Explicit expressions of the generalized Stieltjes polynomial Текст научной статьи по специальности «Математика»

Abstract:

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EXPLICIT EXPRESSIONS OF THE GENERALIZED STIELTJES POLYNOMIAL

Ljijana R. Paunovic

University of Pristina - Kosovska Mitrovica, Faculty of Education in Prizren - Leposavic, Leposavic, Republic of Serbia, -a

e-mail: [email protected],

ORCID iD: http://orcid.org/0000-0002-5449-9367 g

http://dx.doi.org/10.5937/vojtehg65-13355 &

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FIELD: Mathematics ARTICLE TYPE: Review Paper °

ARTICLE LANGUAGE: English

The existence and uniqueness of a Kronrod type extension to the well-known Gauss-Turan quadrature formulas were proved by Li (1994, pp.7183). For the generalized Chebyshev weight functions and for the Gori-Micchelli weight function, we found explicit formulas of the corresponding generalized Stieltjes polynomials. General real Kronrod extensions of the Gaussian quadrature formulas with multiple nodes are introduced. In some cases, the explicit expressions of the polynomials, whose zeros are the nodes of the considered quadratures, are determined. Keywords: Stieltjes polynomials, Kronrod extension, Gori-Micchelli Q-weight function.

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Introduction

Let o be an integrable weight function on the interval (-1,1). It is wellknown that the Gauss-Turan quadrature formula with multiple nodes

1 n 2s

J o(t)((= At J>(zv )+E^s(f )(n e N;s e N0) (1)

v=1 i=0

" is exact for all algebraic polynomials of degree at most 2(s + 1)n -1, and

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that its nodes are the zeros of the corresponding (monic) s -ortogonal polynomial f) of degree n which minimizes the following integral

f1 2 2 / \ 1 </>{a0, ai,...an_i ) = J_1^„ (t ) ®(t (dt, where Kn (t) = tn + anJn- +... + aj + a0.

In order to minimize </), we must have

-1 25+1

ji®(();rn(() tkdt = 0, k = 0,1,...,n-1. (2)

i

_ In order to minimize è, we must have

Dd T '

LJU J 2^+1

* f1 W (t\ tk, o

^ which are the corresponding orthogonality relations. For s=0, we have a < case of the standard orthogonal polynomials.

Following the well-known idea of Kronrod (Gautschi, Milovanovic, o 1988, pp.16-18), S. Li proposed to extend formula (1) to the formula (Li,

1994, pp.71-83):

0£ ,i n 2 s n+1

« j-( ) (t)dt = J(i> (rv)+ X Kf (r> ^n,s (f), (3)

v=1 i=0 /=1

where zv are the same nodes as in (1), and the new nodes zv and new

w weights aiv,KM are chosen to maximize the degree of exactness of (3). It

d is shown in (Li, 1994, pp.71-83) that we can always obtain the maximum degree 2(s + l)n + n +1 by taking zv to be the zeros of the polynomial

x ft +1, which we call the generalized Stieltjes polynomial, satisfying the

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° orthogonality property j-((();rn+1 (()p(( ftn(f)2s+1 dt = 0, all p e Pn. > At the same time, Li showed that jtn+1 always exists and is unique if it

is monic. In the special case when co{t )=(1 -t2) 12, he determined itn+1

explicitly and obtained the weights in (3) for s = 1 and s = 2.

Consider the following four generalized Chebyshev weight functions

(o{t) = ( ((): ( {t) = (1 -12)12, (2 {t) = (1 -12 )1/2+s, (3 (t) = (1 - t f2 (1 + tf+s, ( (t) =(1 -1 )1/2+s (1 +1 f .

Bernstein (1930, pp.127-177) showed that the monic Chebyshev « polynomial (orthogonal with respect to a>1 (t)Tn (t)/2"_1 ) minimized all

integrals of the form f i^Ldt (k > 0).

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TJ . , sin(n +1)0 , , COS(n + l/)0 sin(n + l/)0

U (t) =—--— V (t) =---W (t) =—--

A} sin0 ' A) cos (0/2) ' sin (0/2) '

respectively, where t = cos 0. It is easy to see that Wn (-1) = (- l)n Vn (t).

¿n+1 () = ^(( (()-(())(t2 -lKi()•

Let first ®()be (). In this case, it is known that nns (t) = Un ()/2n.

We have just proved the previous statement (Milovanovic, Spalevic, 2006, pp.171-195), (Milovanovic et al, 2006b, pp.22-28) and (Milovanovic et al, 2009, pp. 246-250).

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This means that the Chebyshev polynomials Tn are s - orthogonal on (-1,1) for each s > 0. Ossicini and Rosati (1975, pp.224-237) found three other weight functions a>i (t)(i = 2,3,4) for which the s - orthogonal polynomials can be identified as the Chebyshev polynomials of the second, third, and fourth kind: Un,V ,W , which are defined by

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Explicit expressions of the generalized Stieltjes polynomials

For an arbitrary integrable weight function o(t) on [-1,1], Li proved that the generalized Stieltjes polynomial nn+1 exists and is unique up to a constant factor. He considered the case when o(t ) = o1 (t), see (Galjak, cl 2006). In this case, it is known that (t) = Tn (t)/ 2n-1. Li obtained that

¿2 (() = 2[T2(()-To() and for n > 2

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Theorem 1. Let 7tn+1 be the monic polynomial of degree n +1 satisfying the orthogonality relation

f (l-t2 f+'n (t)p(t)nn(T dt = 0, all p e Pn. (4)

J-l n+1

(5)

n+1

Then

^ ¿n+i(t ) = 27Tn+i(t ).

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Proof: In this case, orthogonality conditions (4) have the form

O f1 1 (l -12)nn i(()tk[Un(t)]dt = 0, k = 0,1,...,n. (6)

0 We have

1 (1-12 ) [Un (t)]2s+1 = £ PU^j (t), (7)

n(2 j+1)+2j

* ■■■ P = 2-2s (- 1)j r + ^

Where

s-j )

Conditions (4) can be written in the form

J(t)tk (1-12 ) [Un (f+dt = 0, k = 0,1,...,n. (8)

(3 By using (7), the last conditions (8) obtain the form

^ s r1 I-

° X P J -12 n n+1 (t ))kUn(2 j+1)+2 j (t )dt = 0, k = 0,1,...,n. (9)

5 j=0

o Let n n+1 (t) be Tn+1 (( )/2n. By using (Monegato, 1982, pp.137-158)

o 2Tn (t )Un-1 (t ) = U 2 n-1 (t ) the integral under the sum in (9) for j = 0 has the

form f' V'1 -12Tn+1 (t))kUn (t)dt = - J^1 -12 tkU2n+1 (t)dt, and it is equal to 0

j-1 2 J-1

for k = 0,1,...,2n.

For j =1 , the integral under the sum in (9) has the form J71 -12Tn+1 (t))kU3n+2(()dt, and it is equal to 0 for k = 0,1,...,n, if

3n + 2 > 2n +1, which is always fulfilled. As for j = 1, the same conclusions for j > 2 are obtained. Therefore, conditions (9) are fulfilled. Finally, (5) holds because of the uniqueness of the generalized Stieltjes polynomial.

Let now c(t ) = c3 (() and we have just proved the previous statement

(Galjak, 2006), (Milovanovic, Spalevic, 2006, pp.171-195), (Milovanovic, Spalevic, 2003, pp.1855-1873).

Theorem 2. Let ;rn+1 (t) be the monic polynomial of degree n +1 satisfying the orthogonality relation

£ (1 -1i12 (1 +1)2+s nn+1 (()p(()jrn ()2s+1 dt = 0, all p e Pn. Then

nn+1 (t) = ^(t - 1)P"(V2,-V2)(t), (2n)

where Pn (2,-1/2 '(t ) is the orthogonal polynomial with respect to the weight

(10)

(11)

function ü)(t ) = .

1+t

'1 +1

Proof:: In this case, it is known that the corresponding monic s - orthogonal polynomial of degree n is

nn,s(() = (() = ^Pn(-V2V2)((), where P^"™2>(t) is the ordinary orthogonal polynomial with respect to the weight function co(t ) =.. . In

1 -1

this case, orthogonality conditions (10) have the form

£ (1 -1 ) (1 +1 )2+sn n+1 ( ) [Pn (-V2,V2)(( )]2 s+1 dt = 0,

k = 0,1,..., n.

We have (Ossicini, Rosati, 1975, pp.224-237)

for

(12)

(1 +1 )s

Pn

-1/2,1/2)

(( )] '+'

(t ),

(13)

j=0

c2s+1 r 2s +

where r} = 2-s —n- s - and Cn = (2n) /(22n (n!)2 )

n(2 j+1)+ j J J

The above conditions (12) can be written in the form

£irtn n+1 (t ))k (1 +1 ) )(t )] dt = 0, for k = 0,1,...,n. By using (13), the last conditions (14) obtain the form

(14)

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" £r, U^^ ()kPj2j(d = 0, k = 0,1,...,n. (15)

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Let ttn+i(t) be -A-f ((-lp2' 2j((). By using (2n)!

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eg p2 222j() = ^(X), where kn = ((2n-l)!!/(2n)!!) (Monegato, 1982, pp.137-158), the integral under the sum in (15) for j = 0 has the

yy ^n vrn W 2...........-- ..n „2n - 1 2n

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Ï i-yi-7Pn^%ft^fr))kdt = kn2 j^Vl-^n (())kdt, o

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and it is equal to 0 for k = 0,1...,2« -1. For j = 1,2,...,the integrals < under the sum in (15) have the form

(1 -1 >kP„(1/2'-1/2)(í Pj+J (( ))Ví, and they are equal to 0 for

k = 0,1,...,«, if 2« +1 <3« +1 <«(2j +1)+ j, which is always fulfilled. Using these conclusions, we have that conditions (15) are fulfilled. Finally, (11) holds because of the uniqueness of the generalized Stieltjes

0 polynomial.

1 Let o be an integrable weight function on the interval (a, b). Take o now a sequence of nonnegative integers cr = (s1,s2,...). For any « e N ,

q we denote the c°rresponding finite sequence (s ,s2 ...,Sn ) by an and consider a generalization of the Gaurss-Turan quadrature formula (1) to

rules having nodes with arbitrary multiplicities

b n 2 y _

l œ (t )) (t )t = XIA yf(i} Tv ) + R (f ), (16)

a v=1 1=0

WhereAy = Af^, Tv = z{y-a) (i = 0,1,...,2sv;v = 1,...,n). Such formulas

were derived independently by Chakalov and Popoviciu. A significant theoretical progress in this subject was made by Stancu see (Milovanovic, 2001, pp.267-286).

In this case, it is important to assume that the nodes tv =Ti(n,a'> are ordered, say

Ti <t <... <rn, Tv G [a(17)

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with odd multiplicities 2s1 + 1,2s2 + 1,...,2sn +1, respectively, in order to £

have the uniqueness of the Chakalov-Popoviciu quadrature formula (16) (Karlin, Pinkus, 1976, pp.113-141). Then, this quadrature formula has the

maximum degree of exactness dmax = 2sv+ 2n -1 if and only if

b 2 sv+1

jfl (T tM(()dt = 0, k = 0,1,...,n-1. -

a v=1

The existence of such quadrature rules was proved by Chakalov, Popoviciu, Morelli and Verna, and the existence and uniqueness subject to g (17) by Ghizzetti and Ossicini (Ghizzetti, Ossicini, 1995), (Milovanovic, 2001, pp.267-286), and also by (Milovanovic, Spalevic, 2002, pp.619-637). Conditions (17) define a sequence of polynomials

k^U = nt-TV",a)\ ¿r^T^K..., t^ e[a,b\ 2

v=1

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b n 2sv+1 №

' kCT(()n((-TVn,ff) ) ®(t)dt = 0, k = 0,1,...,n-1. |

General Kronrod extensions of the Chakalov-Popoviciu quadratures

Let a*m = (s*,s*,...,s*m) (s* eN0,u = 1,2,...,m) Following the well-

known idea of Kronrod (Gautschi, Li, 1990, pp.315-329), we extend formula (16) to the interpolatory quadrature formula

b n 2sv n+1 2sU

J f ((M)=ZZKf(i) (Tv)+EZO (j) fe)+^ (f), (18)

a v=1 i=0 u=1 j=0

wheretv are the same nodes as in (16), and the new nodes tu and new weights Biv, Cj are chosen to maximize the degree of exactness of (18) which is greater than or equal to

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such that jV_rvn"<< ) °}(t)dt = °, k = 0,1...,n-1.

a V=1

These polynomials are called j- orthogonal polynomials and they x correspond to the sequence j = (s^ s2,...). We will often write simple r,

instead of tJ"' j <. If we have j = (s, s,...), the above polynomials reduce to JX the s - orthogonal polynomials.

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1) + £(// +1)+ m -1 = 2 2

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+ n + 2m -1.

> We called quadrature formula (18) the Chakalov-Popoviciu-Kronrod

quadrature formula. The particular case is the Gauss-Turan-Kronrod quadrature formula, if s1 = s2 =... = sn = s. In the theory of the Gauss-

lju Kronrod quadrature formulas, the important role is played by the Stieltjes

polynomials En+1 (x) whose zeros are the nodes r**, namely

e„+i( x)=[ ((-r*M ) 1=\

Gori and Micchelli (Li, 1994, pp.71-83) have introduced for each n an £ interesting class of weight functions. Consider a subclass of the Gori-£ Micchelli weight functions,

< rTT /.\-\2l

(( )=

Un-1 (()'

(\ - t2 )M/2, l e{0,\,..., s}. (19)

n

In the particular case l = 0, (19) reduced to the Chebyshev weight

(D / 2 V\/2

function of the first kind con0 =(\ -12) . In this case,

(n > 2,ct!,\ = ((s -1)/ 2, s -1.....s -l, (s -1

* (n > 2,a„*+1 =((s -1 )/ 2, s - /,..., s -/, (s -/) / 2)) when quadrature formula (18) has the form (r* = -1,r+1 = 1)

1 n 2s n 2(s-/) , , s-/ . ,

j f ((h,/ () dt=]t () r)(j) (/(c ; f j (-1)+cin+lf« (1) )+^n (f)

-5 -\ v=\ i=0 11=2 j=0 j=0 (20)

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Theorem 3. In the Kronrod extension (20) of the Gauss-Turan quadrature formula (16) with the weight function (19), and for n > 2, the corresponding generalized Stieltjes polynomial

E[Z\) () = ((s -1) /2, s -1,..., s -1, (s -1) /2)) is given by

Efc\)(t) =((2 -\K-\(t) , i.e., the nodes r'M (1 = 2,...,n) are the zeros of Un-1 (() (the Chebyshev polynomial of the second kind of the degree n -\), and r\ =-\,<+\ = \.

When quadrature formula (18) has the form

o

2 s n+1 2 x , s fO

f/{()(1 -,2 r*'J> -H/O+eeC/ T)+R, (/), (21)

-1 v=1 i=0 ,u=1 j=0

we have just proved the previous statement (Galjak, 2006), (Milovanovic et al, 2006a, pp.291-305), (Milovanovic et al, 2006b, pp. 22-28). |

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Theorem 4. In the Kronrod extension (21) of the Gauss-Turan quadrature formula (16) with the weight function c2(t)=(1 -12)1/2+s the corresponding generalized Stieltjes polynomial E^ '(t) (ct*+j = (s, s,..., s)) is ^ given by E^)(() = Tn+1 (t), i.e., the nodes t* (^ = 1,...,n +1) are the zeros of Tn+1 (() (the Chebyshev polynomial of the first kind of the degree n +1). When quadrature formula (18) has the form (t* = -1)

//(()(1 -t)+s(1+1)-12dt=£fx/«tj+gjrc/t)+£c;,j(j)(-1) +r,(/), (22)

-1 V=1 i=0 n=2 j=0 j=0 (/)

we have just proved the previous statement. s

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Theorem 5. In the Kronrod extension (22) of the Gauss-Turan * quadrature formula (16) with the weight function c3 (t) = (1 -1 )1/2+s (1 +1 )-1/2 the corresponding generalized Stieltjes polynomial

E& )(t) fo = (s/2, s,..., s)) is given by e: )(t )= (t + 1)P^ }(t), ^

i.e., the nodes t* ( = 2,...,n +1) are the zeros of Pn(1/21/2)() (the | Chebyshev polynomial of the fourth kind of the degree n), and t* = -1. ^

When quadrature formula (18) has the form (t*+j = 1).

/ / (t c ()dt = £ ^/ThY tc/T )+£cjn+J(j »(1)+rn (/), (23)

-1 v=1 i=0 n=1 j=0 j=0

where c4 (t) = (1 -1)-1/2 (1 +1)2+s is the Chebyshev weight of the fourth kind, in a similar way as in the previous case, the previous statement can be proved.

Theorem 6. In the Kronrod extension (23) of the Gauss-Turan quadrature formula (16) with the weight function c4 (t )= (1 -1 )-12 (1 +1 )2+s the corresponding generalized Stieltjes polynomial

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ESX<) fc: = (*/2)) is given by E&\t)= (t-l)^-12)(),

i.e., the nodes t*m ( = \,...,n) are the zeros of P„(V2,~1/2)(() (the Chebyshev polynomial of the third kind of the degree n), and r*n+\ = \.

References

Bernstein, S., 1930. Sur les polynomes orthogonaux relatifs a un segment fini. J. Math. Pures Appl., 9, pp.127-177.

Galjak, L., 2006. Kronrod extensions of Gaussian quadratures with multiple nodes. Univesity of Kragujevac. Master work.

Gautschi, W., & Li, S., 1990. The remainder term for analytic functions of Gauss-Radau and Gauss-Lobatto quadrature rules with multiple points. J. Comput. Appl. Math., 33, pp.315-329. Available at: https://doi.org/10.1016/S0377-0427(05)80007-X.

Ghizzetti, A., & Ossicini, A., 1995. Quadrature formulae. Berlin: AkademieVerlag.

Karlin, S., & Pinkus, A., 1976. Gaussian quadrature formulae with multiple nodes. In: S. Karlin, C.A. Micchelli, A. Pinkus, & I.J. Schoenberg Eds., Studies in Spline Functions and Approximation Theory. New York: Academic Press, pp.113-141.

Li, S., 1994. Kronrod extension of Turan formula. Studia Sci. Math. Hungar, 29, pp.71-83.

Milovanovic, G.V., 2001. Quadratures with multiple nodes, power orthogonality, and moment-preserving spline approximation. In: Numerical analysis 2000, vol. V, Quadrature and orthogonal polynomials, (W. Gautschi, F. Marcellan, L. Reichel, Eds). J. Comput. Appl.Math., 127, pp.267-286.

Milovanovic, G.V., & Spalevic, M.M., 2002. Quadrature formulae connected to -orthogonal polynomials. J. Comput. Appl. Math., 140, pp.619-637.

Milovanovic, G.V., & Spalevic, M.M., 2003. Error bounds for Gauss-Turan quadrature formulae of analytic functions. Mathematics of Computation, 72(244), pp.1855-1873. Available at: http://dx.doi.org/10.1090/S0025-5718-03-01544-8.

Milovanovic, G.V., & Spalevic, M.M., 2006. Gauss-Turan quadratures of Kronrod type for generalized Chebyshev weight functions. Calcolo, 43(3), pp.171-195. Available at: http://dx.doi.org/10.1007/s10092-006-0121-9.

Milovanovic, G.V., Spalevic, M.M., & Galjak, L. 2006a. Kronrod extensions of Gaussian quadratures with multiple nodes. Computational Methods in Applied Mathematics, 6(1), pp.291-305.

Milovanovic, G.V., Spalevic, M.M., & Galjak, L., 2006b. Kronrod extensions of Gaussian quadratures with multiple nodes. In: ICNAM, 2016-09-27, Kragujevac, pp.22-28

Milovanovic, G.V., Spalevic, M.M., & Paunovic, L., 2009. Error bounds of S

r-

CP

Gauss-Turan-Kronrod quadratures with Gori-Micchelli weight for analytic functions. In: MIT, 2009, Kopaonik, pp.246-250, August.

Monegato, G., 2001. An overview of the computational aspects of Kronrod quadrature rules. Numer.Algorithms , 26(2), pp.173-196. Available at: -s http://dx.doi.org/10.1023/A:1016640617732. §

Ossicini, A., & Rosati, F., 1975. Funzioni caratteristiche nelle formule di quadrature gaussiane con nodi multipli. Boll.Un. Mat. Ital., 11(4), pp.224-237.

ЭКСПЛИЦИТНЫЕ ВЫРАЖЕНИЯ ДЛЯ ОБОБЩЕННЫХ |

ПОДМНОЖЕСТВ СТИЛТЬЕСА

Лиляна Р. Паунович го

Университет в Приштине-Косовской Митровице, Педагогический факультет в Призрене-Лепосавиче, Лепосавич, Республика Сербия

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ОБЛАСТЬ: математика ВИД СТАТЬИ: обзорная статья °

ЯЗЫК СТАТЬИ: английский §

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Резюме:

Уникальное применение теории расширения функций Конрода относительно квадратурной формулы Гаусс-Тюрана доказал Ли (и, 1994, стр.71-83). Определены эксплицитные выражения для обобщенных подмножеств Стилтьесапо отношению квесовым ш функциям Чебышева и весовым функциям Гори-Мишелли. Определено расширение функций Конрода по квадратурной формуле Гаусса с узлами многочлена. 5

В отдельных случаях определены эксплицитные выражения множеств, нули которых являются узлами исследуемых квадратур.

Ключевые слова: подмножества Стилтьеса, расширение функций Кронрода, Гори-Мишелливесовая функция.

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ЕКСПЛИЦИТНИ ИЗРАЗИ УОПШТЕНИХ СТИЛ^ЕСОВИХ ПОЛИНОМА

Ъиъана Р. Паунови^

Универзитет у Приштини - Косовсщ Митровици, Учите^ски факултет у Призрену - Лепосави^у, Лепосави^, Република Срби]а

ОБЛАСТ: математика

ВРСТА ЧЛАНКА: прегледни чланак

иЕЗИК ЧЛАНКА: енглески

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Сажетак:

Егзистенц^у и }единственост Кронродових екстенз^а за добро познате Гаус-Туранове квадратурне формуле доказао jе Ли (и, 1994, стр.71-83). Одре^ени су експлицитни изрази за уопштене Стилтjесове полиноме у односу на Чебишевъеве тежинске функц^е, као и у односу на тежинску функц^у Гори-Мичели. Дефинисана jе Кронродова екстенз^а за Гаусове квадратурне формуле са вишеструким чворовима. У неким случаjевима одре^ени су експлицитни изрази полинома, ч^е су нуле чворови посматраних квадратура.

Къучне речи: Стилтjесови полиноми, Кронродова екстенз^а, тежинска функц^а Гори-Мичели.

Paper received on / Дата получения работы / Датум приема чланка: 03.03.2017. Manuscript corrections submitted on / Дата получения исправленной версии работы / Датум достав^а^а исправки рукописа: 21.03.2017.

Paper accepted for publishing on / Дата окончательного согласования работы / Датум коначног прихвата^а чланка за об]ав^ива^е: 23.03.2017.

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