ìI¡\'II-IEIL'\TIC,\
-
I'.EYUE D''\NALYSE NUIIEIìIQUE Iì'I' DE TI{ÉORID DE I,'i\PI'ROXII{;\TIONL'ANALYSE
NUTÍÉRIQT]II TìTLå îHÉORIE DII
I,'APPTìOXIT'IATION 1-oure 16,N" l,
1$87, pp.55-G:l
Ä NEW OPEIìÄTO}ì OF I3EIìN-STIIIN TYPIT
(;, ll,ESTÌìO]ANNl ancl lI. lì. OOCOP.SIO (Napoli)
, Abstract, A positivc lirrear. opelator is oo[sietelecl rvhich geuet'alizes Bernsteilr's operator'.
'I'he autho¡s giyc árr eirsy coustluctive lrloccss and LhcS' pl'o\¡e some ll.tot.tototl¡' properties al1cl
cortvexiLy pleserr.ing. Soittc convergonce tlteoLctns at'e givelr alrcl estiutates of the reitlraucler al'e establishecl.
l. Intr.otluctiol. It, is
$'ell-lirt0\\.n tha1,to evcry funclion /
clefinedon the interval 1 : [0,1]
one c¿ùn associâ,1,cthe
Be,rnstein polynornial of clegreerl
:(B,,fl(n) : B,,(l(t) i n) : Ì),,(; a) : y,
tr*,,{*) J(ilnt'),i:0 lVhere
pu,,i(n)
: (';) ,',t -
n)n't-t.lt; is
also rvell knorvn\haL
13,,,is
alinear
ancl posil,ivefor
every function/
which is continuous on -¿(/
e C'"(f ))'iint B*(!i n) :
l@)()p
it
eI'Arl,Ol' and 1,hal, lesultsthat
uniformly or
.f.In
spiteof their
slorv couvergelce,,lì
h'ave been(anc1 l,he¡i
still'are) a
sLrbjectof
stuclyby can
quot'eàlroou
ali:
Rernstein[3],
Berens[2], L,or: 1Bl,
Popo-viciu [18],
Schoenberg[20],
Stanc;.t 127, 221.A lot of
propertiesof
great theorel,ical anclpractical
inl;e.restof
the,Brrt o
ha,\¡e shorv "'lîî5,,iåi'#3ïi"?å.Tiä$** I the
stuclicsof
Bashalrov[1]r I(ö-
¡-ä.,Tning-zeltcr
[9-1, sz,asz[26],
anclthose of stancu, frorn l'hich
spccialmention
cleservc 123, 24, 2Ú).A
conrplei,e ancl upclal,ccl bibliggr':r,ph¡'ou l,hc srrlljeç{, is givcn i.n [7],A NIIW OPER.\'TOR Ol| BIITìNSTEIN 1'YPÉ ir/
56 G. MASTRoIANNI and M. R. occoRslo
'I'ho properties
of i-th itctates of ü,,
defirtecl by(1.1) Bi,ç;,) :
_å (;) Bt;, (e^; fll\I,¡,Ï((\, i > I
¡'lrere
e^(u): rr ald
lJc: r
isthe
itlcntica,loperiltol
ale also n ell-k¡oy,n[9,
16 ].Suitable lincar
conrbinationsof the llcrnstein polynornials
have been sl,udiedtoo;
\\¡e c¿ùnlecall
â,mong thern thoserrhiclì
aie containecl inthe
rvor'lisof Butze¡' [4],
_o_fFrentiu-[01
a,nclof lla;,
[11].Ilicchelli [15]
anclllastloianni-Occolsio ¡fZi
irìtr.otuced ¿r,nd stu- clieclì even if_ sepzllzrtelv,tlte folloving
cornbiriatiónsof
itcrlatcsof
groRernsl,ein pol"r'nornials :
(1,2) tr,n,r$; u) : Ë (-t)'-' ,l')
ni,,1.¡,,r¡.
The last one has
thc
ploperl;r'of imlrroving'the
approxirnationto
incleasc thc_apploxirnatingfunction's leuula¡it¡'.
Iírdeed, ^i-nttre
abgye l¡entionecl rt'orks,il,
haSbccn
detnonstlatclclthai the polj'nonrial
clefinedì:v
(1.2) a,pproxirnatesa
func!,ior1,f e-c2k(r^)
rviur a
r'ônräinclc,r,of o(n-k) tr:pô.
rú has a,Iso been shos'nin
[rì] that florn all
thc¡ cornbinationslike
(1.3)
i:lf ,r,.,Iii,(l;r,), É tr¡,,¡:I
i:l
1ìre lrol¡-tromi:rl (1.2)
is the only
oneto
pclsscssthe
aboyc-¡rentione¿ pr,o-pelt¡' of
conr¡Lìr'gence.[12
] ]Iastroianni
ancl Occolsio lenr¿r,r'l<ec1that
a possiblc goncr,¿ì-lizat'ion of (1.2) could
be(1.3)
ut,,,.^U'; u:): i (- r)'-' (';) n,,rt,,rr,
x-il,h
Àreal
posil,ive number,.The airn of this
x'ork,is to
carl,y out, afilst
stucl¡- ofthe
Br??,7, olle.l,¿ì.tor.which, for' À e (0,11, r'esults positirie anrl pleselves rnan¡,
lrrop"iii.*'of tlió Belnstein
opelator._
f1
S_2, l'e shall give
ur,7.¡âltd in
,s ¡3n'e shall cìemonstrate
sotre
ðonyexit¡, pr,e-sert.ation.
Finall¡', irr
$ 4.. cc of
11,,,,'f6r,'t11, --+
Ø
anclÀ+oo separ,at
atesof t'liò
r.e-maincler'.
2. th9
Zl'*,i operator. l['o ever'¡' ftrnction/
rlefincrl ancl lirnil,eclin f,
rve
associatethe
continuousfunction
8,,,;,1,(Àe.E+) which is
definedthlough the
selies:e.r) 8,,¡$;
c)): Ë (-t)'-' (!,) o,"rr,
O,ru'helc /Jí,
is
the i-1,h itcral,cof thc
Bernstein opcrator' .I3,,.It, is
casy to yerif vthat, if
),is an
inl,eger.posilive unmbcL,
(2.1) gives back (1.2) anclin pilrticLrlar
it, result,s tha,t, 1ì,,,,,- I)*.
l,lorcover,for every
À eE+, \l'c
h:lve :8,,,¡.(f ;
0) :
/(0); l],,.¡.(l;
1):
/(1)we s,ra, -,""";"1"';î;Jl": ;::",,, ,,ct ß,,ir
'Io this
aim,l'c
dcnol,eby Sj
and s.l:, respeetivel¡;,Stiiling's
tìum-bers of first
ancì seconclt¡'pe rlefinctl by
: 2fr(nl:-fi
Now, we
set, :fr-
1 nl,)(
n-l ^1t-j
I
^Sî+;
e:(1): I,
r:qt: r;
i:t) 'll L'
n-l
')tì,
üt:
10
fl
lrz)(n,
u',
. .., r") 1
et,¡,:
(0, il-l^(n-jt
ø"
- I r=0
sî-Y
'ttù'.'--,
I
sllm01
,
O,l)r
e IlÁ 1 sfllnt,2.,. sf,-rlm"-'silrn .,.I'i_zlnt"
"0 0 0 0
0
Â,,
: ñ
00
1(¿)00
1Uren
let
IZ¡.lle the
/cX h matrix having for columns thc
autovectorsai(i: f¡t¡ of the matrix Âr8r,
normalized so asto
allorvthat the
ele-ments of the principal
diagonalof Ir¡ result
equalto
1.ft is
obvious th¿r,t^rs/,
I/r : I'r1'r.
lhcnce,
(2.2) (Â,S,¡i :1tr\.l[t1.
ìforcot'err
lot, us observethat
I/41: tlT,
ryherc(ÄrS,,)tU¡,:A¡,L¡,.Then, it, is
easyto
shorv(see[8])
that,(2.3) B'--t(er) n) : n¡VJ\|,
1V;1u,r.So, siuce (2.1),
by
(,t.1) anct (2.3),it follol's
that,(2.4)
B.,,ic; r) :
å (i) *,rrli r-tl,(, i ,) ;rifr;,tt,,ntr,,te).
No\r', \e
observe thai,æ
(2,5)
i:0E
( 1¿-l-1
Àf : -!-
tt1 -
(1-
,)^,1::^¡(t, À);f
e (0,11, À eR*
i58 G. MASTROIANNI âncl M. R. OCCORSIO
from
1(i)< 1, (i :1,,k),
we h¿ù\¡e:Ë i-tl' f 1. ) nl
G,,,¡,i:o \?-l r,/
'lvlrere G¡,7.
is the
cliagonal ma,t,r'ix nhose clemcnts iùre 1(1{'), '),),i:7,
kfn this rvay
(2.4,) bccornes(2.6)
5t,,,¡.(l;-ù :
ä('I¡
,,,,,,,r*, À)^l;,,/(o),
where(2.7)
(l ,,,t(fr¡ ),)-'.
ït,VìG¡,.,i.V ¡ltt¡,.It is
easyto verify \}lnl
q,,,¡(r,l,) is a polynomial in ø of
clegreenot
greaterthan /¡.
So,frorn
(2.7),iL
follox,s t,hab 8,,,7..fis a polynomial
of degreenot
greaterthan
rn.Fulthernlore, if
'¡t,,is à
polynomial of clegreonot
greaterthan
n,(
,nz,lhe
same happensfor
Bn¿,¡ipn.Another
representationof 1ì,,,¡/
is(2.8) ß,,¡.Q'; ù : f;
F,,,¡(n)s(A:lm),
:l:0 tvhete
(2.e)
s(n): j t-tl,(,
ì ,) ai,U; n); (ßi,J :
f1.By casy
oalculations,r'e obtain in particular
:from 'whiclt rve
deducethat
R,n,t,(et,i
n) : nrVr.ArV;ttt,r,
er(ø):
çþ,where
-4, is the
diagonalmal,rix
lvhose elements are:
[1-
1(Ð]i.For
example, wehave:
(2.10)
E,,7,(ezin) :
-8,,,¡,((ú --
n)2;n):
--n(1.-
n)lm,^(2.1t)
B-,t.((t- n"; n) : - #[r - (u - +l]
Let
us now observethat Bn,t¡ (^
€ -R+), isnot in
general apositive
ope-rator; it is
easyto verify
t'lnat 8,,,,¡is
positivèif 0 <
À< 1'
n'or these valnesof
)',, I?'^,¡ results as a positive operatorof llernstein
type.!'inally,
notethat
13,o,7,with 0 <
À{
1,is a particular
caseof
the operator :n!,,,1
:,å,,-1)n+1+i
1r)r;r, n1ì, 1nrr, neN.
It
isnot rlifficult in any
caseto verify
that, verge to ./.Next,
weshall
shou'that
B-,7. isühe
{n,î,,},'€N
ela,ss convergingto /.
,
R},,,.f does not con-positive
operator ofll.
Onthe
properfies oI -B,,.,, (0<
À<
1).The following
propositionsare
true.Pnoposrt'rox 3.T Ttte operator Iì^,7, keeps the conuenity (coneaaity) of
e,Derl1 order
of
thafunction f.
Incleed.,
from (2.8) we
cleducethat
(B-úI!@) :rØ)'i! n--,,^(*)l ! , u-*--t-, ..., Wrnf, p <nù,
pl k---o^ ' lm ,n m
Jl\{oreover,
if / is
convexof
orclerlt - 1, by
(2.9)it
follorvsthat the
samehappens
for g; so the
assertionis
proven.Pnoposrrroir
3.II lor
eaery ø e(0,1), the
relation'
B,,,+t,¡!ltt) - Il*,¡,ff j n): * n(\ - n)lnt,nz¡
ne)fl(];^-
ø*rrt)
hold,s, uhere æ¿,
(i:\,2,3),
are tltree suitablepoinls
of L Tncleerl,by (2.8) ve
haveA*(l;
n): : B-*rt(l; n) -
R^,t.(f ;n) :
So,
the
operatorá,,
has d.egreeof
exactness 1,4 5 A NEW OPERATOF, OT'BËRNSîËIN TYÞE 59
for r¿>0 the
only8,,,$
)n) :/(0)
-r- 2nr\,,¡,!(0).
[(Ui - 1) ' + z(r -;r) *dfq,,¡10¡;
Br,^(.f t
n) :/(0) +
3nt\,r,r/(0) -F,
[+ (-O * å), *
*+ (,' -,i) ,'fni,,¡ço¡
' [(' i (Íl- ¡it)) , -
; i ( z t- |;:- , ;l) ",'
.n+ (' - ;:)] æ'Lt,te)
Tlren,
let
l?,,,¡- I -
7ì,,,,7. bethc
remainderterm,
rvheleI is the
iilen-tical
opelator. JMc casily have :Bm),:
R!,,:
(1-
B,,)À: î t--rl'(1) r;,.
I
60
Furbhermole,
if / is
convexof first
otclel, grvill
be ¿r,lso so; in
thisciùsc, \\¡e havc ,4,,,(/;
o) + 0 fol
ever.v a; e (0,1).For
Popoviciu's theoremf19],
1,hree pciirt,sntt
nz,r.
e (0,1) e.xist, suchthat
4,,(l; n) :
A,,,(er) m)l,rt,
fi2, ßs)l).
fl'hen, from (2,8) it, follol's
1,bat,A,u(er)
r): - ï(7 - ") l-l - -] I ttt)'
(nt)- . I'
Ð| )
ancl so
the
assertionis
proven.I[oreover, from
Proposition3.If, l'tr may
claimConor,r,lny
S.IIL 1/
the Junct'iott,f
is cotlneü oJthefirst
order,then,for
eaeml ø e (0,l), it
results tJtutB,,+t,t(l; n) < 8.,r.(l) r).
'Since
B-,¡
has clegreeof
exactness1
anclby a theorcm u'hich
has been provettin
[13], the foilos'ing
represental,ionof the
remaincler(3.1) t?,¡.Ui ;') : -1"(t), {
e (0,1),holcls
; b¡'
(3.1),tire
o.l,hcr one canbe
cleduced :(3.2)
Il,,t.(.f; n').- -- "It^,-i)-lrr, Í2,Í¿ill
,ìtrÍr,
Fr,f;,
bcing.suitablepoints of
(0.1).4.
Onthc
eonvcrgenceof f3.,7,. Firsl, of all, let us
suppose that,0 <
),(, 1. In this
case, B¿,,¡verifies l{orovhin's
conclitions, and. conse- cluentl¡'the
secluence {Iì,,,i..f}
convcrgcsuniformly to / on 1. For
someclasses
of
functions, [ìr tnearsule o1' ¿pp¡oximationis
given b¡'TnnonnM 4.1.
lor
euery conti'nu,ou,s Ju,nctíon,f
a,ntl etser1,l À e (0,1) i¿resulls tltat
(4.1) lll-B..,.lll < ¿co(f;1lnr\r'¡'
(4,2)
ll,/- 11.,,,/ll < i
cù', tz)
, J'
ec"(r),
il1,'t "
ultere
tl¿e norn't,is the
u,ni,fotttt, one.To plove
(4.1),l'e
observe tha.bif or(/;)
clenotosthe
nodultLs ofcontinuity of the function /, it
lc'sLrlts thal,'(n
- t)'
Ò'
So, we have:
ll@) - B^,t$; n)l < B.,x(ll@) -
/(¿)l ;ø)
<< B,,r(.(.f ;
l¿ t-- ol;
't,n) * lt * ).^,(f
; A).\ )
Ilence, by
(2,11), n'e deduccthat
l/(ø) - ß*^U; r)l < (1 + ),,t' à), 8Þ0.
In partictrlar, for
8:
e¡-)'tz, (4.1) follorvs.To prove
(4.2), we observethat
A NAW OP¡]RATOR OF BÐÍÌNSTEIN TYJE 6l
ll'&t) -
f'(r)l
d,u:
|î'(a) (t- *) f fr(t, *),
G, MAjSTROTANNI âNd M. R. OCCORSIO
-/(ø) +
/(¿):
J'@)(t- n) +
(4.4)
(4.5)
for
everyn,teI.
So, we
hal elx(t,u) < l¿--rul.("f ilt-øl) < (f,-rt +('-/t).ff;à), ùÞ0,
ll@) -
R,^,t.(!,ü)l : lB^,i!U(t,
ø) ; s)I
<.V -q!#L ('* *
V-sfLrll m^l
o,r("f' ; ù) ;.,ll
lltt'
-trom which,
for
à:
,¡n-)'tz, we have (4.2).thc foregoing
incclunlitiesgive bach, for
À: 1r the
well-knowarelations
correspondingto
1,heBernstein
polynomials.Norv,
let
À>
1.'Ihe
caseof
À as an integer number greaterth*n
l.har
alreatly been stuclietlin
tlei,ailin [12];
so,let us
supposothat
À* : i +
1,, $'here rlis thc maximum
intege,r number suchthat
le<
À aqd-y e (0,1).
ltmonnu 4.II. Let
7':
l,;| .¡
whereh í.s un
dntegørnwnbqr
elní), e (0,1). IJ
f
e Cz*+z(l), thenllw folluning
inequ,øli,tyllr - B,,,lll *
-t-l*!,+|,Ir,oltls, ttth,ere lllllzr
=-r?,1ä lll,t)ll ünd c is a
constant d,epe.nd,ingon h
snil independantol I and
m.fn
f¿sli,by
(3.10), we haselH^,t(j;
ø) I: l&*,r(R^,r|; r)l
<lIft?,,fl,:ll
Emr
o\f;ln-rl)
<1+ .("f ;
à),fot
etery n,
t eI
anrlfol
ever-y ò>0
a
{ì A NE\\¡ OPT'RA1I'OR OI¡ RIARNSTEIN TYPI' 63
['1 ] l). L. ll rt t z c t', I'irtcur aontbínalions ol' ßornslcitt polyrtotnitils, Canarìian J. of. l\Iath.,
rr (19ir3).
[5ì G. I" c ] lt c c l<cr, I'ittcrul:otithi¡tctliottc¡t uon ilcLiulett Bentslein-operaforen, ftIanuscripta ì\Iath., 29, 229-248 (1979).
[6] Àf. Iì r'o n I i n, C)otnbinalii líttiare <Ie polittoame Rernslein çi de opetaloti ÀIírakgan, Stndia l.irir'. 13abcs-Boljai, Scr. i\fath. - trfccll., 15, 63-68 (1970).
[7] Ir. tt. conslia and J. ltcicr, .'lbit:lioyapt4¡on(LpptoÌimalionof funclionsbg
_ ller¡tslcín Iype operulors (10i5-1981), Approxinalion'lhcory, lv, 739-?5g (1gs3i.
[8ì l),. l). I( c I is li ¡' nnd 'I'. .I. R ivI i n, Ilerrtlrs of Ruttstein pol¡¡rtomictls, pacitic j.
llrrtlr., i)1, (3), I-r11- ir20 (1907).
[0] ì(ötrin¡¡--NIcvct'uncl Zcllct', .Bet¡tsLcÌnscltcpolutzrciltn, Sturl.Allth,(1g60).
[10] C], (ì. -t, o l o tr L z, llcrttslet¡t ¡>olr¡trontit:Lls, Ljlrjr,, oI 'ì'olonlo lrr.ess, Tor.t¡rto IiOSS¡.
[11] 0. l). À[ a¡', Sn/ttt'rtliort tLttd ittucrsc llteorcnts for conùinrLlions of ettponenliat tgpe opera- 1ols, Crrrrrìian.I. ol trInth.,9B, 1224-12o0 (1976).
[12] G. l\Irstroi¿tnrric. NL Iì. Occolsio, Una genualizzazioned.cLl'ope¡rtloredi ßcrt'tsleín,I'Ìcud. ;\ccacl. Sc. ÀIat. l'is. Nat. Napoli, Serie s IV, XLI\¡, 1b1-169 (1977).
[13] (]. lI a s L t'o I'a ¡ n i, Stri resti cli alcttttc fotntc lincati ctí approssitttuzione CalcoTó 14, 343- 368 (1e78).
[14] G. ll n stloi attni, Str uttopcrtLloreIinearce posiLiuo, Rencl. Accatl. Sc. l{at. Fis. Nat.
Napol i, 40, 161- 176 (l97'i ).
[15] C. trlicchcIìi, I'ire s(inrelion cluss and ilcrales of ßernsldn polgnontials, J. Approx.
'l'ìr., ll, I -- 18 (1973).
It0l 1:- Nngcl,, tlsgntplolìc propcLties of potucr of ßcrnslein opualors, J. Approx. Th., Zg, ij23 --:ì3J (t 980).
[171 ]rl. P lr s s o tt, Sonrc ¡¡rl¡¡srrr¡l ßernslein ¡tolgrtonriuls, A¡lploxinration Theoly IV, 649-6b2
(1 e83 ).
[18] 'l'. lropovicirt, Sttt L'trpprorintrdiottdcsfuncliottsco¡tuc¡-tl'oril¡cstrpetieur, Xfathcma-
tica, 10, 49- 50 (1935).
[19] 1'. Popovicirt, Sttr lc teslc clotts cerlaincs fotntulcs linóctircs cl'upprorintation¿eI,rur Ir¡.sr, ÀIaihcnraticr, I (241, 95- I42 (1959).
[20] l..r. Schocnbelq, on s¡,liuefuttctions. Itrcr¡talilies (Proc. synp. ohio 1965;ed.
ì)\: O. (lì'<lìô), Nou Yot.ì<, r\carl. Pr.css,255-291 (1$67).
[2 l ] l). l). S t a n c tt, -äartlttnliott of lhe rcíntattcler |eun in appro:tintation formulas by Bcrnslcin lrol¡,non.LiaJs, ùInth. Corlp., l7, 270-278(7963)
[22] D. f ). stnrcu, Applicatiort to t.lrc sludl] of monocitg of thc cle-
riualiur: of lhe seqLtences of s. Calcolo lG, 4SI-445 (1929).
[23] I). l). slancu, tl¡tprot-inmt.Ì nctu cktss ol linear polgnomíal opercttot,
lìcr'. lìourn. ÀIa[h, Pulcs 94 (]968).
l14] l). D. S t a l'ì c tì, (tst ol prohabiLislic tnctl.otls itttl,c lltcorg of ttrtifornt approt;intation of conli¡tttt¡tts funcliorts, Rcr'. .ììor¡nt. ]\[ath. I)ur.cs Appl., 1,1, 673- 691 (1969). : [2rrl D. l). S t n Ìì ('tl, ilpptotinrulirtn rtl frutctions liq nteons o[ somc neu, t'íasses of positiue litrcur rtpe rolol's, Nuncl'iscltc llclhcrlcn rìct' ¡r¡rr oxirrllionslhect.ic I (Irroc. couf. À{ath.
llcs. Irts1. Obct's'oìl¡rch 1071 ; ctì. ìr¡ L. Coììatz, G. Àlcinrlclus) Rasel : Bi¡kauser, 1B?- 203 (r e72 ).
[26] O. S z n s z, (icttetùirnlio¡t o/ S. N. BeLttsl¿i¡t's polynonúals to ittfinile inteructl,.I. Res.
llirr. Slcrrrlr¡tls, .15 (1950).
flcccivcrl 6.\'1980
C iuscppc IIas [r.oìanni
L)iptrrlirtrcrtlo tli X'ltlcnnlíca c Applicazioni - (Jniuersitù
di Napoli
Yia lIc::octt¡trronc, I 801 i,l Nupoli - Ikl¡1
Isl.ilLtlo \/iale per tlpplicazionì tlelLa A. Glonrsci, 5 E0I2Z Napoli X'Ialc¡ttcttíca - ILatu- C.N.R.
rlI¡rlio Rosnl.io Occolsio Diparlimcrúo di. tr,Irtlantalk'a c A pplicctz.iorti
- Uttiuersilà di Napoli
\tkt llczzr¡cuttrlor1c, I 801 :t,1 NapoLi - Ikrlu
I sLilttLo )/iule pcr ¡1 À . pplícarioni Gromscì ,,i dclla ÃIulttnalicn E0122 Nupoli - Ilaly- C.N.1l.
ß2) G: MAS.¡ROIANNÍ ancl M. R. OCCORSIO
rlr(l
Ïr'omIim
7,' æ(2.7),
On
llre
otlrer side,il ,[e
0zt'+21/),it
result,s t]ral, (sec f,l4-1,Th,
ti.1)ll( R,,.t,)"
ll < ";-
ll.fllr,,n1,"
anrl llrc
Íìss('t'lionis
lllovcrr.tr'inzr,ll¡', r\'0 prorrc
fllnnor¿nrr
4.tlI.
f .f ,fi:
definerl onl,
[,hen, tlte rekdion,liri
8,,,,,.(fi r') - L,,(l;
n')/.+æ
Iloltls, mltere I',,(.f
; n) is
tlLe I)a,qrcl11,{Je itùte.r'poh,tin,ç¡ polynontiu'l cotreEton.rl' inç¡to In
the orclel Jutctiotr,to plovc tho f 'nil lo
tlt,ethcoletn,
ltnots{ilnt}, \\'c
ol:lselt'c¡i -
0,, nt.thal,, if ve
substibul,esj þ¡.
Stirling)s nurnbclsof Íilst
olcleÌ ÀSj,in the
clcmen1,sof the rìâtri.\, Br,
tt'cobllirr
tlLcrri¿ltl'ir I
t. = - S t,t.
'Norv,
b¡'
(2,6)l-e
obt,ainlj,,,r(l;
r,): I
h:silt(i) ll':'
Ç*,t(n,)')l
^Íi,,,/(o),l'l'Jtt,,,t./;
,?')--
^ä (il)
â,¿lll(lim c,,,).) t/i,rrr,¡af¡,,,/(0).Fur'1,heluroro,
siìlc(' 0 < (l
._'1(¿)).{ 1,
lt.r, (2.5),rvc
tlecluco lìi¿ttLi':c''' - r\t''
-[Icnoe,
br- (2.2), it,
r'csults t,hat(,1 6) Tirn ß,,¡,(f
i r) - T
t|?, ør( ^¿/S¿ ) -1,rt,¡,4 f¡,,,./(0).(
t,S1n.qe
,.'^'(r\
ìr.
.. ,:
r¡(;\pS ^7-'11¡,,=,=
{né/t,A;;tu,t.:
i,;
'and from
('1.6),i.lie
¿ìssertiolìis
ilrttlrerlirr,i,t-'.tìlltrrtIìllNcES
[1] \'.,,\. Ilnskukor', .{rr ínsla¡tt:¡: ol tLscr¡ncnt'c ol lItretu'¡tosilluco¡tcrolorsofaonlíni.ioirs Ittttt'lions, Dol<1, r\liarl. Nauli. SSStì, 113,249 .251 '(1057) (in lìtrssirrt).
[2] ì'I. ß(,Ì'oììs ¡i¡cl lì. [)cvole, AcharrLr:lct'i:uliottol']']crnslaittpol ttttontiols,Àppt'o,ri- rnatiorr 'l'ìrcoly¡ I l.lr 21il+215 ,(1980). :
[]l S. N. Il elns íei¡, DénrctislraLiott tlu llrcor'cnudellteicÌsltasc, foirtl(e sttr lc crtLt'nl tlcs proltabì.Li|tis, (ìournuri, Soc. ùIaLìr. I(arko\', lt, (2), 7 _2 (191'2).