Constructive Theory of Functions of Several Variables: Proceedings of a Conference Held at Oberwolfach April 25 – May 1, 1976

297 Pages·1977·3.71 MB·English

by  Prof. Dr. J. Albrecht

Lecture Notes in
Mathematics
Edited by A Dold and B. Eckmann
571
Constructive Theory of
Functions of Several Variables
Proceedings of a Conference
Held at 0 berwolfach
April 25 - May 1, 1976
Edited by
w.
Schempp and K. Zeller
Springer-Verlag
Berlin' Heidelberg· New York 1977
Editors
Prof. Dr. Walter Schempp
Lehrstuhl fOr Mathematik I
Universitat Siegen
H6lderlinstra6e 3
5900 Siegen 211BRD
Prof. Dr. Karl Zeller
Mathematisches Institut
Universitiit Tubingen
Auf der Morgenstelle 10
7400 TObingen lIBRD
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:0.0.3.126 DC. 511 (W31.,1 51O',S. ('l:i'.84] n·l912
AMS Subject Classifications (1970): 41A63
ISBN 3·540-06069·4 SpringerNeriag Berlin' Heidelberg· New York
ISBN o-387-{)6069'4 Springer·Verlag New York' Heidelberg' Berlin
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Preface
The conference ·Constructive Theory of Functions of Several
was held at the Oberwolfach Mathematical Research
Variables~
Institute, April 25 - May 1, 1976. It was attended by 46 mathe
maticians from seven countries. These proceedings contain most
of the papers presented during the conference. The topics
treated cover different problems on multivariate approximation
theory such as new results in the theory of spline approxi
mation, multivariate interpolation, cubature, poSitive operators,
intermediate spaces, special functions of mathematical physics,
etc.
wo take this opportunity to oxpross our thanks to all those
who participated in the conference or contributed to this volume.
OUr particular thankS are due to Prof. Or. G. Meinardus (Siegenj
for his interest in this conference although he was unable to
participate, to Dr. F.J. Delvos \Siegenj , who did a large part
of the editorial and organizational work, to the Oberwolfach
Mathematical Research Institute for financial assistance and
for the facilities provided, and to the staff of Springer Verlag
for their courtesy and valuable co-operation.
September 1976 Walter Schempp, Slegen
Karl Zeller, TUbingen
CONTENTS
J. ALBRECHT and H. ENGELS
Zur numerischen Integration Uber ~reisbereichen .... . . 1
B.O. BJORNESTAL
Stability of Steiner Points 6
K. BOHMER and GH. COMAN
Blending Interpolation Schemes on Triangles with
Error Bounds 14
J. BOMAN
Comparison Theorems for Generalized Moduli of
Continuity. Vector-Valued Measures ..............•.... 38
F.J. DELVOS and H. POSOORF
N-th Order Blending ..............•....•••............ 53
B. DRESELER
On Summation Processes of Fourier Expansions for
Spherical Functions ,................................. 65
J. DUCHON
Splines Minimizing Rotation-Invariant Semi-Norms in
85
Sobolev Spaces .................... ..............•....
W. HAUSSMANN and P. POTTINGER
On Multivariate Approximation by Continuous Linear
Operators 101
K. JETTER and F. LOCHER
A Note on Numerical Fourier Analysis and Uniform
Approximation on Cubes •. " .........•.. , •.•.•.•..••... 109
H. JOHNEN and K. SCHERER
On the Equivalence of the ~-Functional and Moduli of
Continuity and Some Applications ...............•...•. 119
T. KOORNWINDER
Harmonics and Spherical Functions on Grassmann Mani
folds of Rank Two and Two-Variable Analogues of Jacobi
polynomials ••.••..•............•............•........ 14-1
y,
H.M. MOLLER
Hermite Interpolation in Several Variables Using
Ideal-Theoretic Methods .. , ............••...... ..•.... 155
W. NIETHAMMER
On the Numerical Analytic Continuation of Power
Series 164
K. REIMER
Clenshaw Sums in Several Variables................... 176
A. SARO
Function Spaces for Analysis 186
W. SCHXFER and W. SCHEMPP
Error Bounds for Bivariate Spline Interpolation.. .... 196
W. SCHEMPP
Bernstein Polynomials in Several Variables ......•.•.. 212
W. SCHEMPP
Approximation in G-Homogeneous Banach Spaces ......••. 220
H.J. SCHMID
Interpblatlon of Harmonic Functions . ..•...•.•.•••.•.• 226
H.S. SHAPIRO
Convergence Almost Everywhere of Convolution Integrals
with ill Dilation Parameter .•......•..•.•••.•••.. •... .. 250
D.O. STANCU
Use of Biermann's Interpolation Formula for Con-
structing a Class of Positive Linear Operators for
Approximating Multivariate Functions . ......•..••••... 267
w. TREBELS
Estimates for Moduli of Continuity of Functions Given
by their Fourier Transform ...•.... ••. •••••..••. .•.... 277
List of Authors .... .••••••• ••••... ••••• ••••••• ••..... 289
ZUR NUMERISCHEN INTEGRATION USER KREISSEREICHEN
J. Albrecht und H. Engels
Der Gau8'schen Idee gemaa ist man auch zur numerischen Integration
Uber Kreisbereichen
N(n)
J J u(x,y)dxdy=w ~ a u(x ,y ) V u(x,y)cP4n_l
X 2 +y 2 ~, " =1 " \I \I (n-':2 :3,4 :5,6,7,8: •.• )
bestrebt'), minimale StUtzstellenzahl N(n) zu erre1chen, jedoch so, d~a
2
x +y2<1 und a >0. Ein1ge Kubaturformeln mit diesen beiden Eigenschaften,
" " "
bei denen allerdings offen ist, ob N(n) minimal ist, verwenden die Ecken
regelm~8iger Polygone al5 5tijtz5tellen:
Nr(n ) nr
". . , a "u (x ", y ") -,,-15 4 54(0'4 )
,\I .V
mit
52 (0')- 2rm u(O'cosJl"- ,,,sln~-II1 (2m=4)
m \1=1 m m
(2m_4;8;16)
bezeichnet die Menge aller Polynome vom H5chstgrad 4n-1 .
,
un"
, ,
•
n 2 ] 6 7
•
12 28 76 108 1<0 17'
Lit. [,] [ .]
2
Zur Herleitunq der Formeln: Aus den Taylorentwicklungen )
r
f J -:-k , P
u(x,y)dxdy:., p! (P+1) ,6. u(O,O)
22 p=02 P
x +'/ !,.1
,
,/p p-lm
(p-l.Jn)! (p+lml! 1121m u (0,0)
22p
,
1 2p r... 1 ~r 2p p-l~
-22p -p'I-pll!P u+2 1=1( -1) palm ,2p (p-lm)T! (p+lm) IliUm (0,0)
ergeben sieh mit den Abkilrzungen
(n-' ;2;3,4 ;5,6,7,8; ... )
n-,
1, r 2p
,pi" 4 v- l t 4,\1 t 4,v (n=2;3,4 ;5,6,7,8; ... )
(p-o(1)2n-l)
r2 ,p -8 nvI-- 2l t 82p,\ 1 t 8,v (n'"3,4;5,6,7 ,8; . . . J
n-'
r 2p
4,p- 16.c"L , T16,v t 16,v (n-5,6,7,8: ••. )
2JOie Beweise der Taylorentwlcklungen von S2m(a) und T (t) sind
2m J,
- elnschlleBlich der Definition der Operatoren lI~q - in [, [2]
zu finden; der KUrze halber wird hier die Konvergenz dcr Taylor
[11,
relhen vorausgesetzt (Vgl. [2] 1.
3
durch Taylorabgleich folgende Bedinqung~n:
10 IP+ 1, ,p+ 1, ,p +1, ,p+ ••• - ..', (p-o(,) 2n-1)
IQ,p-L"p (p-2 (') 2n-1)
• 0
Lo,p+I"p-I ,p
(p-4(1)2n-l)
2 • 0
Lo,p+I"p+L2,p-I4,p • 0 (ps6(1)2n-l)
(n-, ;2;3,4;5,6,7,8: ... J.
Aus Ihnen tolgt, dae die
Abstandspar~ter
Nullstellen bestimmter Polynome
(n-1;2;3,4;5,6,7,81
(n"'3,4;5,6,7,8)
(n-S,6,7,8)
sind und daB (nach daren Berechnungl die Gewlchte 54,v' t 4,v' ts,v'
t 16•v LQsungen linearer Gleichungssysteme sind. &1n1ge Beispiele:
, ,
PO,2-S2,04 -54o4+9
P",-4T4-3
1
0,0" 2"1
Io.,· ;i
•
PO,3a6317094Q:-l00222450!+4149900o;-336375
• 2
P,,2-1S3T4-2S0T4+15
2
P2,,-6Ta-S
\' 168899 " 98617
l.o,o· 337500 L, .0" 331500
\' 7661
"0,,· 45000
PO,4-220SBOO8a!-47224002a:[phone]a!-73827000;+352125
, • 2
P,.3-392T4-763T4+450T4-75
• 2
P2,2=28Ta-42Ts+15
r 5057 t 9'J:l7
o•o• T'25oO [.1,e'" 337500
'\' 6389 '\' 111541
La.,· 45000 [.1,,""35000
\' 193
[.1.:'= ~
Die Ergebnisse fUr n-S,6,7,B wurden mit Hilfe der Programmiersprache
FORMAC errnittelt [3J; aie wurden auf andere Weise auch von A. Haegemans
berechnet [4J.
5
Literatur
Albrecht, J., Collat2, L.: Zur numerischen Auswertung mehrdimen
sionaler Integrale, ZAMM 38 (1958) 1-15.
2 ALbrecht, J.: YormeLn zur numerischen Integration Uber Kreisbe
reiche, ZAM.M 40 ( 1960) 514-517.
3 Engels, H. • Numerical Quadrature and Cubature, Academic Press,
London 1977.
4 Haegemans, A.: Circularly symmetrical integration formulas for
two-dimensional circularly symmetrical regions, BIT 16
(1976) 52-59.
5 Ham~er, p,e. , Stroud, A.H., Numerical evaluation of multiple
integrals II, MTAC 12 (1958) 272-279.
6 v. Mises, R.: Formules de cubature, Revue mathematlque de l'Unlon
Interbalkanique, Athen 1936, 17-31.
7 Mysowskich, I.P.: On the construction of cubature formulas for the
simplest regions, Z. Vyelsl. Mat. i Mat. Fiz. 4 ( 1964) 3-14.
8 Stroud, A.M.: Approximate Caleulat~on of Multiple Integrals,
Prentieo Hall, Englewood Cliffs 1971, 4315.
Prof. Dr. J. Albrecht
Technische Universit~t Clausthal
Institut flir Mathematik
Erz:straBe 1
3392 Clausthal-Zellerfeld
Prof. Dr . H. Engels
Technische Universlt~t Aachen
Institut fUr Geometrle und
Praktische Mathematik
Templergraben 55
5100 Aachen