Analytic Theory of Continued Fractions II: Proceedings of a Seminar-Workshop held in Pitlochry and Aviemore, Scotland June 13–29, 1985

304 Pages·1986·2.845 MB·English

by  Christopher Baltus, William B. Jones (auth.), Wolfgang J. Thron (eds.)

Lecture Notes ni
Mathematics
Edited by .A Dold and B. Eckmann
1199
Analytic Theory .
of Continued Fractions II
Proceedings of a Seminar-Workshop
held ni Pitlochry and Aviemore, Scotland
June 13-29, 1985
Edited by W.J. Thron
galreV-regnirpS
Berlin Heidelberg NewYork London Paris Tokyo
Ecl~r
Wolfgang .J Thron
Colorado, of Boulder of Mathematics, University Department
Campus xoB 426, Boulder, Colorado 80309, USA
(1980): Subject Classification Mathematics 30 B 70, 33 A 40, 65 D 99
ISBN 3-540-16768-4 galreV-regnirpS Heidelberg Berlin New kroY
ISBN 0-387-16768-4 galreV-regnirpS New Heidelberg York nilreB
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PREFACE
The success of the workshop held in Loen, Norway in the summer of
1981 (see Springer Lecture Notes in Mathematics No. 932) encouraged us
to arrange a second workshop in Pitlochry and Aviemore, Scotland in the
summer of 1985. Most of the organizational work was done by Haakon
Waadeland. Local arrangements were in the hands of John McCabe.
There were both continuity and progress in the topics treated at
the two conferences. In these proceedings most contributions fall into
two subareas. They are: convergence theory of continued fractions and
continued fraction methods in the solution of strong moment problems.
Under the first topic limit periodic continued fractions with n ÷ a -I/4
or ~ receive most attention. Modified convergence also plays an
important role. In the proofs element regions and value (limit) regions
are frequently used. Many of the element regions are Cartesian ovals.
Truncation error estimates are obtained whenever feasible. In the
second subarea Stieltjes, Hamburger and trigonometric moment problems
are studied and various types of continued fractions, useful in solving
these problems, are investigated. These continued fractions correspond
to power series both at 0 and at .~ The two-point Pad~ tables, known
as M-tables are analyzed for some of these continued fractions. Szeg~
polynomials coming up in connection with trigonometric moment problems
are orthogonal on the unit circle. In addition there are contributions
dealing with the location of the zeros of polynomials and multi-point
Pad~ tables and applications to special functions. Applicability of
results is emphasized in almost all articles.
There is one survey article in this volume (pp. 127 - 158). All
other papers contain original research. All contributions were refereed
and we appreciate the efforts of those who helped with this task.
Grateful acknowledgement is made for the financial support of the
Seminar-Workshop from a number of sources. Support for various
individuals was received from their respective universities and, in some
instances, from the Norges Allmen Vitenskaplige Forskningsr~d, the
London Mathematical Society, the U.S. National Science Foundation and
the Fridtjof Nansen Foundation. The latter organization also
contributed to the workshop as a whole. Finally, we would like to thank
Professor B. Eckmann for accepting this volume for publication in the
LECTURE NOTES IN MATHEMATICS.
CONTENTS
A family of best value regions for
modified continued fractions
Christopher Baltus and William B. Jones
On M-tables associated with strong moment 21
problems
S. Clement Cooper
A strategy for numerical computation of 37
limit regions
Roy M. Istad and Haakon Waadeland
On the convergence of limit periodic continued 48
1
fractions K(an/1), where n a ÷ - 4 " -- Part II.
Lisa Jacobsen
A theorem on simple convergence regions for 59
continued fractions
Lisa Jacobsen
Further results on the computation of incomplete 67
gamma functions
Lisa Jacobsen, William B. Jones, and Haakon Waadeland
Oval convergence regions and circular limit 90
regions for continued fractions K(an/1)
L. Jacobsen and W.J. Thron
Schur fractions, Perron-Carath6odoryfractions 127
and Szeg~ polynomials, a survey
W~B. Jones, O. Njastad o and W.J. Thron
Equimodular limit periodic continued fractions 159
N.J. Kalton and L.J. Lange
Continued fraction applications to zero location 220
L.J. Lange
A multi-point Pade approximation problem 263
Olav Njastad
J-fractions and strong moment problems 269
A. Sri Ranga
On the convergence of a certain class of 285
continued fractions K(an/l ) with n ÷ a
Ellen S~rsdal and Haakon Waadeland
A note on partial derivatives of continued 294
fractions
Haakon Waadeland
LIST OF CONTRIBUTORS AND PARTICIPANTS
CHRISTOPHER BALTUS, Department of Mathematics, University of Northern
Colorado, Greeley, Colorado 80639, USA.
SANDRA CLEMENT COOPER, Department of Mathematics, Colorado State
University, Fort Collins, Colorado 80523, USA.
ROY .M ISTAD, Department of Mathematics and Statistics, University of
Trondheim (AVH), N-7055 Dragvoll, Norway.
LISA JACOBSEN, Department of Mathematics and Statistics, University of
Trondheim (AVH), N-7055 Dragvoll, Norway.
WILLIAM B. JONES, Department of Mathematics, University of Colorado,
Boulder, Colorado 80309, USA.
N.J. KALTON, Department of Mathematics, University of Missouri,
Columbia, Missouri 65211, USA.
L.J. LANGE, Department of Mathematics, University of Missouri, Columbia,
Missouri 65211, USA.
ARNE MAGNUS, Department of Mathematics, Colorado State University,
Fort Collins, Colorado 80523, USA.
JOHN McCABE, Department of Applied Mathematics, University of St.
Andrews, North Haugh, St. Andrews, Fife, Scotland, KY169SS, U.K.
OLAV NJ~STAD, Department of Mathematics, University of Trondheim (NTH),
N-7034 Trondheim, Norway.
A. SRI RANGA, Department of Mathematics, Universidade de Sao Paulo,
Instituto de Ciencias Matematicas de Sao Carlos, Av. Dr. Carlos
Botelho, 1465, 13560-S~o Carlos, SP, Brazil.
ELLEN S~RSDAL, Department of Mathematics and Statistics, University of
Trondheim (AVH), N-7055 Dragvoll, Norway.
W.J. THRON, Department of Mathematics, University of Colorado, Boulder,
Colorado 80309, USA.
HAAKON WAADELAND, Department of Mathematics and Statistics, University
of Trondheim (AVH), N-7055 Dragvol, Norway.
A FAMILY OF BEST VALUE REGIONS FOR
MODIFIED CONTINUED FRACTIONS
Christopher Baltus William B. Jones*
Department of Mathematics and Department of Mathematics
and Applied Statistics Campus Box 426
University of Northern Colorado University of Colorado
Greeley, CO 80639 U.S.A. Boulder, CO 80309-0426 U.S.A.
.I Introduction. In the analytic theory of continued fractions,
value regions have played an important role both for convergence theory
and truncation error analysis. A sequence E = {En} of non-empty
subsets of ~ is called a sequence of element regions for a continued
fraction K(an/1) if n n, 6 n E a ~ .I A sequence V = {Vn} of
subsets of ~ is called a sequence of value regions corresponding to
E if
E
n
V and ~ n = 1,2,3,.
En n-1 I+V Vn-1' ""
n
If fn denotes the n th approximant of K(an/1) and
I a 2 a an_ I n a
Sn(W):= --] + ~ + "'" + 1 + l'+'W '
and if we assume that 0 6 V ,n n > ,0 then it can be readily seen that
fn = Sn(0) 6 Sn(V n) ~ Sn_1(Vn_1), n = 1,2,3,...,
where S0(w):= w. It follows that {Sn(Vn) } is a nested sequence of
non-empty sets such that, for n = 1,2,3,... and m = 0,1,2,...,
fn+m Sn(Vn) and Ifn+m-fnl ! diam Sn(V n)
In many cases, suitable conditions have been found for the element
regions E to insure that lim diam Sn(V n) = ,0 thus proving the
n+~
convergence of the continued fraction to the value f = lim fn [I,
Chap. 4]. Moreover, sharp estimates for diam Sn(V n) provide useful
bounds for the truncation error If-fnl of the n th approximant f n
,I[ Chap. 8]. Value regions have also been used to investigate the
numerical stability of the backward recurrence algorithm for evaluating
continued fraction approximants ,I[ Chap. 10].
For a modified continued fraction K(an,1; w n) (with reference
continued fraction K(an/1)), one considers the n th approximant
gn =: Sn(Wn), where the modifying factor n w may or may not be zero.
Since
*W.B.J. was supported in part by the U.S. National Science Foundation
under Grant DMS-8401717.
f = lim f ~ = S , (f,n,)f
n n
n÷~
where f(n) denotes the n th tail of K(an/1),
f(n) an+1 an+2 an+3 ...
I + I + 1 +
we see that, if n w is closer to f(n) than 0 is, then gn = Sn(Wn)
may be a better approximation of f = Sn(f(n)) than fn = Sn(0) is.
It has been shown (see, for example, [3]), that a judicious choice of
the n w can accelerate the convergence of the continued fraction; that
is, {Sn(0) } and {Sn(Wn) } satisfy
f - Sn(Wn)
lim - 0 .
f - S (0)
n÷® n
Value regions also play an important role in the study of modified
continued fractions. The purpose of this paper is to investigate value
regions for modified continued fractions, particularly best value
regions. In Section 2 we give a characterization of best value regions
(Theorem 2.1) and a result (Theorem 2.2) that is helpful in proving
that value regions are best. Section 3 is used to investigate a family
of circular value regions corresponding to sequences of element regions
that are bounded by Cartesian ovals. It is shown (Theorem 3.1) that
these value regions are best. Some interesting properties of Cartesian
ovals are described in Lemmas 3.3, 3.4, 3.5, 3.7, and 3.8. Examples of
ovals are illustrated by Figure 2. It is expected that the results of
this paper will be directly applicable to the problem of finding best
truncation error bounds for modified continued fractions. We now
summarize some basic definitions and concepts that are subsequently
used.
By a modified continued fraction we mean an ordered pair
(1.1) <<{an}l, {bn}0; {Wn}0 >, {gn}0 >
where the elements n a and n b are complex numbers with n ~ a 0, the
n th modifying factor n w lies in ~:= ~ U [-] and the n th
approximant gn is defined by
(1.2) gn := Sn(Wn) , n = 0,I,2,...,
where {Sn} is the sequence of linear fractional transformations
(l.f.t.) defined by
a
n
(1 .3a) s O(w) := 0 b + w, s n(w) :- b +w ' n = 1,2,3,...,
n
(1.3b) S0(w):= s0(w) , Sn(W):= Sn_1(Sn(W)), n = 1,2,3, ....
For convenience we denote the modified continued fraction (1.1) by one
of the symbols
(1.4) 0 + b K(an,bn; Wn) or 0 + b K (an,bn; w n)
n=1
The n th approximant gn may be expressed by the symbol
I a 2 a an_ I n a
(1.5)
gn = b0 + ~-b + bq + ''" + bn_ I + bn+W n
A modified continued fraction (1.1) is said to converge to a value g
if its sequence of approximants {gn} converges in the larger sense
to g in ~.
Every modified continued fraction (1.1) has an underlying
reference continued fraction
a I a 2 a 3 a
(1.6) 0 b + : 0 b ÷ + bq + j--b +
n
The n th numerator A and denominator B of (1.6) are defined by
n n
the difference equations [I, (2.1.6)]
(1.7a) A_I:= ,I A0:= b0, B_I:= 0, B0:= ,I
(1.7b) n A bnAn_ I + anAn_2, n 1,2,3 .... ,
(1.7c) Bn = b B n n-1 + anBn_ 2 ' n = 1 ' 2,3 '''"
In terms of A and B we have the well known expressions
n n
A + wA
n n-1
(1.8) Sn(W ) - B + wB , n = 1,2,3,...
n n-1
and the determinant formulas [I, (2.1.9)]
(1.9) AnBn-1 - An-IBn = (-1)n- I n E aj, n = 1,2,3, ....
j=1
For m = 1,2,3,..., the m th tail of a modified continued
fraction (1.1) is defined to be the modified continued fraction
® = ® (m)}~+1 >
K (an,bn; Wn)= <<{an}m+1, {bn}m+1>, {Wn}m+1>, t, g n
n=m+1
The O th tail of (1.1) is defined to be (1.1) itself• The n th
approximant g~m) of the m th tail is given by
I( 10a) ur.)0lg S~0)(w0 )''v _(m) = s(m), ) m,n = 1,2,3,...,
• "= Yn n ~Wn+m '
where
(1.10b) S(0)(w): = S (w), n = 0,1,2 ....
n n
am+ I am+ 2 am+n-1 am+ n
(1.10C) s(m)(w): = - -
n bm+ I + bm+ 2 + + bm+n_ I + bm+n+W '
m,n = 1,2,3, ....
It can be seen that
(1.11) Sm+n(W) = Sm(S~m)(w)), m,n = 1,2,3 ....
and hence
(1.12) gm+n := Sm+n(Wm+n ) = Sm(S(m) n ( Wm+n )) = Sm(g~m) )
m,n = 1,2,3, ....
If an m th tail converges to a limit g (m):= lim _(m) then all tails
~n '
n+~
converge and we have
(1.13) g:= lim gn = Sm(g(m))' m = 1,2,3,...
n+~
Our attention in this paper is restricted to modified continued
fractions (1.1) where 0 b = 0 and n b = I, n = 1,2,3,...; that is,
modified continued fractions of the form K(an,1; Wn).
A sequence E = {En} I of subsets of ~ such that
(1.14) E - ]0[ ~ ~, n = 1,2,3 ....
n
is called a sequence of element regions for a modified continued
fraction K(an,1; w n) if
(1.15) n a E En, n = 1,2,3, ....
A sequence V = {Vn} 0 of subsets of is called a sequence of value
regions corresponding to a sequence of element regions E = {En} tt__a
W = {Wn} if
E E
n ~ and n
(1.16) 1+w Vn-1 ~ ~ Vn-l' n = 1,2,3, ....
n n
We denote by (E,W) the family of all sequences of value regions
corresponding to E at W. The following statements are readily
verified:
E
(1.17) [Wn_ I E Vn_ I and __n_n ~I+V Vn-1' n >_ ]I => {Vn}E F(E,W),
n
(1.18) {Vn} E F(E,W) => {C(Vn) } E F(E,W),
(1.19) (~)} 6 [~v v(F,w) E ~ A]--> {N (~)} V C ~(E,w)
" n ' A n
Here c(V ) n denotes the closure of V . If
n
(1.20a) {Un(E,W) } 6 V(E,W)
and
(1.20b) {Vn} 6 F(E,W) => Un(E,W ~ ) Vn, n = 0,1,2 .....
then {Un(E,W) } is called the best sequence of value regions cor-
responding to E at W. We shall denote the best sequence of value
regions corresponding to E at W by
U(E,W) :: {Un (E,W) }
In Section 2 (Theorem 2.1) it is shown that U(E,W) exists and is
unique. Theorem 2.2 provides a set of sufficient conditions on {Vn}
to insure that
(1.21) n = V CUn(E,W), n = 0,1,2, ....
This result is used in Section 3 to establish (1.21) for certain
sequences of disks n V and Cartesian ovals n E (Theorem 3.1).
By diam S and c(S) we mean the diameter and closure,
respectively, of the set S.
2. Best Value Regions. Throughout this section we let E = {En}
denote a given sequence of element regions and W = {Wn} a sequence
of modifying factors for modified continued fractions K(an,1; Wn).