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ìI¡\'II-IEIL'\TIC,\

-

I'.EYUE D''\NALYSE NUIIEIìIQUE Iì'I' DE TI{ÉORID DE I,'i\PI'ROXII{;\TION

L'ANALYSE

NUTÍÉRIQT]II TìT

Lå 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 /

clefined

on the interval 1 : [0,1]

one c¿ùn associâ,1,c

the

Be,rnstein polynornial of clegree

rl

:

(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

a

linear

ancl posil,ive

for

every function

/

which is continuous on -¿

(/

e C'"(f ))'

iint B*(!i n) :

l@)

()p

it

eI'Arl,Ol' and 1,hal, lesults

that

uniformly or

.f.

In

spite

of their

slorv couvergelce,

,lì

h'ave been

(anc1 l,he¡i

still'are) a

sLrbject

of

stucly

by can

quot'e

àlroou

ali:

Rernstein

[3],

Berens

[2], L,or: 1Bl,

Popo-

viciu [18],

Schoenberg

[20],

Stanc;.t 127, 221.

A lot of

properties

of

great theorel,ical ancl

practical

inl;e.rest

of

the

,Brrt o

ha,\¡e shorv "'lîî5,,iåi'#3ïi"?å.Tiä$** I the

stuclics

of

Bashalrov

[1]r I(ö-

¡-ä.,T

ning-zeltcr

[9-1, sz,asz

[26],

ancl

those of stancu, frorn l'hich

spccial

mention

cleservc 123, 24, 2Ú).

A

conrplei,e ancl upclal,ccl bibliggr':r,ph¡'ou l,hc srrlljeç{, is givcn i.n [7],

(2)

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

is

the

itlcntica,l

operiltol

ale also n ell-k¡oy,n

[9,

16 ].

Suitable lincar

conrbinations

of the llcrnstein polynornials

have been sl,udied

too;

\\¡e c¿ùn

lecall

â,mong thern those

rrhiclì

aie containecl in

the

rvor'lis

of Butze¡' [4],

_o_f

Frentiu-[01

a,ncl

of lla;,

[11].

Ilicchelli [15]

ancl

llastloianni-Occolsio ¡fZi

irìtr.otuced ¿r,nd stu- clieclì even if_ sepzllzrtelv,

tlte folloving

cornbiriatións

of

itcrlatcs

of

gro

Rernsl,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

approxirnation

to

incleasc thc_apploxirnating

function's leuula¡it¡'.

Iírdeed, ^i-n

ttre

abgye l¡entionecl rt'orks,

il,

haS

bccn

detnonstlatclcl

thai the polj'nonrial

clefined

ì:v

(1.2) a,pproxirnates

a

func!,ior1

,f e-c2k(r^)

rviur a

r'ônräinclc,r,

of o(n-k) tr:pô.

rú has a,Iso been shos'n

in

[rì

] that florn all

thc¡ cornbinations

like

(1.3)

i:l

f ,r,.,Iii,(l;r,), É tr¡,,¡:I

i:l

1ìre lrol¡-tromi:rl (1.2)

is the only

one

to

pclsscss

the

aboyc-¡rentione¿ pr,o-

pelt¡' of

conr¡Lìr'gence.

[12

] ]Iastroianni

ancl Occolsio lenr¿r,r'l<ec1

that

a possiblc goncr,¿ì-

lizat'ion of (1.2) could

be

(1.3)

ut,,,.^U'; u:)

: i (- r)'-' (';) n,,rt,,rr,

x-il,h

À

real

posil,ive number,.

The airn of this

x'ork,is to

carl,y out, a

filst

stucl¡- of

the

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 ¡3

n'e shall cìemonstrate

sotre

ðonyexit¡, pr,e-

sert.ation.

Finall¡', irr

$ 4

.. cc of

11,,,,'f6r,

't11, --+

Ø

ancl

À+oo separ,at

ates

of t'liò

r.e-

maincler'.

2. th9

Zl'*,i operator. l['o ever'¡' ftrnction

/

rlefincrl ancl lirnil,ecl

in f,

rve

associate

the

continuous

function

8,,,;,1,

(Àe.E+) which is

defined

thlough the

selies:

e.r) 8,,¡$;

c))

: Ë (-t)'-' (!,) o,"rr,

O,

ru'helc /Jí,

is

the i-1,h itcral,c

of thc

Bernstein opcrator' .I3,,.

It, is

casy to yerif v

that, if

),

is an

inl,eger.

posilive unmbcL,

(2.1) gives back (1.2) ancl

in pilrticLrlar

it, result,s tha,t, 1ì,,,,,

- I)*.

l,lorcover,

for every

À e

E+, \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,e

by Sj

and s.l:, respeetivel¡;,

Stiiling's

tìum-

bers of first

ancì seconcl

t¡'pe rlefinctl by

: 2

fr(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

-Y

'ttù'

.'--,

I

sllm

01

,

O,l)r

e IlÁ 1 sfllnt,2.,. sf,-rlm"-'

silrn .,.I'i_zlnt"

"

0 0 0 0

0

Â,,

: ñ

00

1(¿)

00

1

Uren

let

IZ¡.

lle the

/c

X h matrix having for columns thc

autovectors

ai(i: f¡t¡ of the matrix Âr8r,

normalized so as

to

allorv

that the

ele-

ments of the principal

diagonal

of Ir¡ result

equal

to

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 observe

that

I/41

: tlT,

ryherc

(ÄrS,,)tU¡,:A¡,L¡,.Then, it, is

easy

to

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:0

E

( 1

¿-l-1

À

f : -!-

t

t1 -

(1

-

,)^,1

::^¡(t, À);f

e (0,11, À e

R*

i

(3)

58 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,

k

fn 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

easy

to verify \}lnl

q,,,¡(r,

l,) is a polynomial in ø of

clegree

not

greater

than /¡.

So,

frorn

(2.7),

iL

follox,s t,hab 8,,,7..f

is a polynomial

of degree

not

greater

than

rn.

Fulthernlore, if

'¡t,,

is à

polynomial of clegreo

not

greater

than

n,

(

,nz,

lhe

same happens

for

Bn¿,¡ipn.

Another

representation

of 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

deduce

that

R,n,t,(et,i

n) : nrVr.ArV;ttt,r,

er(ø)

:

çþ,

where

-4, is the

diagonal

mal,rix

lvhose elements are

:

[1

-

1(Ð]i.

For

example, we

have:

(2.10)

E,,7,(ezi

n) :

-8,,,¡,((ú --

n)2;

n):

--n(1.

-

n)lm,^

(2.1t)

B-,t.((t

- n"; n) : - #[r - (u - +l]

Let

us now observe

that Bn,t¡ (^

€ -R+), is

not in

general a

positive

ope-

rator; it is

easy

to verify

t'lnat 8,,,,¡

is

positivè

if 0 <

À

< 1'

n'or these valnes

of

)',, I?'^,¡ results as a positive operator

of llernstein

type.

!'inally,

note

that

13,o,7,

with 0 <

À

{

1,

is a particular

case

of

the operator :

n!,,,1

:,å,,-1)n+1+i

1r)r;r, n1ì, 1nrr, neN.

It

is

not rlifficult in any

case

to verify

that, verge to ./.

Next,

we

shall

shou'

that

B-,7. is

ühe

{n,î,,},'€N

ela,ss converging

to /.

,

R},,,.f does not con-

positive

operator of

ll.

On

the

properfies oI -B,,.,, (0

<

À

<

1).

The following

propositions

are

true.

Pnoposrt'rox 3.T Ttte operator Iì^,7, keeps the conuenity (coneaaity) of

e,Derl1 order

of

tha

function f.

Incleed.,

from (2.8) we

cleduce

that

(B-úI!@) :rØ)'i! n--,,^(*)l ! , u-*--t-, ..., Wrnf, p <nù,

pl k---o^ ' lm ,n m

J

l\{oreover,

if / is

convex

of

orcler

lt - 1, by

(2.9)

it

follorvs

that the

same

happens

for g; so the

assertion

is

proven.

Pnoposrrroir

3.II lor

eaery ø e

(0,1), the

relation

'

B,,,+t,¡!lt

t) - Il*,¡,ff j n): * n(\ - n)lnt,nz¡

ne)f

l(];^-

ø*rrt)

hold,s, uhere æ¿,

(i:\,2,3),

are tltree suitable

poinls

of L Tncleerl,

by (2.8) ve

have

A*(l;

n)

: : B-*rt(l; n) -

R^,t.(f ;

n) :

So,

the

operator

á,,

has d.egree

of

exactness 1,

4 5 A NEW OPERATOF, OT'BËRNSîËIN TYÞE 59

for r¿>0 the

only

8,,,$

)

n) :/(0)

-r- 2n

r\,,¡,!(0).

[(Ui - 1) ' + z(r -;r) *dfq,,¡10¡;

Br,^(.f t

n) :/(0) +

3n

t\,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. be

thc

remainder

term,

rvhele

I is the

iilen-

tical

opelator. JMc casily have :

Bm),:

R!,,

:

(1

-

B,,)À

: î t--rl'(1) r;,.

(4)

I

60

Furbhermole,

if / is

convex

of first

otclel, g

rvill

be ¿r,lso so

; in

this

ciùsc, \\¡e havc ,4,,,(/;

o) + 0 fol

ever.v a; e (0,1).

For

Popoviciu's theorem

f19],

1,hree pciirt,s

ntt

nz,

r.

e (0,1) e.xist, such

that

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

assertion

is

proven.

I[oreover, from

Proposition

3.If, l'tr may

claim

Conor,r,lny

S.IIL 1/

the Junct'iott,

f

is cotlneü oJthe

first

order,then,

for

eaeml ø e (0,

l), it

results tJtut

B,,+t,t(l; n) < 8.,r.(l) r).

'

Since

B-,¡

has clegree

of

exactness

1

ancl

by a theorcm u'hich

has been provett

in

[13

], the foilos'ing

represental,ion

of the

remaincler

(3.1) t?,¡.Ui ;') : -1"(t), {

e (0,1),

holcls

; b¡'

(3.1),

tire

o.l,hcr one can

be

cleduced :

(3.2)

Il,,t.(.f; n')

.- -- "It^,-i)-lrr, Í2,Í¿ill

,ìtr

Ír,

Fr,

f;,

bcing.suitable

points of

(0.1).

4.

On

thc

eonvcrgence

of 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

}

convcrgcs

uniformly to / on 1. For

some

classes

of

functions, [ìr tnearsule o1' ¿pp¡oximation

is

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

', tz)

, J'

e

c"(r),

il1,'t "

ultere

tl¿e norn't,

is the

u,ni,fotttt, one.

To plove

(4.1),

l'e

observe tha.b

if or(/;)

clenotos

the

nodultLs of

continuity 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 deducc

that

l/(ø) - ß*^U; r)l < (1 + ),,t' à), 8Þ0.

In partictrlar, for

8

:

e¡-)'tz, (4.1) follorvs.

To prove

(4.2), we observe

that

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

every

n,teI.

So, we

hal e

lx(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

llt

t'

-trom which,

for

à

:

,¡n-)'tz, we have (4.2).

thc foregoing

incclunlities

give bach, for

À

: 1r the

well-knowa

relations

corresponding

to

1,he

Bernstein

polynomials.

Norv,

let

À

>

1.

'Ihe

case

of

À as an integer number greater

th*n

l.

har

alreatly been stuclietl

in

tlei,ail

in [12];

so,

let us

supposo

that

À

* : i +

1,, $'here rl

is thc maximum

intege,r number such

that

le

<

À aqd-

y e (0,1).

ltmonnu 4.II. Let

7'

:

l,;

|

where

h í.s un

dntegør

nwnbqr

elní

), e (0,1). IJ

f

e Cz*+z(l), then

llw folluning

inequ,øli,ty

llr - B,,,lll *

-t-l*!,+|,

Ir,oltls, ttth,ere lllllzr

=-r?,1ä lll,t)ll ünd c is a

constant d,epe.nd,ing

on h

snil independant

ol I and

m.

fn

f¿sli,

by

(3.10), we hase

lH^,t(j;

ø) I

: l&*,r(R^,r|; r)l

<

lIft?,,fl,:ll

Emr

o\f;ln-rl)

<

1+ .("f ;

à),

fot

et

ery n,

t e

I

anrl

fol

ever-y ò

>0

(5)

a

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. 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'om

Iim

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'lion

is

lllovcrr.

tr'inzr,ll¡', r\'0 prorrc

fllnnor¿nrr

4.tlI.

f .f ,f

i:

definerl on

l,

[,hen, tlte rekdion,

liri

8,,,,,.(f

i 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,e

thcoletn,

ltnots

{ilnt}, \\'c

ol:lselt'c¡

i -

0,, nt.

thal,, if ve

substibul,e

sj þ¡.

Stirling)s nurnbcls

of Íilst

olcleÌ ÀSj,

in the

clcmen1,s

of the rìâtri.\, Br,

tt'c

obllirr

tlLc

rri¿ltl'ir I

t. = - S t,

t.

'

Norv,

b¡'

(2,6)

l-e

obt,ain

lj,,,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¿tt

Li':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

¿ìssertiolì

is

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).

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