On an optimal quadrature formula in Sobolev space L (m) 2 (0,1)
aa r X i v : . [ m a t h . NA ] D ec On an optimal quadrature formula in Sobolev space L ( m )2 (0 , Kh.M.Shadimetov, A.R.Hayotov, F.A.Nuraliev
Abstract
In this paper in the space L ( m )2 (0 ,
1) the problem of construction of optimal quadrature formulas isconsidered. Here the quadrature sum consists on values of integrand at nodes and values of first derivativeof integrand at the end points of integration interval. The optimal coefficients are found and norm of theerror functional is calculated for arbitrary fixed N and for any m ≥
2. It is shown that when m = 2 and m = 3 the Euler-Maclaurin quadrature formula is optimal. MSC:
Keywords: optimal quadrature formulas, error functional, extremal function, Sobolev space,optimal coefficients.
It is known, that numerical integration formulae, or quadrature formulae, are methods for theapproximate evaluation of definite integrals. They are needed for the computation of thoseintegrals for which either the antiderivative of the integrand cannot be expressed in terms ofelementary functions or for which the integrand is available only at discrete points, for examplefrom experimental data. In addition and even more important, quadrature formulae provide abasic and important tool for the numerical solution of differential and integral equations.There are various methods in the theory of quadrature, which allow us approximately calcu-late integrals with the help of finite number values of integrand. Present work also is devotedto one of such methods, i.e. to construction of optimal quadrature formulas for approximateevaluation of definite integrals in the space L ( m )2 (0 ,
1) equipped with the norm k ϕ ( x ) k L ( m )2 (0 , = Z ( ϕ ( m ) ( x )) dx / . Consider following quadrature formula Z ϕ ( x ) dx ∼ = N X β =0 C [ β ] ϕ [ β ] + Aϕ ′ [0] + Bϕ ′ [ N ] (1 . ith the error functional ℓ ( x ) = ε [0 , ( x ) − N X β =0 C [ β ] δ ( x − hβ ) + Aδ ′ ( x ) + Bδ ′ ( x −
1) (1 . L ( m )2 (0 , C [ β ], β = 0 , N , A and B are the coefficients of the formula (1.1),[ β ] = hβ , h = N , N = 1 , , ... , ε [0 , ( x ) is the indicator of interval [0,1], δ ( x ) is the Diracdelta-function.The difference ( ℓ ( x ) , ϕ ( x )) = Z ϕ ( x ) dx − N X β =0 C [ β ] ϕ [ β ] − Aϕ ′ [0] − Bϕ ′ [ N ] . is called the error of the quadrature formula (1.1)Error of the formula (1.1) is estimated with the help of norm of the error functional (1.2) inthe conjugate space L ( m ) ∗ (0 , (cid:13)(cid:13)(cid:13) ℓ ( x ) | L ( m ) ∗ (cid:13)(cid:13)(cid:13) = sup ‚‚‚ ϕ ( x ) | L ( m )2 ‚‚‚ =1 | ( ℓ ( x ) , ϕ ( x )) | . Furthermore, norm of the error functional ℓ ( x ) depends on the coefficients C [ β ] , A and B .Choice of the coefficients when nodes are fixed is linear problem. Therefore we minimize normof the functional ℓ ( x ) by coefficients, i.e. we find (cid:13)(cid:13)(cid:13) ◦ ℓ ( x ) | L ( m ) ∗ (cid:13)(cid:13)(cid:13) = inf C [ β ] ,A,B (cid:13)(cid:13)(cid:13) ℓ ( x ) | L ( m ) ∗ (cid:13)(cid:13)(cid:13) . (1 . (cid:13)(cid:13)(cid:13) ◦ ℓ ( x ) | L ( m ) ∗ (cid:13)(cid:13)(cid:13) is found then the functional ◦ ℓ ( x ) is said to be correspond to the optimal quadra-ture formula (1.1) in L ( m )2 and corresponding coefficients are called optimal . Thus we get fol-lowing problems. Problem 1.
Find norm of the error functional ℓ ( x ) of quadrature formula of the form (1.1)in the space L ( m ) ∗ (0 , . Problem 2.
Find coefficients C [ β ] , A and B which satisfy the equality (1.3). Problem 2 for quadrature formulas of the form N Z ϕ ( x ) dx ∼ = N X k =0 p k ϕ ( k ) n L ( m )2 first considered by A.Sard [1]. By A.Sard and S.D.Meyers [2] the solution of thisproblem was obtained for the following cases: m = 1 for arbitrary fixed N ; m = 2 for N ≤ m = 3 for N ≤ m = 4 for N ≤ N → ∞ , i.e. for formula ofthe form ∞ Z ϕ ( x ) dx ∼ = ∞ X k =0 B ( m ) k ϕ ( k )is considered. In [3] an algorithm for finding of the coefficients B ( m ) k is given with the help ofspline of degree 2 m −
1. In the cases m = 2 , , ..., B ( m ) k are calculated using aComputer. Calculation of these coefficients up to m = 30 were done by F.Ya.Zagirova [4].In [5] in the space L ( m )2 considered quadrature formula of the form N + η Z − η ω ( x ) ϕ ( x ) dx ∼ = N X β =0 C [ β ] ϕ [ β ] , (1 . ≤ η j < ω ( x ) is weight function, C [ β ] are the coefficients. In [5] the algorithm forfinding optimal coefficients C [ β ] of quadrature formulas of the form (1.4) is given and for the op-timal coefficients the system of 2 m − N and for any m in the space L ( m )2 (0 , In this section we give some definitions and formulas which are necessary in the proofs of mainresults. uler polynomials E k ( x ) , k = 1 , , ... is defined by following formula [8] E k ( x ) = (1 − x ) k +2 x (cid:18) x ddx (cid:19) k x (1 − x ) , (2 . E ( x ) = 1.For Euler polynomials following identity hold E k ( x ) = x k E k (cid:18) x (cid:19) , (2 . Theorem 2.1 [9].
Polynomial P k ( x ) which determined by formula P k ( x ) = ( x − k +1 k +1 X i =0 ∆ i k +1 ( x − i (2 . is the Euler polynomial (2.1) of degree k , i.e. P k ( x ) = E k ( x ) . Following formula is valid [10]: n − X γ =0 q γ γ k = 11 − q k X i =0 (cid:18) q − q (cid:19) i ∆ i k − q n − q k X i =0 (cid:18) q − q (cid:19) i ∆ i γ k | γ = n , (2 . i γ k is finite difference of order i of γ k , ∆ i k = ∆ i γ k | γ =0 .At last we give following well known formulas from [11] β − X γ =0 γ k = k +1 X j =1 k ! B k +1 − j j ! ( k + 1 − j )! β j , (2 . B k +1 − j are Bernoulli numbers,∆ α x ν = ν X p =0 νp ∆ α p x ν − p . (2 . To solve problem 1, i.e. for finding norm of the error functional (1.2) in the space L ( m )2 (0 , ψ ℓ ( x ) is said to be extremal function f the error functional (1.2) (see [12]) if following equality holds( ℓ ( x ) , ψ ℓ ( x )) = (cid:13)(cid:13)(cid:13) ℓ | L ( m ) ∗ (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) ψ ℓ | L ( m )2 (cid:13)(cid:13)(cid:13) . In the space L ( m )2 (0 ,
1) the extremal function ψ ℓ ( x ) of the error functional ℓ ( x ) is found byS.L.Sobolev. This extremal function have the form ψ ℓ ( x ) = ( − m ℓ ( x ) ∗ G ( x ) + P m − ( x ) , (3 . G ( x ) = x m − signx m − , P m − ( x ) is a polynomial of degree m −
1. Since the functional ℓ ( x )belongs to the space L ( m ) ∗ (0 ,
1) therefore following holds( ℓ ( x ) , x α ) = 0 , α = 0 , , ..., m − . (3 . L ( m )2 (0 ,
1) is Hilbert space, then by using (3.1), taking into account ofRiesz theorem about common form of a linear continuous functional on Hilbert space, we get k ℓ k = ( ℓ, ψ ℓ ) = ( − m +1 (cid:20) A · B (2 m − − A Z x m − m − dx − B Z ( x − m − m − dx ++2 N X β =0 C [ β ] (cid:18) A ( hβ ) m − m − − B ( hβ − m − m − (cid:19) + 2 N X β =0 C [ β ] Z | x − hβ | m − m − dx −− N X β =0 N X γ =0 C [ β ] C [ γ ] | hβ − hγ | m − m − − Z Z ( x − y ) m − sign ( x − y )2(2 m − dxdy (cid:21) . Thus, the problem 1 is solved for quadrature formulas of the form (1.1) in the space L ( m )2 (0 , Now we investigate problem 2. For finding of minimum of the k ℓ k under the conditions (3.2)Lagrange method of undetermined multipliers is used. For this we consider following functionΨ = k ℓ k + 2 · ( − m +1 m − X α =0 λ α ( ℓ, x α ) . quating to zero partial derivatives by coefficients C [ β ] , A and B, together with conditions(3.2) we get following system of linear equations N X γ =0 C [ γ ] | hβ − hγ | m − m − − A ( hβ ) m − m − B ( hβ − m − m − m − X α =0 λ α ( hβ ) α = Z | x − hβ | m − m − dx, β = 0 , N , (4 . N X γ =0 C [ γ ] ( hγ ) m − m − B m − − λ = 12(2 m − , (4 . N X γ =0 C [ γ ] ( hγ − m − m − − A m − m − X α =1 αλ α = 12(2 m − , (4 . N X γ =0 C [ γ ] = 1 , (4 . N X γ =0 C [ γ ] hγ + A + B = 12 , (4 . N X γ =0 C [ γ ]( hγ ) α + αB = 1 α + 1 , α = 2 , m − . (4 . system of Wiener-Hof type for the optimal coefficients [12].In the system (4.1)-(4.6) coefficients C [ β ] , β = 0 , N , A and B, and also λ α , α = 0 , m − L ( m )2 (0 ,
1) for quadrature formulasof the form (1.4) (see [13]).
In present section we study solution of the system (4.1)-(4.6). In the solution of this systemwe use the approach which used in solution of the linear system for optimal coefficients ofquadrature formulas of the form (1.4) in [6]. .1 The coefficients of optimal quadrature formulas It is easy to prove following lemma for the coefficients C [ β ] of quadrature formulas of the form(1.1). Lemma 5.1.
The optimal coefficients C [ β ] , ≤ β ≤ N − , of quadrature formulas of theform (1.1) have following form C [ β ] = h m − X k =1 (cid:16) d k q βk + p k q N − βk (cid:17)! , ≤ β ≤ N − , (5 . where d k, p k are unknowns, q k are roots of the Euler polynomial E m − ( q ) , | q k | < . Lemma is proved as lemma 3 of the work [9] and in the proof the discrete analogue D m [ β ]of the polyharmonic operator d m dx m is used. The discrete analogue D m [ β ] of the polyharmonicoperator d m dx m is constructed in [14].We need following lemmas in proof of main results. Lemma 5.2.
Following identity is take placed α X i =0 dq + pq N + i ( − i +1 ( q − i +1 ∆ i α = ( − α +1 α X i =0 dq i + pq N +1 ( − i +1 (1 − q ) i +1 ∆ i α , (5 . here α and N are natural numbers, ∆ i α is finite difference of order i of γ α at the point 0. Proof.
For convenience left and right sides of (5.2)we denote by L L respectively, i.e. L = α X i =0 dq + pq N + i ( − i +1 ( q − i +1 ∆ i α L = ( − α +1 α X i =0 dq i + pq N +1 ( − i +1 (1 − q ) i +1 ∆ i α . First consider L . By using the equality (2.3) and identity (2.2) for L consequently we get L = α X i =0 dq + pq N + i ( − i +1 ( q − i +1 ∆ i α = dq ( q − α +1 E α − ( q ) + pq N + α ( − α +1 ( q − α +1 E α − (cid:18) q (cid:19) == dq ( q − α +1 E α − ( q ) + pq N + α ( − α +1 ( q − α +1 E α − ( q ) q α − = dq + pq N +1 ( − α +1 ( q − α +1 E α − ( q ) . (5 . L by using (2.3) and (2.2) we have L = α X i =0 dq i + pq N +1 ( − i +1 (1 − q ) i +1 ∆ i α = dq α ( q − α +1 E α − (cid:18) q (cid:19) + pq N +1 ( q − α +1 E α − ( q ) = dq (1 − q ) α +1 E α − ( q ) + pq N +1 ( q − α +1 E α − ( q ) = dq ( − α +1 + pq N +1 ( q − α +1 E α − ( q ) == ( − α +1 dq + pq N +1 ( − α +1 ( q − α +1 E α − ( q ) . (5 . L = ( − α +1 L . Lemma 5.2 is proved.We denote Z p = m − X k =1 p X i =0 d k q N + ik + p k q k ( − i +1 (1 − q k ) i +1 ∆ i p . (5 . ∗ ) Lemma 5.3.
Following identities are valid m − X j =1 ( − j − ( j − m − − j X i =1 B m − j − i h m − j − i i ! (2 m − j − i )! == m X j =3 B j h j j ! m − X i =0 ( − i i ! (2 m − − j − i )! + m − X j = m +1 B j h j j ! m − − j X i =0 ( − i i ! (2 m − − j − i )!and m − X j =1 ( − j − ( j − m − − j X p =2 h p +1 Z p p ! (2 m − − j − p )! == m +1 X j =3 h j Z j − ( j − m − X l =0 ( − l l ! (2 m − − j − l )! + m − X j = m +2 h j Z j − ( j − m − − j X l =0 ( − l l ! (2 m − − j − l )! . The proof of lemma is obtained by expansion in powers of h of left sides of given identities.For the coefficients of optimal quadrature formulas of the form (1.1) following theorem holds. Theorem 5.1.
Among quadrature formulas of the form (1.1) with the error functional (1.2)there exists unique optimal formula which coefficients are determined by following formulas C [0] = h
12 + m − X k =1 p k q Nk − d k q k − q k ! , (5 . C [ β ] = h m − X k =1 ( d k q βk + p k q N − βk ) ! , β = 1 , N − , (5 . C [ N ] = h
12 + m − X k =1 d k q Nk − p k q k − q k ! , (5 . A = h − m − X k =1 d k q k + p k q N +1 k (1 − q k ) ! , (5 . = h −
112 + m − X k =1 d k q N +1 k + p k q k (1 − q k ) ! , (5 . where d k and p k satisfy following system m − linear equations: m − X k =1 j X i =0 d k q k + p k q N + ik ( − i +1 ( q k − i +1 ∆ i j = B j +1 j + 1 , j = 2 , m − , (5 . m − X k =1 2 m − X i =0 d k q k + p k q N + ik ( − i +1 ( q k − i +1 ∆ i m − = 0 , (5 . m − X k =1 j X i =0 (1 − q Nk ) ( − i +1 d k q ik − p k q k ( q k − i +1 ∆ i j = 0 , j = 2 , m − . (5 . m − X k =1 2 m − X i =0 ( − i +1 d k q N + ik + p k q k ( q k − i +1 ∆ i m − = 0 . (5 . Here B α are Bernoulli numbers, ∆ i γ j is difference of order i of γ j , ∆ i j = ∆ i γ j | γ =0 , q k areroots of Euler polynomial of degree m − , | q k | < . Proof.
First we give plan of proof.From (5.1) clear that instead of unknowns C [ β ], β = 1 , N − d k , p k , k = 1 , m −
1. The coefficients C [0], C [ N ], A , B and λ α , α = 0 , m − d k and p k , k = 1 , m −
1. So if we find d k and p k , then the system (4.1)-(4.6) is solved completely.Substituting the equality (5.1) to equation (4.1) we get polynomial of degree 2 m of hβ on bothsides of (4.1). Equating coefficients of same degrees of hβ we find λ α , α = 0 , m − C [0], A andsystem (5.10) for d k , p k . Taking account of (5.1), (5.5), (5.9), from conditions (4.4) and (4.5)we get (5.7), (5.9), i.e. we obtain C [ N ] and B . Further, by using (5.1), (5.9) and expressionfor λ , from (4.2) we get the equation (5.11). System of equations (5.12) for unknowns d k , p k ,we obtain from equation (4.6), using (5.1), (5.5)-(5.9). Finally, taking into account (5.1), (5.8)and λ α , α = 1 , m − S = N X γ =0 C [ γ ] | hβ − hγ | m − m − C [0] ( hβ ) m − (2 m − β X γ =1 C [ γ ] ( hβ − hγ ) m − (2 m − − N X γ =0 C [ γ ] ( hβ − hγ ) m − m − . Lat two sums of the expression S we denote S = β X γ =1 C [ γ ] ( hβ − hγ ) m − (2 m − , S = N X γ =0 C [ γ ] ( hβ − hγ ) m − m − S we have S = β X γ =0 h m − X k =1 (cid:16) d k q γk + p k q N − γk (cid:17)! ( hβ − hγ ) m − (2 m − h m (2 m − " β − X γ =0 γ m − + m − X k =1 d k q βk β − X γ =0 q − γk γ m − + p k q N − βk β − X γ =0 q γk γ m − ! == h m (2 m − " m X j =1 (2 m − B m − j j ! · (2 m − j )! β j + m − X k =1 " d k q βk ( q k q k − m − X i =0 ∆ i m − ( q k − i −− q − βk q k − m − X i =0 ∆ i β m − ( q k − i ) + p k q N − βk ( − q k m − X i =0 (cid:18) q k q k − (cid:19) i ∆ i m − −− q βk − q k m − X i =0 (cid:18) q k q k − (cid:19) i ∆ i β m − ) . Taking into account that q k is a root of Eulaer polynomial E m − ( q ) an using formulas (2.3),(2.6) the expression for S we reduce to following form S = ( hβ ) m (2 m )! + h · ( hβ ) m − (2 m − B + h m m − X j =1 B m − j j !(2 m − j )! β j ++ h m m − X j =0 β m − − j j !(2 m − − j )! m − X k =1 j X i =0 − d k q k + p k q N + ik ( − i ( q k − i +1 ∆ i j . (5 . S . By using conditions of orthogonality (4.4)-(4.6) the expression S we rewriteby powers of hβS = N X γ =0 C [ γ ] ( hβ − hγ ) m − m − m − X j =2 ( hβ ) m − − j j !(2 m − − j )! (cid:18) j + 1 − jB (cid:19) − ( hβ ) m − m − (cid:18) − A − B (cid:19) + ( hβ ) m − m − m − X j = m ( hβ ) m − − j j !(2 m − − j )! N X γ =0 C [ γ ]( − hγ ) j . (5 . Z | x − hβ | m − m − dx = ( hβ ) m (2 m )! + m − X j =0 ( − hβ ) m − − j m − − j )!( j + 1)! (5 . S into equation (4.1) and using (5.14), (5.15) we have( hβ ) m (2 m )! + C [0] ( hβ ) m − (2 m − h ( hβ ) m − (2 m − B + m − X j =1 B m − j h m − j ( hβ ) j j !(2 m − j )! ++ m − X j =0 h j +1 ( hβ ) m − − j j !(2 m − − j )! m − X k =1 j X i =0 − d k q k + p k q N + ik ( − i ( q k − i +1 ∆ i j −− m − X j =2 ( hβ ) m − − j ( − j j !(2 m − − j )! (cid:18) j + 1 − jB (cid:19) + ( hβ ) m − m − (cid:18) − A − B (cid:19) −− ( hβ ) m − m − − m − X j = m ( hβ ) m − − j j !(2 m − − j )! N X γ =0 C [ γ ]( − hγ ) j − A ( hβ ) m − m − B m − X j =0 ( hβ ) m − − j ( − j · j ! · (2 m − − j )! + m − X α =0 λ α ( hβ ) α = ( hβ ) m (2 m )! + m − X j =0 ( − hβ ) m − − j · (2 m − − j )! · ( j + 1)! . Hence equating coefficients of same powers of hβ gives λ j = 1(2 m − − j )! · j ! (cid:18) ( − m − j m − j ) − B m − j h m − j m − j −− h m − j m − X k =1 2 m − − j X i =0 − d k q k + p k q N + ik ( − i ( q k − i +1 ∆ i m − − j ++ 12 N X γ =0 C [ γ ]( − hγ ) m − − j − (2 m − − j ) · B · ( − m − − j (cid:19) , j = 1 , , ..., m − , (5 . λ = 12 · (2 m )! + 12 · (2 m − N X γ =0 C [ γ ]( − hγ ) m − − B · (2 m − , (5 . m − X k =1 j X i =0 d k q k + p k q N + ik ( − i +1 ( q k − i +1 ∆ i j = B j +1 j + 1 , j = 2 , m − , (5 . [0] = h
12 + m − X k =1 p k q Nk − d k q k − q k ! , (5 . A = h − m − X k =1 d k q k + p k q N +1 k (1 − q k ) ! . (5 . d k and p k .Substituting expressions (5.20) and (5.21) into (4.4) and (4.5), also taking into account (5.1),we find C [ N ] and B , which have following form C [ N ] = h
12 + m − X k =1 d k q Nk − p k q k − q k ! , (5 . B = h −
112 + m − X k =1 d k q N +1 k + p k q k (1 − q k ) ! . (5 . λ from (5.17) when j = 1 into (4.2), we get one moreequation with respect to unknowns d k and p k : m − X k =1 2 m − X i =0 d k q k + p k q N + ik ( − i +1 ( q k − i +1 ∆ i m − = 0 , (5 . α = 2 , m − N X γ =0 C [ γ ]( hγ ) α + α B = N − X γ =1 C [ γ ]( hγ ) α + C [ N ] + α B = 1 α + 1 . (5 . L = N − P γ =1 C [ γ ]( hγ ) α . Using (5.1), (2.4), (2.5) for L we get L = h α +1 N − X γ =1 γ α + m − X k =1 d k N − X γ =1 q γk γ α + p k q Nk N − X γ =1 q − γk γ α !! == α +1 X j =1 α ! B α +1 − j j ! ( α + 1 − j )! h α +1 − j + h α +1 m − X k =1 α X i =0 d k q ik + p k q N +1 k ( − i +1 (1 − q k ) i +1 ∆ i α −− h α +1 m − X k =1 α X i =0 d k q N + ik + p k q k ( − i +1 (1 − q k ) i +1 ∆ i N α . ence taking into account(2.6) and grouping in powers of h , we have L = 1 α + 1 + α X j =1 α ! h j ( j − α + 1 − j )! B j j − m − X k =1 j − X i =0 d k q N + ik + p k q k ( − i +1 (1 − q k ) i +1 ∆ i j − ! ++ h α +1 m − X k =1 α X i =0 d k q ik + p k q N +1 k ( − i +1 (1 − q k ) i +1 ∆ i α − m − X k =1 α X i =0 d k q N + ik + p k q k ( − i +1 (1 − q k ) i +1 ∆ i α ! . Substitution of obtained expression of L to (5.25) gives1 α + 1 + α X j =1 α ! h j ( j − α + 1 − j )! B j j − m − X k =1 j − X i =0 d k q N + ik + p k q k ( − i +1 (1 − q k ) i +1 ∆ i j − ! ++ h α +1 m − X k =1 α X i =0 (1 − q Nk ) ( − i +1 d k q ik − p k q k ( q k − i +1 ∆ i α ! + C [ N ] + αB = 1 α + 1 . Hence keeping in mind (5.22), (5.23) we get α X j =3 α ! h j ( j − α + 1 − j )! B j j − m − X k =1 j − X i =0 d k q N + ik + p k q k ( − i +1 (1 − q k ) i +1 ∆ i j − ! ++ h α +1 m − X k =1 α X i =0 (1 − q Nk ) ( − i +1 d k q ik − p k q k ( q k − i +1 ∆ i α ! = 0 . (5 . α + 1 with respect to h . From (5.26)we obtain that each coefficient of this polynomial is equal to zero, i.e. m − X k =1 j − X i =0 d k q N + ik + p k q k ( − i +1 (1 − q k ) i +1 ∆ i j − = B j j , j = 3 , α (5 . m − X k =1 α X i =0 (1 − q Nk ) ( − i +1 d k q ik − p k q k ( q k − i +1 ∆ i α = 0 . (5 . α = 2 , m − m − X k =1 j X i =0 d k q N + ik + p k q k ( − i +1 (1 − q k ) i +1 ∆ i j = B j +1 j + 1 , j = 2 , m − . m − X k =1 α X i =0 d k q ik + p k q N +1 k ( − i +1 (1 − q k ) i +1 ∆ i α = m − X k =1 α X i =0 d k q N + ik + p k q k ( − i +1 (1 − q k ) i +1 ∆ i α , α = 2 , m − . (5 . ence by using (5.30) and lemma 5.2 it is easy to show the system of equations (5.29) is thepart of (5.19). Thus here we get only system of equations (5.30) which is (5.12).Now consider the last equation, i.e. the equation (4.3). Difference of the left and the rightsides of equation (4.3) we denote by K , i.e. K = N X γ =0 C [ γ ] ( hγ − m − m − − A m − m − X α =1 αλ α − m − K = 0.Applying binomial formula for first expression of K and taking into account (4.4)-(4.6), aftersome simplifications K = − m − m − − m − (cid:18) − B (cid:19) + m − X j =1 ( − j j !(2 m − − j )! ++ m − X j =1 ( − j m − j )!( j − − m − X j = m − ( − j B j !(2 m − − j )! −− m − X j =1 ( − j B m − − j )!( j − m − X j =1 ( − j − ( j − m − − j )! N X γ =0 C [ γ ]( hγ ) m − − j ++ m − X j =1 h m − j (2 m − − j )! ( j − " m − X k =1 2 m − − j X i =0 d k q k + p k q N + ik ( − i +1 ( q k − i +1 ∆ i m − − j − B m − j m − j . (5 . N P γ =0 C [ γ ]( hγ ) m − − j in (5.31). For this sum using formulas (5.1), (2.4)-(2.6) after some simplifications we obtain N X γ =0 C [ γ ]( hγ ) m − − j = m − j X i =1 (2 m − − j )! B m − j − i i ! (2 m − j − i )! h m − j − i ++ h m − j m − X k =1 2 m − − j X i =0 d k q ik + p k q N +1 k ( − i +1 (1 − q k ) i +1 ∆ i m − − j −− m − X k =1 2 m − − j X i =0 d k q N + ik + p k q k ( − i +1 (1 − q k ) i +1 ∆ i m − − j ! −− m − − j X p =0 (2 m − − j )! h p +1 p ! (2 m − − j − p )! m − X k =1 p X i =0 d k q N + ik + p k q k ( − i +1 (1 − q k ) i +1 ∆ i p . (5 . ubstituting (5.32) into (5.31) we get polynomial of degree 2 m − h . It is easyto see that constant term and coefficients in front of h and h are zero. Then for K we obtain K = m − X j =1 ( − j − ( j − m − − j )! m − − j X i =1 (2 m − − j )! B m − j − i i ! (2 m − j − i )! h m − j − i −− m − − j X p =2 (2 m − − j )! h p +1 p ! (2 m − − j − p )! m − X k =1 p X i =0 d k q N + ik + p k q k ( − i +1 (1 − q k ) i +1 ∆ i p ! ++ m − X j =1 h m − j ( j − m − − j )! ( − j − m − X k =1 2 m − − j X i =0 d k q ik + p k q N +1 k ( − i +1 (1 − q k ) i +1 ∆ i m − − j ++( − j m − X k =1 2 m − − j X i =0 d k q N + ik + p k q k ( − i +1 (1 − q k ) i +1 ∆ i m − − j ++ m − X k =1 2 m − − j X i =0 d k q k + p k q N + ik ( − i +1 (1 − q k ) i +1 ∆ i m − − j ! − m − X j =1 h m − j B m − j (2 m − j )!( j − . Hence if take into account lemma 5.2 and (5.4*), then the expression in the second parenthesisare simplified and K has the form K = m − X j =1 ( − j − ( j − m − − j X i =1 B m − j − i h m − j − i i ! (2 m − j − i )! − m − X j =1 h m − j B m − j (2 m − j )! ( j − −− m − X j =1 ( − j − ( j − m − − j X p =2 h p +1 p ! (2 m − − j − p )! Z p . (5 . K = m X j =3 B j h j j ! m − X i =0 ( − i i ! (2 m − − j − i )! + m − X j = m +1 B j h j j ! m − − j X i =0 ( − i i ! (2 m − − j − i )! −− m − X j =1 h m − j B m − j (2 m − j )! ( j − − m +1 X j =3 h j Z j − ( j − m − X i =0 ( − i i ! (2 m − − j − i )! −− m − X j = m +2 h j Z j − ( j − m − − j X i =0 ( − i i ! (2 m − − j − i )! = 0 . Hence using (5.19) after simplifications we have K = h m ( m − (cid:18) B m m − Z m − (cid:19) m − X i =0 ( − i i ! ( m − − i )! + m − X j = m +1 B j h j j ! m − − j X i =1 ( − i i !(2 m − − j − i )! −− m − X j = m +2 h j Z j − ( j − m − − j )! m − − j X i =0 (2 m − − j )!( − i i ! (2 m − − j − i )! − h m − Z m − (2 m − h m ( − m [( m − (cid:18) B m m − Z m − (cid:19) + m − X j = m +1 B j h j j ! (2 m − − j )! (cid:0) ( − j − (cid:1) − h m − Z m − (2 m − . Here the middle sum is equal to zero because when j is even ( − j − j is oddBernoulli numbers B j = 0. Therefore finally for K we get K = − h m − Z m − (2 m − h m ( − m [( m − (cid:18) B m m − Z m − (cid:19) = 0 . It means hence taking into account (5.4*) we get following equation m − X k =1 m − X i =0 d k q N + ik + p k q k ( − i +1 (1 − q k ) i +1 ∆ i m − = B m m , (5 . m − X k =1 2 m − X i =0 d k q N + ik + p k q k ( − i +1 (1 − q k ) i +1 ∆ i m − = 0 . (5 . α = 1 and taking into account lemma 5.2 it is easy to show , that theequation (5.34) coincide by equation of the system (5.19) when α = 1. Thus, we have obtainedthe last equation (5.35) for d k and p k , which is the same as (5.13). Theorem is proved. For square of norm of the error functional (1.2) of optimal quadrature formulas of the form(1.1) take placed following theorem.
Theorem 5.2.
For square of norm of the error functional (1.2) of optimal quadratureformula of the form (1.1) following is valid (cid:13)(cid:13)(cid:13) ℓ | L ( m ) ∗ (0 , (cid:13)(cid:13)(cid:13) = ( − m +1 (cid:20) B m h m (2 m )! + h m +1 (2 m )! m − X k =1 2 m X i =0 (1 − q Nk )( d k q ik + ( − i p k q k )(1 − q k ) i +1 ∆ i m , where d k, p k are determined from system (5.10)-(5.13), B m are Bernoulli numbers, q k are rootsof Euler polynomial of degree m − , | q k | < . roof. Computing defined integrals in the expression || ℓ || we get || ℓ || = ( − m +1 " A · B (2 m − − A − B (2 m − N X β =0 C [ β ] A ( hβ ) m − − B ( hβ − m − m − N X β =0 C [ β ] F ( hβ ) + N X β =0 C [ β ] ( F ( hβ ) − N X γ =0 C [ γ ] | hβ − hγ | m − m − A ( hβ ) m − − B ( hβ − m − m − (cid:27) − m + 1)! (cid:21) , where F ( hβ ) is determine by formula (5.16). As is obvious from here according to (4.1) theexpression into curly brackets is equal to the polynomial P m − ( hβ ) = m − P α =0 λ α ( hβ ) α . Then || ℓ || have the form || ℓ || = ( − m +1 " A · B (2 m − − A − B (2 m − N X β =0 C [ β ] A ( hβ ) m − − B ( hβ − m − m − N X β =0 C [ β ] F ( hβ ) + N X β =0 C [ β ] P m − ( hβ ) − m + 1)! (cid:21) . Hence using (4.2) and (4.3) we get || ℓ || = ( − m +1 (cid:20) A · B (2 m − − A − B (2 m − A · (cid:18) m − λ − B m − (cid:19) −− B · m − − m − X α =1 αλ α + A m − ! ++ N X β =0 C [ β ] F ( hβ ) + N X β =0 C [ β ] P m − ( hβ ) − m + 1)! (cid:21) . (5 . || ℓ || = ( − m +1 (cid:20) B − A m − A (2 m − · − m −
1) + 12 N X γ =0 C [ γ ]( hγ ) m − ++ (2 m − B (cid:19) + B · m − X j =1 m − − j )!( j − (cid:18) ( − m − j m − j ) − B m − j h m − j m − j −− h m − j m − X k =1 2 m − − j X i =0 − d k q k + p k q N + ik ( − i ( q k − i +1 ∆ i m − − j + 12 N X γ =0 C [ γ ]( − hγ ) m − − j − (2 m − − j ) B ( − j (cid:19) + N X β =0 C [ β ] F ( hβ ) + N X β =0 C [ β ] P m − ( hβ ) − m + 1)! . Hence, taking into account (5.16), (5.17), (5.18) and using (4.4)-(4.6), after some calculationswe obtain || ℓ || = ( − m +1 (cid:20) B (2 m − − m + 1)! − m − X j =0 B ( − j (2 m − − j )! ( j + 1)! + m − X j =0 ( − j ( j + 1)! (2 m − j )! −− m − X j =2 h m − j (2 m − − j )!( j + 1)! B m − j m − j + m − X k =1 2 m − − j X i =0 − d k q k + p k q N + ik ( − i ( q k − i +1 ∆ i m − − j ! ++ m − X j =0 ( − m − − j (2 m − − j )!( j + 1)! N X γ =0 C [ γ ]( hγ ) m − − j + N X β =0 C [ γ ] ( hβ ) m (2 m )! . (5 . α > m − N X γ =0 C [ γ ]( hγ ) α = 1 α + 1 + α − X j =1 α ! B α +1 − j j !( α + 1 − j )! h α +1 − j ++ h α +1 m − X k =1 α X i =0 d k q ik + p k q N +1 k ( − i +1 (1 − q k ) i +1 ∆ i α −− α X j =1 α ! h j +1 j !( α − j )! m − X k =1 j X i =0 d k q N + ik + p k q k ( − i +1 (1 − q k ) i +1 ∆ i j . (5 . || ℓ || = ( − m +1 " m − X j = m B m − j h m − j (2 m − j )!( j + 1)! + B m h m (2 m )! ++ h m +1 (2 m )! m − X k =1 2 m X i =0 d k q ik + p k q N +1 k ( − i +1 (1 − q k ) i +1 ∆ i m ++ m − X α = m ( − α (2 m − α )! α − X j =1 B α +1 − j h α +1 − j j !( α + 1 − j )! −− m X α = m ( − α (2 m − α )! α X j =2 h j +1 j !( α − j )! m − X k =1 j X i =0 d k q N + ik + p k q k ( − i +1 (1 − q k ) i +1 ∆ i j . ence last two sums regrouping in powers of h , using designation (5.4*) and keepin in mind Z α − = B α α , α = 3 , m , we have || ℓ || = ( − m +1 " m − X j = m B m − j h m − j (2 m − j )!( j + 1)! + B m h m (2 m )! ++ h m +1 (2 m )! m − X k =1 2 m X i =0 d k q ik + p k q N +1 k ( − i +1 (1 − q k ) i +1 ∆ i m ++ m X α =3 B α h α α ! m − X j = m ( − j (2 m − j )! ( j − α + 1)! + m − X α = m +1 B α h α α ! m − X j = α ( − j (2 m − j )! ( j − α + 1)! −− m X α =3 B α h α α ! m X j = m ( − j (2 m − j )! ( j − α + 1)! − m +1 X α = m +1 Z α − h α ( α − m X j = α − ( − j (2 m − j )! ( j − α + 1)! . (5 . m X j = α − ( − j (2 m − j )! ( j − α + 1)! = ( − α − (2 m − α − − m − α +1 = 0 , m + 1 ≤ α ≤ m and m − X j = α ( − j (2 m − j )! ( j − α + 1)! = − m − α + 1)! (( − α − + 1) , then, using these equalities, from (5.38) we obtain k ℓ k = ( − m +1 (cid:20) B m h m (2 m )! + h m +1 (2 m )! m − X k =1 2 m X i =0 (1 − q Nk )( d k q ik + ( − i p k q k )(1 − q k ) i +1 ∆ i m −− m − X j = m +1 B α (( − α − + 1) α ! (2 m − α + 1)! . Hence taking into account that when α is even ( − α − + 1 = 0 and when α is odd B α = 0,(since α = 1), we get the statement of theorem 5.2. Theorem 5.2 is proved. Corollary 5.1.
In the space L (2)2 (0 , among quadrature formulas of the form (1.1) withthe error functional (1.2) there exists unique optimal formula which coefficients are determinedby following formulas C [ β ] = h , β = 0 , N,h, β = 1 , N − , = h , B = − h . Furthermore for square of norm of the error functional following is valid (cid:13)(cid:13)(cid:13) ℓ | L (2) ∗ (0 , (cid:13)(cid:13)(cid:13) = h . Corollary 5.2.
In the space L (3)2 (0 , among quadrature formulas of the form (1.1) withthe error functional (1.2) there exists unique optimal formula which coefficients are determinedby following formulas C [ β ] = h , β = 0 , N,h, β = 1 , N − ,A = h , B = − h . Furthermore for square of norm of the error functional following is valid (cid:13)(cid:13)(cid:13) ℓ | L (3) ∗ (0 , (cid:13)(cid:13)(cid:13) = h . Proofs of Corollaries 5.1 and 5.2 we get immediately from theorems 5.1 and 5.2 when m = 2and m = 3 respectively. The second author gratefully acknowledges the Abdus Salam School of Mathematical Sciences(ASSMS), GC University, Lahore, Pakistan for providing the Post Doctoral Research Fellow-ship.
References [1] A.Sard. Best approximate integration formulas, best approximate formulas, American J.of Math. V.71, No 1, (1949), pp.80-91.
2] I.F.Meyers, A.Sard. Best approximate integration formulas, J. Math and Phys. XXIX,(1950), pp.118-123.[3] I.J.Schoenberg and S.D.Silliman. On semicardinal quadrature formulae, Math. Comp.,V.126, (1974), pp.483-497.[4] F.Ya.Zagirova. On construction of optimal quadrature formulas with equal spaced nodes.-Novosibirsk, (1982), 28 p. (Preprint No 25, Institute of Mathematics SD of AS of USSR).[5] S.L.Sobolev. The coefficients of optimal quadrature formulas, Selected Works ofS.L.Sobolev. Springer, (2006). pp.561-566.[6] Kh.M.Shadimetov. Optimal quadrature formulas in the L ( m )2 (Ω) and L ( m )2 ( R ), Dokl. ANRUz, No 3, (1983), pp.5-8. (in Russian)[7] Kh.M.Shadimetov. Construction of weight optimal quadrature formulas in the space L ( m )2 (0 , N ), Siberian Journal of Computational Mathematics, V.5, No 3, (2002), pp.275-293. (in Russian)[8] S.L.Sobolev. On the roots of Euler Polynomials. Selected Works of S.L.Sobolev. Springer,(2006). pp.567-572.[9] Kh.M.Shadimetov. Optimal Formulas of Approximate Integration for differentiable Func-tions, Candidate dissertation. -Novosibirsk, (1983). 140p. (in Russian)[10] R.W.Hamming. Numerical Methods for Scientists and Engenerees, NY, McGraw Bill BookCompany, Inc. USA. 1962. 411p.[11] A.O.Gelfond. Calculus of finite differences. - Moscow. Nauka, (1967). 376 p. (in Russian)[12] S.L.Sobolev. Introduction to the Theory of Cubature Formulas. Moscow. Nauka, (1974)808 p.[13] S.L.Sobolev, V.L.Vaskevich. The Theory of Cubature Formulas, Kluwer Academic Pub-lishers Group, Dordrecht, (1997). 416 p.
14] Kh.M.Shadimetov. Discrete analogue of the differential operator d m dx m and its construction,Problems of Computational and Applied Mathematics. -Tashkent, (1985). pp.22-35. (inRussian) Shadimetov Kholmat Mahkambaevich
Institute of Mathematics and Information TechnologiesUzbek Academy of Sciences29, Durmon yuli strTashkent, 100125, Uzbekistan
Hayotov Abdullo Rahmonovich
Institute of Mathematics and Information TechnologiesUzbek Academy of Sciences29, Durmon yuli strTashkent, 100125, Uzbekistane-mail: [email protected], abdullo [email protected]
Nuraliev Farhod Abduganievich