung über Numerische Methoden der Approximationstheorie
<1975, Oberwolfach>; Mathematisches Forschungsinstitut <Oberwolfach>
Nachdruck verboten
Alle Rechte, insbesondere das der übersetzung in fremde Sprachen und der Reproduktion
auf photostatischem Wege oder durch Mikrofilm, vorbehalten.
© Springer Basel AG 1976
Originally published by Birkhäuser Verlag Basel in 1976
ISBN 978-3-7643-0824-7 ISBN 978-3-0348-7692-6 (eBook)
DOI 10.1007/978-3-0348-7692-6
Vorwort
Der vorliegende Band stellt Vortragsmanuskripte einer am Mathematischen
Forschungsinstitut, Oberwolfach, in der Zeit vom 25. bis 31. Mai 1975 veran
stalteten Tagung zusammen, die unter der Leitung der Unterzeichner stand.
Die letzten dieser Tagungen über numerische Methoden der Approximations
theorie fanden 1971 und 1973 statt - der Schwerpunkt lag bei Fragen der
Numerik von Algorithmen zur Darstellung von Funktionen -, ließen aber
bereits ein wachsendes Interesse an Anwendungen erkennen. Die diesjährige
Tagung war gekennzeichnet durch die Behandlung praktischer AufgabensteI
lungen sowie durch die Einbeziehung der Anwendungen aus Nachbargebie
ten bzw. die Verwendung der Methoden dieser Gebiete in der Approxima
tionstheorie, insbesondere wurde auch auf die Beziehungen von Optimierung
und Kontrolltheorie zu speziellen approximationstheoretischen Aufgaben
eingegangen. Der starke Einfluß auf die numerischen Methoden zur Behand
lung von Differentialgleichungen wurde etwa bei der Methode der finiten
Elemente oder bei Kollokationsaufgaben deutlich. So ist zu hoffen, daß auch
diese Tagung dazu beigetragen hat, Theorie und Anwendungen wieder
stärker zu verbinden.
Die spezifische Atmosphäre des Forschungsinstituts stimulierte einen intensi
ven, durch die breite internationale Streuung der Tagungsteilnehmer verstärk
ten, fruchtbaren Gedankenaustausch. Zum Erfolg der Tagung trug - wie
immer - die hervorragende Betreuung durch die Mitarbeiter und Angestellten
des Forschungsinstituts und das verständnisvolle Entgegenkommen von
Herrn Kollege Barner bei.
Unser besonderer Dank gilt ferner dem Birkhäuser Verlag für die sehr gute
Ausstattung des Buches.
L. COLLATZ, G. MEINARDUS, H. WERNER
Inhaltsverzeichnis
ANSELONE, P.M., LEE, J. W.:
Double Approximation Methods for the Solution of Fredholm Integral
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
ARNDT, H., EICKENSCHEIDT, B.:
Zur Konvergenz von Splines .................................... 35
BARNHILL, R. E.:
Blending Function Interpolation: A Survey and Some New Results . . . . 43
BRASS, H.:
Interpolation und E III-Abschätzung . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
n
BRAESS, D.:
Über das Anzahlproblem bei der rationalen L -Approximation ....... 109
2
de BOOR, c., LYCHE, T., SCHUMAKER, L. L.:
On Calculation with B-Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 123
BREDENDIEK, E., COLLATZ, L.:
Simultanapproximation bei Randwertaufgaben .................... 147
CROMME, L.:
Bemerkungen zur Numerischen Behandlung nichtlinearer Aufgaben
der Tschebyscheff-Approximation ............................... , 175
DELvos, F.J.:
Approximative Flächeninterpolation ............................. 187
KAsMA, S., WETTERLING, W.:
Überbestimmte Eigenwertaufgaben .............................. 197
KAUFMANN, E.H., TAYLOR, G.D.:
An Application of a Restricted Range Version of the Differential
Correction Algorithm to the Design of Digital Systems. . . . . . . . . . . . . .. 207
KLOTZ, V.:
Ein Aufstiegsverfahren zur Konstruktion polynomialer Minimallösun-
gen in der komplexen Ebene .................................... 233
KRABS, W.:
Ein Kontroll-Approximationsproblem für die schwingende Saite ...... 257
LEMPIO, F.:
Minimumnormprobleme und zeitoptimale Steuerungen ............. 277
RIVLIN, Th. J.:
Bounds for Trigonometrie Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . .. 299
WHITEMAN, J. R., SCHIFF, B.:
Finite Element Approximation of Singular Functions . . . . . . . . . . . . . . .. 319
DOUBLE APPROXIMATION METHODS FOR THE SOLUTION
OF FREDHOLM INTEGRAL EQUATIONS
P. M. Anselone and J. W. Lee
A double approximation method with error bounds is
presented for the numerical solution of Fredholm integral
equations. The method embraces standard finite rank and
projections methods which, when applied to the Fredholm
equation (I - K)x = y, yield approximations (I-Km)xm = y
equivalent to matrix problems. However, the matrix
elements are integrals which almost always must be done
numerically. This is equivalent to a double approximation
scheme (I - ~)xm = y, where xmn is the solution com-
n n
puted numerically. Under typical conditions, K~ + K
pointwise as m, n + 00, K is compact, {K~} is collec
tively compact, and an available operator approximation
theory yields convergence theorems and error bounds for
the approximate solutions. Optimal choices for m
relative to n are also considered.
1. Introduction and Summary
Consider a Fredholm integral equation
(1.1) x(s) - ~Olk(S,t)X(t)dt = y(s), 0 ~ s ~ 1,
where, for the present, all functions are continuous on
closed domains. Later more general Fredholm equations
with discontinuous or singular kernels defined in Rn
will be considered, as weIl as extensions to nonlinear
problems. Widely used numerical methods for the solution
of (1.1) are based on the approximation of the kernel k
10 ANSELONE, 'et al.'
by a sequence of kerneis km, m = 1, 2, •••• Thus,
approximate solutions xm of (1.1) satisfy integral
equations,
m Jl m m
(1. 2) x (s) - k (s,t)x (t)dt = y(s).
o
If km is degenerate then the corresponding integral
operator has finite rank (finite dimensional range) and
(1.2) reduces to a matrix problem. Most projection
methods fall into this category. Further specialization
yields collocation, Ritz-Galerkin, moment, and least
squares procedures.
Consider (1.1) and (1.2) in operator form
(I - K) x = y,
on e[O,l] with the max norm. If km + k uniformly,
which is often the case, then ~ Km - K 11 + O. Standard
operator approximation theory relates the existence and
operator norm convergence of the inverses of I - K and
I - ~, plus error bounds. There are also comparisons of
eigenvalues, eigenfunctions, and other spectral proper
ties of K and Km. Particular approximation methods,
such as collocation or Ritz-Galerkin, require more
detailed analysis which utilizes results from the theory
of approximation of functions, e.g., polynomial or spline
approximation.
Numerical integration yields approximations to (1.1)
of the form
n
(1. 3) xn(s) - E wnJ' k(S,tnJ,)xn(tnJ') = y(s)
j=l
with, say, a convergent quadrature rule. Consider (1.1)
and (1.3) in operator form
(I - K) x = y,
The operators K and Kn satisfy the conditions:
Kn + K pointwise, {Kn} collectively compact, K compact.
ANSELONE, 'et. al.' 11
An operator approximation theory presented in [1] and
summarized in §4 below relates the existence and point
wise convergence of the inverses of I - K and I - Kn,
plus error bounds. Other results compare eigenvalues,
eigenfunctions, and other spectral properties of K and
Kn•
Since (1.3) reduces to a matrix problem when
s = tni, the xn can be calculated by means of various
matrix algorithms. This is seldom true for the approxi
mations xm in (1.2). Even if km is degenerate, the
coefficients and right members in the corresponding
matrix problem are integrals which ordinarily must be
estimated numerically. Unless k is rather simple,
efficient kernel approximation methods, including most
projection methods, produce approximate solutions xm
which in turn must be approximated. This means that the
theoretical convergence results and error bounds pertain
to approximate solutions which in fact are not available
computationally.
Numerical integration in (1.2) yields equations of
the form
m m =
(1. 4) w . k (s,t .)x (t .) y(s).
nJ nJ n nJ
When km is degenerate, this is equivalent to the use of
numerical integration in the matrix problem corresponding
to (1.2). Consider (1.1) and (1.4) in operator form
(I - K) x = y, (I - Knm )Xmn = y.
It will be shown that ~ ~ K pointwise and {K~} is
collectively compact. The cOllectively compact operator
approximation theory applies without essential change to
double sequences. It yields convergence results and
error bounds for the approximations x~ to x.
There are several practical advantages to the fore
going method. The approximations x~ can be calculated.
In fact, such approximations are obtained when (1.2) is
12 ANSELONE, let. al. 1
treated computationally. It is possible to infer the
existence and uniqueness of the solution x of (1.1)
and to estimate it directly from numerical results. The
error bounds depend in the correct qualitative manner on
the smoothness of the given functions. Perhaps most
important, it is possible to balance the errors fram the
kernel approximations and fram the numerical integration
so as to reduce computation. In fact, the error bounds
for IIx - x~1/ given below lead to asymptotically
"optimal" choices for m relative to n for many
practical numerical procedures. An important area for
future study is the extent to which practical use can be
made of these asymptotic results. It would be especially
useful to identify classes of kernels with associated
choices of kernel approximation methods and numerical
integration rules for which the asymptotic error estimates
provide nearly optimal choices for m and n over the
range of values commonly used in the numerical solution
of integral equations.
2. Notation and Conventions
The operator approximation theory to be applied will
have an abstract setting. Let X be areal or complex
Banach space and ß = {x EX: Ilxll ~ I}, the closed unit
ball in X. Let [X] denote the space of bounded linear
operators T: X + X with the usual operator norm,
" T 11 = sup{ I Tx 11 : x E 'l~}. Operator norm convergence will
be expressed by IITm - TII + 0 with the understanding that
m + ~ through the positive integers. Pointwise or strong
convergence, IITnX - Txll + 0 for each x E X, will be
expressed by T + T. Operator norm convergence is
n
equivalent to pointwise convergence uniform on 2 or any
bounded set. Any pointwise convergent sequence is
bounded and converges uniformlyon each relatively
compact (equivalently, totally bounded) set.
