Generalizations of Guessab-Schmeisser formula via Fink type identity with applications to quadrature rules
aa r X i v : . [ m a t h . C A ] M a r GENERALIZATIONS OF GUESSAB–SCHMEISSER FORMULAVIA FINK TYPE IDENTITY WITH APPLICATIONS TOQUADRATURE RULES
MOHAMMAD W. ALOMARI
Abstract.
In this work, an expansion of Guessab–Schmeisser two points for-mula for n -times differentiable functions via Fink type identity is established.Generalization of the main result for harmonic sequence of polynomials isestablished. Several bounds of the presented results are proved. As applica-tions, some quadrature rules are elaborated and discussed. Error bounds ofthe presented quadrature rules via Chebyshev-Gr¨uss type inequalities are alsoprovided. Introduction
For a continuous function f defined on [ a, b ], the integral mean-value theorem(IMVT) guarantees an x ∈ [ a, b ] such that f ( x ) = 1 b − a Z ba f ( t ) dt. (1.1)In order to measure the difference between any value of f in [ a, b ] and its weightedvalue, Ostrowski in his celebrated work [45] established a very interesting inequalityfor differentiable functions with bounded derivatives which in connection with (1.1),which reads: Theorem 1.
Let f : I ⊂ R → R be a differentiable function on I ◦ , the interiorof the interval I, such that f ′ ∈ L [ a, b ] , where a, b ∈ I with a < b . If k f ′ k ∞ =sup x ∈ [ a,b ] | f ′ ( x ) | ≤ ∞ . Then, the inequality (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( b − a ) f ( x ) − Z ba f ( u ) du (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ " ( b − a ) (cid:18) x − a + b (cid:19) k f ′ k ∞ , (1.2) holds for all x ∈ [ a, b ] . The constant is the best possible in the sense that it cannotbe replaced by a smaller ones. In 1976 Milovanovi´c and Peˇcari´c [43] presented their famous generalization of(1.1) via Taylor series, where they proved that: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n f ( x ) + n − X k =1 F k ( x ) ! − b − a Z ba f ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( n, ∞ , x ) (cid:13)(cid:13)(cid:13) f ( n ) (cid:13)(cid:13)(cid:13) ∞ , (1.3) Date : October 15, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Ostrowski inequality, Euler–Maclaurin formula, Quadrature formula,Approximations, Expansions. such that F k ( x ) = n − kn ! f ( k − ( a ) ( x − a ) k − f ( k − ( b ) ( x − b ) k b − a . (1.4)In fact, Milovanovi´c and Peˇcari´c proved the case that C ( n, ∞ , x ) = ( x − a ) n +1 + ( b − x ) n +1 ( b − a ) n ( n + 1)! . In 1992, Fink studied (1.3) in different point of view, he introduced a new rep-resentation of real n -times differentiable function whose n -th derivative ( n ≥ n f ( x ) + n − X k =1 F k ! − b − a Z ba f ( y ) dy = 1 n ! ( b − a ) Z ba ( x − t ) n − p ( t, x ) f ( n ) ( t ) dt, for all x ∈ [ a, b ], where p ( t, x ) = (cid:26) t − a, t ∈ [ a, x ] t − b, t ∈ [ x, b ] . (1.6)In the same work, Fink proved the following bound of (1.5). (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n f ( x ) + n − X k =1 F k ( x ) ! − b − a Z ba f ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( n, p, x ) (cid:13)(cid:13)(cid:13) f ( n ) (cid:13)(cid:13)(cid:13) p (1.7)where k·k r , 1 ≤ r ≤ ∞ are the usual Lebesgue norms on L r [ a, b ], i.e., k f k ∞ := ess sup t ∈ [ a,b ] | f ( t ) | , and k f k r := Z ba | f ( t ) | r dt ! /r , ≤ r < ∞ , such that C ( n, p, x ) = h ( x − a ) nq +1 + ( b − x ) nq +1 i /q ( b − a ) n ! B /q (( n − q + 1 , q + 1) , for 1 < p ≤ ∞ , B ( · , · ) is the beta function, and for p = 1 C ( n, , x ) = ( n − n − ( b − a ) n n n ! max { ( x − a ) n , ( b − x ) n } . All previous bounds are sharp.Indeed Fink representation can be considered as the first elegant work (afterDarboux work [39], p.49) that combines two different approaches together, so thatFink representation is not less important than Taylor expansion itself. So that,many authors were interested to study Fink representation approach, more detailedand related results can be found in [1],[2],[13],[14] and [20].
ENERALIZATIONS OF GUESSAB–SCHMEISSER FORMULA 3
In 2002 and the subsequent years after that, the Ostrowski’s inequality enteredin a new phase of modifications and developments. A new inequality of Ostrowski’stype was born, where Guessab and Schmeisser in [36] discussed an inequality fromalgebraic and analytic points of view which is in connection with Ostrowski inequal-ity; called ‘ the companion of Ostrowski’s inequality ’ as suggested later by Dragomirin [26]. The main part of Guessab–Schmeisser inequality reads the difference be-tween symmetric values of a real function f defined on [ a, b ] and its weighed value,i.e., f ( x ) + f ( a + b − x )2 − b − a Z ba f ( t ) dt, x ∈ (cid:20) a, a + b (cid:21) . Namely, in the significant work [36] we find the first primary result is that:
Theorem 2.
Let f : [ a, b ] → R be satisfies the H¨older condition of order r ∈ (0 , .Then for each x ∈ [ a, a + b ] , the we have the inequality (1.8) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( x ) + f ( a + b − x )2 − b − a Z ba f ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Mb − a (2 x − a ) r +1 + ( a + b − x ) r +1 r ( r + 1) . This inequality is sharp for each admissible x . Equality is attained if and only if f = ± M f ∗ + c with c ∈ R and f ∗ ( t ) = ( x − t ) r , if a ≤ t ≤ x ( t − x ) r , if x ≤ t ≤ a + b f ∗ ( a + b − t ) , if a + b ≤ t ≤ b . In the same work [36], the authors discussed and investigated (1.8) for othertype of assumptions. Among others, a brilliant representation (or identity) of n -times differentiable functions whose n -th derivatives are piecewise continuous wasestablished as follows: Theorem 3.
Let f be a function defined on [ a, b ] and having there a piecewisecontinuous n -th derivative. Let Q n be any monic polynomial of degree n such that Q n ( t ) = ( − n Q n ( a + b − t ) . Define K n ( t ) = ( t − a ) n , if a ≤ t ≤ xQ n ( t ) , if x ≤ t ≤ a + b − x ( t − b ) n , if a + b − x ≤ t ≤ b . Then, Z ba f ( t ) dt = ( b − a ) f ( x ) + f ( a + b − x )2 + E ( f ; x )(1.9) M.W. ALOMARI where, E ( f ; x ) = n − X ν =1 " ( x − a ) ν +1 ( ν + 1)! − Q ( n − ν − n ( x ) n ! f ( ν ) ( a + b − x ) + ( − ν f ( x ) i + ( − n ! Z ba K n ( t ) f ( n ) ( t ) dt. This generalization (1.9) can be considered as a companion type expansion ofEuler–Maclaurin formula that expand symmetric values of real functions. In thisway, families of various quadrature rules can be presented, as shown -for example-in [38]. Therefore, since 2002 and after the presentation of (1.8), several authorshave studied, developed and established new presentations concerning (1.8) usingseveral approaches and different tools, for this purpose see the recent survey [25].Far away from this, in the last thirty years the concept of harmonic sequence ofpolynomials or Appell polynomials have been used at large in numerical integrationsand expansions theory of real functions. Let us recall that, a sequence of polynomi-als { P k ( t, · ) } ∞ k =0 satisfies the Appell condition (see [12]) if ∂∂t P k ( t, · ) = P k − ( t, · )( ∀ k ≥
1) with P ( t, · ) = 1, for all well-defined order pair ( t, · ). A slightly differentdefinition was considered in [42].In 2003, motivated by work of Mati´c et. al. [42], Dedi´c et. al. in [20], introducedthe following smart generalization of Ostrowski’s inequality via harmonic sequenceof polynomials:(1.10) 1 n " f ( x ) + n − X k =1 ( − k P k ( x ) f ( k ) ( x ) + n − X k =1 f F k ( a, b ) = ( − n − ( b − a ) n Z ba P n − ( t ) p ( t, x ) f ( n ) ( t ) dt, where P k is a harmonic sequence of polynomials satisfies that P ′ k = P k − with P = 1, f F k ( a, b ) = ( − k ( n − k ) b − a h P k ( a ) f ( k − ( a ) − P k ( b ) f ( k − ( b ) i (1.11)and p ( t, x ) is given in (1.5). In particular, if we take P k ( t ) = ( t − x ) k k ! then we referto Fink representation (1.5).In 2005, Dragomir [26] proved the following bounds of the companion of Os-trowski’s inequality for absolutely continuous functions. ENERALIZATIONS OF GUESSAB–SCHMEISSER FORMULA 5
Theorem 4.
Let f : I ⊂ R → R be an absolutely continuous function on [ a, b ] .Then we have the inequalities (1.12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( x ) + f ( a + b − x )2 − b − a Z ba f ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ " + 2 (cid:18) x − a + b b − a (cid:19) ( b − a ) k f ′ k ∞ , f ′ ∈ L ∞ [ a, b ] /q ( q +1) /q "(cid:16) x − ab − a (cid:17) q +1 − (cid:18) a + b − xb − a (cid:19) q +1 /q ( b − a ) /q k f ′ k [ a,b ] ,p ,p > , p + q = 1 , and f ′ ∈ L p [ a, b ] (cid:20) + (cid:12)(cid:12)(cid:12)(cid:12) x − a + b b − a (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) k f ′ k [ a,b ] , for all x ∈ [ a, a + b ] . The constants and are the best possible in (1.12) in thesense that it cannot be replaced by smaller constants. The author of this paper have took a serious attention to Guessab–Schmeisserinequality in the works [3]–[11]. For other related results and generalizations con-cerning Ostrowski’s inequality and its applications we refer the reader to [13]–[18],[24], [26]–[28], [37], [40], [47] and [48].In the last fifteen years, constructions of quadrature rules using expansion of anarbitrary function in Bernoulli polynomials and Euler–Maclaurin’s type formulaehave been established, improved and investigated. These approaches permit manyresearchers to work effectively in the area of numerical integration where severalerror approximations of various quadrature rules presented with high degree ofexactness. Mainly, works of Dedi´c et al. [20]–[24], Aljinovi´c et al. [1], [2], Kova´cet al. [38] and others, received positive responses and good interactions from otherfocused researchers. Among others, Franji´c et al. in several works (such as [30]–[34]) constructed several Newton-Cotes and Gauss quadrature type rules using acertain expansion of real functions in Bernoulli polynomials or Euler–Maclaurin’stype formulae.Unfortuentaley, the expansions (1.5), (1.9) and (1.10) have not been used to con-struct quadrature rules yet. It seems these expansions were abandoned or neglectedin literature because most of authors are still use the classical Euler–Maclaurin’sformula and expansions in Bernoulli polynomials.This work has several aims and goals, the first aim is to generalize Guessab–Schmeisser two points formula for n -times differentiable functions via Fink typeidentity and provide several type of bounds for the remainder formula. The secondgoal, is to highlight the importance of these expansions and give a serious attentionto their applicable usefulness in constructing various quadrature rules. The thirdaim, is to spotlight the role of ˇCebyˇsev functional in integral approximations.This work is organized as follows: in the next section, a Guessab–Schmeissertwo points formula for n -times differentiable functions via Fink type identity isestablished. Bounds for the remainder term of the presented formula are proved.In section 3, bounds for the remainder term via Chebyshev-Gr¨uss type inequalitiesare presented. In section 4, generalizations of the obtained results to harmonic M.W. ALOMARI sequence of polynomials are given. In section 5, representations of some quadraturerules are introduced and their errors are explored.2.
The Results
Guessab–Schmeisser formula via Fink type identity.Theorem 5.
Let I be a real interval, a, b ∈ I ◦ ( a < b ) . Let f : I → R be n -times differentiable on I ◦ such that f ( n ) is absolutely continuous on I ◦ with ( · − t ) n − S ( t, · ) f ( n ) ( t ) is integrable. Then we have the representation (2.1) 1 n f ( x ) + f ( a + b − x )2 + n − X k =1 G k ! − b − a Z ba f ( y ) dy = 1 n ! ( b − a ) Z ba ( x − t ) n − S ( t, x ) f ( n ) ( t ) dt, for all x ∈ (cid:2) a, a + b (cid:3) , where (2.2) G k := G k ( x ) = ( n − k ) k ! ( b − a ) · n ( x − a ) k h f ( k − ( a ) + ( − k +1 f ( k − ( b ) i + (cid:16) − k +1 (cid:17) (cid:18) a + b − x (cid:19) k f ( k − (cid:18) a + b (cid:19)) , and S ( t, x ) = t − a, t ∈ [ a, x ] t − a + b , t ∈ ( x, a + b − x ) t − b, t ∈ [ a + b − x, b ] . (2.3) Proof.
Fix x ∈ [ a, b ]. Starting with Taylor series expansion for f along (cid:2) a, a + b (cid:3) f ( x ) = f ( y ) + n − X k =1 f ( k ) ( y ) k ! ( x − y ) k + 1( n − Z xy ( x − t ) n − f ( n ) ( t ) dt. (2.4)Integrating with respect to y along (cid:2) a, a + b (cid:3) , we have(2.5) b − a f ( x ) = Z a + b a f ( y ) dy + n − X k =1 k ! Z a + b a ( x − y ) k f ( k ) ( y ) dy + 1( n − Z a + b a (cid:18)Z xy ( x − t ) n − f ( n ) ( t ) dt (cid:19) dy. Also, for x ∈ (cid:2) a + b , b (cid:3) , f has the representation(2.6) f ( a + b − x ) = f ( y ) + n − X k =1 f ( k ) ( y ) k ! ( a + b − x − y ) k + 1( n − Z a + b − xy ( a + b − x − t ) n − f ( n ) ( t ) dt. ENERALIZATIONS OF GUESSAB–SCHMEISSER FORMULA 7
Integrating with respect to y along (cid:2) a + b , b (cid:3) , we have(2.7) b − a f ( a + b − x ) = Z b a + b f ( y ) dy + n − X k =1 k ! Z b a + b ( a + b − x − y ) k f ( k ) ( y ) dy + 1( n − Z b a + b Z a + b − xy ( a + b − x − t ) n − f ( n ) ( t ) dt ! dy Adding (2.5) and (2.7), we get( b − a ) f ( x ) + f ( a + b − x )2= Z ba f ( y ) + n − X k =1 I k (2.8) + 1( n − Z a + b a (cid:18)Z xy ( x − t ) n − f ( n ) ( t ) dt (cid:19) dy + Z ba + b Z a + b − xy ( a + b − x − t ) n − f ( n ) ( t ) dt ! dy , where, I k = J k + h k , I = R ba f ( y ) dy , J k = k ! R a + b a ( x − y ) k f ( k ) ( y ) dy and h k = k ! R b a + b ( a + b − x − y ) k f ( k ) ( y ) dy ( k ≥ n − k ) ( J k − J k − ) = − ( b − a ) D k , (1 ≤ k ≤ n − , (2.9)where, D k = ( n − k ) k ! · ( x − a ) k f ( k − ( a ) − (cid:0) x − a + b (cid:1) k f ( k − (cid:0) a + b (cid:1) b − a . Similarly we have( n − k ) ( ℓ k − ℓ k − ) = − ( b − a ) L k , (1 ≤ k ≤ n − L k = ( n − k ) k ! · (cid:0) a + b − x (cid:1) k f ( k − (cid:0) a + b (cid:1) − ( a − x ) k f ( k − ( b ) b − a . Therefore, by adding (2.9) and (2.10) we get( n − k ) ( I k − I k − ) = − ( b − a ) G k , (1 ≤ k ≤ n − G k = D k + L k .Summing the terms in (2.11) form k = 1 up to k = n −
1, simplifications lead usto write n − X k =1 I k = − ( b − a ) n − X k =1 G k + ( n − I . (2.12) M.W. ALOMARI
Substituting (2.12) in (2.8) and rearrange the terms we get that(2.13) 1 n f ( x ) + f ( a + b − x )2 + n − X k =1 G k ! − b − a Z ba f ( y ) dy = 1 n ! ( b − a ) Z a + b a (cid:18)Z xy ( x − t ) n − f ( n ) ( t ) dt (cid:19) dy + Z ba + b Z a + b − xy ( a + b − x − t ) n − f ( n ) ( t ) dt ! dy . To simplify the right hand side, we write Z a + b a dy Z xy dt = Z xa dy Z xy dt + Z a + b x dy Z xy dt = Z xa dt Z ta dy − Z a + b x dy Z xy dt = Z xa dt Z ta dy − Z a + b x dt Z a + b t dy, (2.14)and Z b a + b dy Z a + b − xy dt = Z a + b − x a + b dy Z a + b − xy dt + Z ba + b − x dy Z a + b − xy dt = Z a + b − x a + b dt Z t a + b dy − Z b a + b dy Z a + b − xy dt = Z a + b − x a + b dt Z t a + b dy − Z ba + b − x dt Z bt dy. (2.15)Adding (2.14) and (2.15), we get(2.16) Z a + b a dy Z xy dt + Z b a + b dy Z a + b − xy dt = Z xa dt Z ta dy − Z a + b x dt Z a + b t dy + Z a + b − x a + b dt Z t a + b dy − Z ba + b − x dt Z bt dy. In viewing(2.16), the right hand side of (2.13) becomes1 n ! ( b − a ) Z a + b a (cid:18)Z xy ( x − t ) n − f ( n ) ( t ) dt (cid:19) dy + Z ba + b Z a + b − xy ( a + b − x − t ) n − f ( n ) ( t ) dt ! dy = 1 n ! ( b − a ) Z ba ( x − t ) n − S ( t, x ) f ( n ) ( t ) dt, ENERALIZATIONS OF GUESSAB–SCHMEISSER FORMULA 9 where S ( t, x ) = t − a, t ∈ [ a, x ] t − a + b , t ∈ ( x, a + b − x ) t − b, t ∈ [ a + b − x, b ] . Thus, the identity (2.13) becomes(2.17) 1 n f ( x ) + f ( a + b − x )2 + n − X k =1 G k ( x ) ! − b − a Z ba f ( y ) dy = 1 n ! ( b − a ) Z ba ( x − t ) n − S ( t, x ) f ( n ) ( t ) dt for all x ∈ (cid:2) a, a + b (cid:3) . (cid:3) Theorem 6.
Under the assumptions of Theorem 5. We have (2.18) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n f ( x ) + f ( a + b − x )2 + n − X k =1 G k ! − b − a Z ba f ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( n, p, x ) (cid:13)(cid:13)(cid:13) f ( n ) (cid:13)(cid:13)(cid:13) p holds, where M ( n, p, x ) = n · n !( b − a ) (cid:0) n − n (cid:1) n − (cid:2) b − a + (cid:12)(cid:12) x − a + b (cid:12)(cid:12)(cid:3) n , if p = 1 /q n !( b − a ) h ( x − a ) nq +1 + (cid:0) a + b − x (cid:1) nq +1 i /q × B q (( n − q + 1 , q + 1) , if 1 < p ≤ ∞ , q = pp − . (2.19) The constant C ( n, p, x ) is the best possible in the sense that it cannot be replacedby a smaller ones.Proof. Utilizing the triangle integral inequality on the identity (2.1) and employingsome known norm inequalities we get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n f ( x ) + f ( a + b − x )2 + n − X k =1 G k ( x ) ! − b − a Z ba f ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ n ! ( b − a ) Z ba | x − t | n − | S ( t, x ) | (cid:12)(cid:12)(cid:12) f ( n ) ( t ) (cid:12)(cid:12)(cid:12) dt ≤ (cid:13)(cid:13) f ( n ) (cid:13)(cid:13) sup a ≤ t ≤ b n | x − t | n − | k ( t, x ) | o , p = 1 (cid:13)(cid:13) f ( n ) (cid:13)(cid:13) p (cid:16)R ba | x − t | ( n − q | k ( t, x ) | q dt (cid:17) /q , < p < ∞ (cid:13)(cid:13) f ( n ) (cid:13)(cid:13) ∞ R ba | x − t | n − | k ( t, x ) | dt, p = ∞ . It is easy to find that for p = 1, we havesup a ≤ t ≤ b n | x − t | n − | S ( t, x ) | o = 1 n (cid:18) n − n (cid:19) n − max (cid:26) ( x − a ) n , (cid:18) a + b − x (cid:19) n (cid:27) = 1 n (cid:18) n − n (cid:19) n − (cid:20) b − a (cid:12)(cid:12)(cid:12)(cid:12) x − a + b (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) n , and for 1 < p < ∞ , we have Z ba | x − t | ( n − q | S ( t, x ) | q dt = Z xa | x − t | ( n − q ( t − a ) q dt + Z a + b − xx | x − t | ( n − q (cid:12)(cid:12)(cid:12)(cid:12) t − a + b (cid:12)(cid:12)(cid:12)(cid:12) q dt + Z ba + b − x | x − t | ( n − q ( b − t ) q dt = 2 " ( x − a ) nq +1 + (cid:18) a + b − x (cid:19) nq +1 (1 − s ) ( n − q s q ds (cid:19) = 2 " ( x − a ) nq +1 + (cid:18) a + b − x (cid:19) nq +1 B (( n − q + 1 , q + 1)where, we use the substitutions t = (1 − s ) a + sx , t = (1 − s ) x + s ( a + b − x ) and t = (1 − s ) ( a + b − x ) + sb ; respectively.The third case, p = ∞ holds by setting p = ∞ and q = 1, i.e., Z ba | x − t | ( n − | S ( t, x ) | dt = 2 " ( x − a ) n +1 + (cid:18) a + b − x (cid:19) n +1 B ( n, , where B ( · , · ) is the Euler beta function. To argue the sharpness, we consider firstwhen 1 < p ≤ ∞ , so that the equality in (2.1) holds when f ( n ) ( t ) = | x − t | ( n − q − | S ( t, x ) | q − sgn n ( x − t ) n − S ( t, x ) o , thus the inequality (2.18) holds for 1 < p ≤ ∞ . In case that p = 1, setting g ( t, x ) = ( x − t ) n − S ( t, x ) ∀ x ∈ (cid:2) a, a + b (cid:3) , let t be the point that gives the supremum. If t = x +( n − an , we take f ( n ) ε ( t ) = (cid:26) ε − , t ∈ ( t − ε, t )0 , otherwise . Since (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ba g ( t, x ) f ( n ) ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 1 ε (cid:12)(cid:12)(cid:12)(cid:12)Z t t − ε g ( t, x ) dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε Z t t − ε | g ( t, x ) | dt ≤ sup t − ε ≤ t ≤ t | g ( t, x ) | · ε Z t t − ε dt = | g ( t , x ) | , ENERALIZATIONS OF GUESSAB–SCHMEISSER FORMULA 11 also, we have lim ε → + ε Z t t − ε | g ( t, x ) | dt = | g ( t , x ) | = C ( n, , x )proving that C ( n, , x ) is the best possible. (cid:3) Corollary 1.
Under the assumptions of Theorem 4.(1) If k is even and f ( k − ( a ) = f ( k − ( b ) = 0 , for all k = 1 , · · · , n − . Then, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n (cid:18) f ( x ) + f ( a + b − x )2 (cid:19) − b − a Z ba f ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( n, p, x ) (cid:13)(cid:13)(cid:13) f ( n ) (cid:13)(cid:13)(cid:13) p . (2.20) (2) If f ( k − ( a ) = f ( k − (cid:0) a + b (cid:1) = f ( k − ( b ) = 0 , for all k = 1 , · · · , n − .Then the inequality (2.20) holds. Remark 1.
In Theorem 6, if one assumes that f ( n ) is n -convex, r -convex, quasi-convex, s -convex, P -convex, or Q -convex; we may obtain other new bounds involv-ing convexity. Bounds via Chebyshev-Gr¨uss type inequalities
The celebrated ˇCebyˇsev functional(3.1) C ( h , h )= 1 d − c Z dc h ( t ) h ( t ) dt − d − c Z dc h ( t ) dt · d − c Z dc h ( t ) dt. has multiple applications in several subfields including Numerical integrations,Probability Theory & Statistics, Functional Analysis, Operator Theory and oth-ers. For more detailed history see [44].The most famous bounds of the ˇCebyˇsev functional are incorporated in thefollowing theorem: Theorem 7.
Let f, g : [ c, d ] → R be two absolutely continuous functions, then |C ( h , h ) | ≤ ( d − c ) k h ′ k ∞ k h ′ k ∞ , if h ′ , h ′ ∈ L ∞ ([ c, d ]) , proved in [19] ( M − m ) ( M − m ) , if m ≤ h ≤ M , m ≤ h ≤ M , proved in [35] ( d − c ) π k h ′ k k h ′ k , if h ′ , h ′ ∈ L ([ c, d ]) , proved in [41] ( d − c ) ( M − m ) k h ′ k ∞ , if m ≤ h ≤ M, h ′ ∈ L ∞ ([ c, d ]) , proved in [45](3.2) The constants , , π and are the best possible. In this section, we highlight the role of ˇCebyˇsev functional in integral approxi-mations by using the ˇCebyˇsev–Gr¨uss type inequalities (3.2).
Setting h ( t ) = n ! f ( n ) ( t ) and h ( t ) = ( x − t ) n − k ( t, x ), we have C ( h , h ) = 1 n ! ( b − a ) Z ba ( x − t ) n − k ( t, x ) f ( n ) ( t ) dt − n ! · b − a Z ba ( x − t ) n − k ( t, x ) dt · b − a Z ba f ( n ) ( t ) dt = 1 n ! ( b − a ) Z ba ( x − t ) n − k ( t, x ) f ( n ) ( t ) dt − n ! ( b − a ) " ( x − a ) n +1 + (cid:18) a + b − x (cid:19) n +1 B ( n, × f ( n − ( b ) − f ( n − ( a ) b − a which means C ( h , h ) = 1 n f ( x ) + f ( a + b − x )2 + n − X k =1 G k ( x ) ! − b − a Z ba f ( y ) dy − n + 1)! n ( b − a ) " ( x − a ) n +1 + (cid:18) a + b − x (cid:19) n +1 × f ( n − ( b ) − f ( n − ( a ) b − a := P ( f ; x, n ) . Theorem 8.
Let I be a real interval, a, b ∈ I ◦ ( a < b ) . Let f : I → R be ( n + 1) -times differentiable on I ◦ such that f ( n +1) is absolutely continuous on I ◦ with ( · − t ) n − k ( t, · ) f ( n ) ( t ) is integrable. Then, for all n ≥ we have (3.3) |P ( f ; x, n ) |≤ ( b − a ) (cid:0) n − n (cid:1) n − n − n +212 n · ( n !) (cid:2) b − a + (cid:12)(cid:12) x − a + b (cid:12)(cid:12)(cid:3) n − · (cid:13)(cid:13) f ( n +1) (cid:13)(cid:13) ∞ , if f ( n +1) ∈ L ∞ ([ a, b ]) (cid:0) n − n (cid:1) n − n − n +24 n · ( n !) (cid:0) − n − − − n − (cid:1) ( b − a ) n − · ( M − m ) , if m ≤ f ( n ) ≤ M, b − a ( n !) π q A ( n ) ( x − a ) n − + B ( n ) (cid:0) a + b − x (cid:1) n − · (cid:13)(cid:13) f ( n +1) (cid:13)(cid:13) , if f ( n +1) ∈ L ([ a, b ]) , ( b − a ) (cid:0) n − n (cid:1) n − n − n +28 n · ( n !) (cid:2) b − a + (cid:12)(cid:12) x − a + b (cid:12)(cid:12)(cid:3) n − · ( M − m ) , if m ≤ f ( n ) ≤ M, (cid:0) n − n (cid:1) n − n − n +28 n · ( n !) (cid:0) − n − − − n − (cid:1) ( b − a ) n · (cid:13)(cid:13) f ( n +1) (cid:13)(cid:13) ∞ , if f ( n +1) ∈ L ∞ ([ a, b ]) , holds for all x ∈ (cid:2) a, a + b (cid:3) , where A ( n ) = 2 ( n − (2 n −
1) (2 n −
2) (2 n − and B ( n ) = 2 n − (2 n −
1) (2 n −
2) + 4 n (2 n −
1) + 2 n (2 n −
1) (2 n −
2) (2 n − ENERALIZATIONS OF GUESSAB–SCHMEISSER FORMULA 13 ∀ n ≥ .Proof. • If f ( n +1) ∈ L ∞ ([ a, b ]): Applying the first inequality in (3.2), it is notdifficult to observe that sup a ≤ t ≤ b {| h ′ ( t ) |} = n ! (cid:13)(cid:13) f ( n +1) (cid:13)(cid:13) ∞ andsup a ≤ t ≤ b {| h ′ ( t ) |} = (cid:18) n − n (cid:19) n − n − n + 2 n (cid:20) b − a (cid:12)(cid:12)(cid:12)(cid:12) x − a + b (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) n − , ∀ n ≥ . So that |P ( f ; x, n ) | ≤ ( b − a ) (cid:18) n − n (cid:19) n − n − n + 212 n · n ! (cid:20) b − a (cid:12)(cid:12)(cid:12)(cid:12) x − a + b (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) n − · n ! (cid:13)(cid:13)(cid:13) f ( n +1) (cid:13)(cid:13)(cid:13) ∞ . • If m ≤ f ( n ) ( t ) ≤ M , for some m, M >
0: Applying the second inequality in(3.2), we get |P ( f ; x, n ) | ≤ n − n + 24 n · n ! (cid:18) n − n (cid:19) n − (cid:0) − n − − − n − (cid:1) ( b − a ) n − · n ! ( M − m ) . • If f ( n +1) ∈ L ([ a, b ]): Applying the third inequality in (3.2), we get |P ( f ; x, n ) | ≤ ( b − a ) n ! π · s A ( n ) ( x − a ) n − + B ( n ) (cid:18) a + b − x (cid:19) n − · n ! (cid:13)(cid:13)(cid:13) f ( n +1) (cid:13)(cid:13)(cid:13) ∀ n ≥
2, where A ( n ) and B ( n ) are defined above. • If m ≤ f ( n ) ( t ) ≤ M , for some m, M >
0: Applying the forth inequality in(3.2), we get |P ( f ; x, n ) | ≤ ( b − a ) (cid:18) n − n (cid:19) n − n − n + 28 n · n ! (cid:20) b − a (cid:12)(cid:12)(cid:12)(cid:12) x − a + b (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) n − · n ! ( M − m ) . By applying the forth inequality again the with dual assumptions, i.e., f ( n +1) ∈ L ∞ ([ a, b ]), we have |P ( f ; x, n ) | ≤ n − n + 28 n · n ! (cid:18) n − n (cid:19) n − (cid:0) − n − − − n − (cid:1) ( b − a ) n · n ! (cid:13)(cid:13)(cid:13) f ( n +1) (cid:13)(cid:13)(cid:13) ∞ . Hence the proof is completely established. (cid:3)
Corollary 2.
Let assumptions of Theorem 8 hold. If moreover, f ( n − ( a ) = f ( n − ( b ) ( n ≥ , then the inequality (3.4) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n f ( x ) + f ( a + b − x )2 + n − X k =1 G k ! − b − a Z ba f ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( b − a ) (cid:0) n − n (cid:1) n − n − n +212 n · ( n !) (cid:2) b − a + (cid:12)(cid:12) x − a + b (cid:12)(cid:12)(cid:3) n − · (cid:13)(cid:13) f ( n +1) (cid:13)(cid:13) ∞ , if f ( n +1) ∈ L ∞ ([ a, b ]) (cid:0) n − n (cid:1) n − n − n +24 n · ( n !) (cid:0) − n − − − n − (cid:1) ( b − a ) n − · ( M − m ) , if m ≤ f ( n ) ≤ M, b − a ( n !) π q A ( n ) ( x − a ) n − + B ( n ) (cid:0) a + b − x (cid:1) n − · (cid:13)(cid:13) f ( n +1) (cid:13)(cid:13) , if f ( n +1) ∈ L ([ a, b ]) , ( b − a ) (cid:0) n − n (cid:1) n − n − n +28 n · ( n !) (cid:2) b − a + (cid:12)(cid:12) x − a + b (cid:12)(cid:12)(cid:3) n − · ( M − m ) , if m ≤ f ( n ) ≤ M, (cid:0) n − n (cid:1) n − n − n +28 n · ( n !) (cid:0) − n − − − n − (cid:1) ( b − a ) n · (cid:13)(cid:13) f ( n +1) (cid:13)(cid:13) ∞ , if f ( n +1) ∈ L ∞ ([ a, b ]) , holds for all x ∈ (cid:2) a, a + b (cid:3) , where A ( n ) = 2 ( n − (2 n −
1) (2 n −
2) (2 n − and B ( n ) = 2 n − (2 n −
1) (2 n −
2) + 4 n (2 n −
1) + 2 n (2 n −
1) (2 n −
2) (2 n − ∀ n ≥ . Remark 2.
By setting h ( t ) = n ! f ( n ) ( t ) k ( t, x ) and h ( t ) = ( x − t ) n − , we obtainthat C ( h , h ) = 1 n f ( x ) + f ( a + b − x )2 + n − X k =1 G k ( x ) ! − b − a Z ba f ( y ) dy − n ! · ( x − a ) n − ( x − b ) n n ( b − a ) · f ( n ) ( x ) + f ( n ) ( a + b − x )2:= Q ( f ; x, n ) . (3.5) Applying Theorem 7 Chebyshev type bounds for Q ( f ; x, n ) can be proved. We shallomit the details. Generalizations of the results
In this section, generalization of the identity (2.1) via Harmonic sequence ofpolynomials through Fink’s approach is considered. Generalizations of Guessab–Schmeisser formula integral formula (1.9) which is of Euler–Maclaurin type forsymmetric values of real functions are established. Some norm inequalities of thesegeneralized formulae with some special cases which are of great interests are alsoprovided.
Theorem 9.
Let I be a real interval, a, b ∈ I ◦ ( a < b ) . Let P k be a harmonicsequence of polynomials and let f : I → R be such that f ( n ) is absolutely contin-uous on I for n ≥ with P n − ( t ) S ( t, · ) f ( n ) ( t ) is integrable. Then we have the ENERALIZATIONS OF GUESSAB–SCHMEISSER FORMULA 15 representation (4.1) 1 n " f ( x ) + f ( a + b − x )2 + n − X k =1 n T k ( x ) + f F k ( a, b ) o − b − a Z ba f ( y ) dy = ( − n − ( b − a ) n Z ba P n − ( t ) S ( t, x ) f ( n ) ( t ) dt, for all x ∈ (cid:2) a, a + b (cid:3) , where T k ( x ) = ( − k n P k ( x ) f ( k ) ( x ) + P k ( a + b − x ) f ( k ) ( a + b − x ) o (4.2) f F k ( a, b ) is given in (1.10) and S ( t, x ) as given in Theorem 4.Proof. Fix x ∈ [ a, b ]. In the representation (1.9), replace b by a + b we get1 n " f ( x ) + n − X k =1 ( − k P k ( x ) f ( k ) ( x ) + 2 n − X k =1 f F k (cid:18) a, a + b (cid:19) − b − a Z a + b a f ( y ) dy (4.3)= 2 ( − n − ( b − a ) n Z a + b a (cid:18)Z xy P n − ( t ) f ( n ) ( t ) dt (cid:19) dy, where F k i given in (1.10). As a second step, in the same formula (1.9) we replaceevery x by a + b − x and a by a + b for all x ∈ (cid:2) a + b , b (cid:3) , then f has the representation(4.4) 1 n " f ( a + b − x ) + n − X k =1 ( − k P k ( a + b − x ) f ( k ) ( a + b − x )+2 n − X k =1 f F k (cid:18) a + b , b (cid:19) − b − a Z b a + b f ( y ) dy = 2 ( − n − ( b − a ) n Z b a + b Z a + b − xy P n − ( t ) f ( n ) ( t ) dt ! dy, Multiplying (4.3) and (4.4) by and then adding the corresponding equations, weget1 n (cid:20) f ( x ) + f ( a + b − x )2(4.5) + 12 n − X k =1 ( − k n P k ( x ) f ( k ) ( x ) + P k ( a + b − x ) f ( k ) ( a + b − x ) o + n − X k =1 (cid:26)f F k (cid:18) a, a + b (cid:19) + f F k (cid:18) a + b , b (cid:19)(cid:27) − b − a Z ba f ( y ) dy = ( − n − ( b − a ) n "Z a + b a (cid:18)Z xy P n − ( t ) f ( n ) ( t ) dt (cid:19) dy + Z b a + b Z a + b − xy P n − ( t ) f ( n ) ( t ) dt ! dy . But since f F k (cid:18) a, a + b (cid:19) + f F k (cid:18) a + b , b (cid:19) = ( − k ( n − k ) b − a (cid:20) P k ( a ) f ( k − ( a ) − P k (cid:18) a + b (cid:19) f ( k − (cid:18) a + b (cid:19)(cid:21) + (cid:20) P k (cid:18) a + b (cid:19) f ( k − (cid:18) a + b (cid:19) − P k ( b ) f ( k − ( b ) (cid:21) = f F k ( a, b ) , then (4.5) becomes1 n " f ( x ) + f ( a + b − x )2 + n − X k =1 n T k ( x ) + f F k ( a, b ) o − b − a Z ba f ( y ) dy (4.6)= ( − n − ( b − a ) n "Z a + b a (cid:18)Z xy P n − ( t ) f ( n ) ( t ) dt (cid:19) dy + Z b a + b Z a + b − xy P n − ( t ) f ( n ) ( t ) dt ! dy . Also, the right hand-side can be simplified as shown in (2.14)–(2.16), i.e., we have Z a + b a (cid:18)Z xy P n − ( t ) f ( n ) ( t ) dt (cid:19) dy + Z b a + b Z a + b − xy P n − ( t ) f ( n ) ( t ) dt ! dy = Z ba P n − ( t ) S ( t, x ) f ( n ) ( t ) dt, where S ( t, x ) = t − a, t ∈ [ a, x ] t − a + b , t ∈ ( x, a + b − x ) t − b, t ∈ [ a + b − x, b ] . for all x ∈ (cid:2) a, a + b (cid:3) , which gives the desired representation in (4.1). (cid:3) Corollary 3.
Under the assumptions of Theorem 9, we have (4.7) 1 n " f ( x ) + f ( a + b − x )2 + n − X k =1 { W ( x, y ) + F k ( y ) } − b − a Z ba f ( y ) dy = 1( b − a ) n ! Z ba ( y − t ) n − S ( t, x ) f ( n ) ( t ) dt where W ( x, y ) = ( − k n ( x − y ) k f ( k ) ( x ) + ( a + b − x − y ) k f ( k ) ( a + b − x ) o for all x ∈ (cid:2) a, a + b (cid:3) and all y ∈ [ a, b ] , where S ( t, x ) is given in Theorem 4 and F k ( y ) is given in (1.4) Proof.
In (4.1), choose P k ( t ) = ( t − y ) k k ! , we get the desired representation (4.7). (cid:3) A Guessab–Schmeisser like expansion (see Theorem 3) may be deduced as fol-lows:
ENERALIZATIONS OF GUESSAB–SCHMEISSER FORMULA 17
Corollary 4.
Under the assumptions of Theorem 9. Additionally if P k ( t ) =( − k P k ( a + b − t ) , ∀ t ∈ [ a, b ] , then (4.8) 1 n " f ( x ) + f ( a + b − x )2 + n − X k =1 (cid:26)ff T k ( x ) + f F k ( a, b ) (cid:27) − b − a Z ba f ( y ) dy = ( − n − ( b − a ) n Z ba P n − ( t ) S ( t, x ) f ( n ) ( t ) dt where ff T k ( x ) = ( − k P k ( x ) h f ( k ) ( x ) + ( − k f ( k ) ( a + b − x ) i , for all x ∈ (cid:2) a, a + b (cid:3) .Proof. Since P k ( t ) = ( − k P k ( a + b − t ) , ∀ t ∈ [ a, b ], substituting in (4.1) we getthe required result. (cid:3) It is conveient to remark here, from (4.8) we can deduce (4.1) by substituting P k ( t ) = ( t − x ) k k ! in (4.8), so that we get(4.9) 1 n " f ( x ) + f ( a + b − x )2 + n − X k =1 f F k ( a, b ) − b − a Z ba f ( y ) dy = 1 n ! ( b − a ) Z ba ( x − t ) n − S ( t, x ) f ( n ) ( t ) dt. Clearly, the desired deduction is finished once we observe that f F k ( a, b ) = G k ( x ).Since P k ( b ) = ( − k P k ( a ), then f F k ( a, b ) = ( − k ( n − k ) b − a P k ( a ) h f ( k − ( a ) − ( − k f ( k − ( b ) i = ( − k ( n − k ) b − a ( a − x ) k k ! h f ( k − ( a ) − ( − k f ( k − ( b ) i = ( n − k ) b − a ( x − a ) k k ! h f ( k − ( a ) + ( − k +1 f ( k − ( b ) i . Also, we note that P k (cid:18) a + b (cid:19) = (cid:0) a + b − x (cid:1) k k ! = ( − k (cid:0) x − a + b (cid:1) k k != ( − k P k (cid:18) a + b (cid:19) = ( − k P k (cid:18) a + b − a + b (cid:19) , this gives that0 = P k (cid:18) a + b (cid:19) − ( − k P k (cid:18) a + b (cid:19) = (cid:16) − k +1 (cid:17) P k (cid:18) a + b (cid:19) . By our choice of P k ; we have P k (cid:0) a + b (cid:1) = ( a + b − x ) k k ! , for all x ∈ (cid:2) a, a + b (cid:3) , thereforewe can write f F k ( a, b ) + 0 = f F k ( a, b ) + (cid:16) − k +1 (cid:17) P k (cid:18) a + b (cid:19) = ( n − k )( b − a ) k ! h ( x − a ) k (cid:16) f ( k − ( a ) + ( − k +1 f ( k − ( b ) (cid:17) + (cid:16) − k +1 (cid:17) (cid:18) a + b − x (cid:19) k = G k ( x ) , which is given in (2.2) . Hence, the representation (4.8) reduces to (4.1).
Theorem 10.
Under the assumptions of Theorem 9, we have (4.10) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n " f ( x ) + f ( a + b − x )2 + n − X k =1 n T k ( x ) + f F k ( a, b ) o − b − a Z ba f ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ N ( f ; x, a, b ) · (cid:13)(cid:13)(cid:13) f ( n ) (cid:13)(cid:13)(cid:13) p , ∀ p ∈ [1 , ∞ ] and all x ∈ (cid:2) a, a + b (cid:3) , where N ( f ; x, a, b ) := 1 n ( b − a ) sup a ≤ t ≤ b {| P n − ( t ) | | S ( t, x ) |} , p = 1 (cid:16)R ba | P n − ( t ) | q | S ( t, x ) | q dt (cid:17) /q , < p < ∞ R ba | P n − ( t ) | | S ( t, x ) | dt, p = ∞ , (4.11) where T k ( x ) is given in (4.2) and f F k ( a, b ) is given in (1.10) .Proof. Utilizing the triangle integral inequality on the identity (4.1) and employingsome known norm inequalities we get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n " f ( x ) + f ( a + b − x )2 + n − X k =1 n T k ( x ) + f F k ( a, b ) o − b − a Z ba f ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ n ( b − a ) Z ba | P n − ( t ) | | S ( t, x ) | (cid:12)(cid:12)(cid:12) f ( n ) ( t ) (cid:12)(cid:12)(cid:12) dt ≤ n ( b − a ) (cid:13)(cid:13) f ( n ) (cid:13)(cid:13) sup a ≤ t ≤ b {| P n − ( t ) | | S ( t, x ) |} , p = 1 (cid:13)(cid:13) f ( n ) (cid:13)(cid:13) p (cid:16)R ba | P n − ( t ) | q | S ( t, x ) | q dt (cid:17) /q , < p < ∞ (cid:13)(cid:13) f ( n ) (cid:13)(cid:13) ∞ R ba | P n − ( t ) | | S ( t, x ) | dt, p = ∞ . = N ( f ; x, a, b ) (cid:13)(cid:13)(cid:13) f ( n ) (cid:13)(cid:13)(cid:13) p , ∀ p, ≤ p ≤ ∞ ENERALIZATIONS OF GUESSAB–SCHMEISSER FORMULA 19 where N ( f ; x, a, b ) is defined in (4.11), and this completes the proof. (cid:3) Corollary 5.
Under the assumptions of Theorem 10, we have (4.12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n " f ( x ) + f ( a + b − x )2 + n − X k =1 n T k ( x ) + f F k ( a, b ) o − b − a Z ba f ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ n "
14 + (cid:12)(cid:12) x − a + b (cid:12)(cid:12) b − a · k P n − k q · (cid:13)(cid:13)(cid:13) f ( n ) (cid:13)(cid:13)(cid:13) p , ∀ p, q ≥ with p + q = 1 and all x ∈ (cid:2) a, a + b (cid:3) , where k P n − k q = sup a ≤ t ≤ b {| P n − ( t ) |} , q = ∞ (cid:16)R ba | P n − ( t ) | q dt (cid:17) /q , < q < ∞ R ba | P n − ( t ) | dt, q = 1 Proof.
In (4.10), it is easy to verify that N ( f ; x, a, b ) ≤ n ( b − a ) sup a ≤ t ≤ b {| S ( t, x ) |} · k P n − k q = 1 n "
14 + (cid:12)(cid:12) x − a + b (cid:12)(cid:12) b − a · k P n − k q , ∀ q ∈ [1 , ∞ ] and all x ∈ (cid:2) a, a + b (cid:3) . (cid:3) Remark 3.
In Theorem 10, if one assumes that f ( n ) is convex, r -convex, quasi-convex, s -convex, P -convex, or Q -convex; we may obtain other new bounds involv-ing convexity. Remark 4.
Bounds for the generalized formula (4.1) via Chebyshev-Gr¨uss type in-equalities can be done by setting h ( t ) = ( − n − n f ( n ) ( t ) and h ( t ) = P n − ( t ) S ( t, x ) ,therefore we have C ( h , h )= ( − n − n ( b − a ) Z ba P n − ( t ) S ( t, x ) f ( n ) ( t ) dt − b − a Z ba P n − ( t ) S ( t, x ) dt × ( − n − n ( b − a ) Z ba f ( n ) ( t ) dt = 1 n ( b − a ) Z ba P n − ( t ) S ( t, x ) f ( n ) ( t ) dt − b − a Z ba P ′ n ( t ) S ( t, x ) dt × ( − n − n · f ( n − ( b ) − f ( n − ( a ) b − a = ( − n − n ( b − a ) Z ba P n − ( t ) S ( t, x ) f ( n ) ( t ) dt − (cid:20) P n ( x ) + P n ( a + b − x )2 − P n +1 ( b ) − P n +1 ( a ) b − a (cid:21) × ( − n − n · f ( n − ( b ) − f ( n − ( a ) b − a = L ( f, P n , x ) . We left the representations to the reader.
Theorem 11.
Let I be a real interval, a, b ∈ I ◦ ( a < b ) . Let f : I → R be ( n + 1) -times differentiable on I ◦ such that f ( n +1) is absolutely continuous on I ◦ with ( · − t ) n − k ( t, · ) f ( n ) ( t ) is integrable. Then, for all n ≥ we have (4.13) |L ( f, P n , x ) |≤ ( b − a ) n k P n − + P n − S ( · , x ) k ∞ · (cid:13)(cid:13) f ( n +1) (cid:13)(cid:13) ∞ , if f ( n +1) ∈ L ∞ ([ a, b ]) n ( M − m ) ( M − m ) , if m ≤ f ( n ) ≤ M , b − aπ n D ( n, x ) · (cid:13)(cid:13) f ( n +1) (cid:13)(cid:13) , if f ( n +1) ∈ L ([ a, b ]) , b − a n k P n − + P n − S ( · , x ) k ∞ · ( M − m ) , if m ≤ f ( n ) ≤ M , b − a n ( M − m ) · (cid:13)(cid:13) f ( n +1) (cid:13)(cid:13) ∞ , if f ( n +1) ∈ L ∞ ([ a, b ]) , holds for all x ∈ (cid:2) a, a + b (cid:3) , where M := max a ≤ t ≤ b { P n − ( t ) S ( t, x ) } , m := min a ≤ t ≤ b { P n − ( t ) S ( t, x ) } and D ( n, x ) = Z ba | P n − ( t ) + P n − ( t ) S ( t, x ) | dt ! / ∀ n ≥ . Proof.
The proof of the result follows directly by applying Theorem 3.2 to thefunctions h ( t ) = ( − n − n f ( n ) ( t ) and h ( t ) = P n − ( t ) S ( t, x ) as shown previouslyin Remark 4 and the rest of the proof done using Theorem 7. (cid:3) ENERALIZATIONS OF GUESSAB–SCHMEISSER FORMULA 21
Corollary 6.
Let assumptions of Theorem 11 hold. If moreover, f ( n − ( a ) = f ( n − ( b ) ( n ≥ , then the inequality (4.14) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n " f ( x ) + f ( a + b − x )2 + n − X k =1 n T k ( x ) + f F k ( a, b ) o − b − a Z ba f ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( b − a ) n k P n − + P n − S ( · , x ) k ∞ · (cid:13)(cid:13) f ( n +1) (cid:13)(cid:13) ∞ , if f ( n +1) ∈ L ∞ ([ a, b ]) n ( M − m ) ( M − m ) , if m ≤ f ( n ) ≤ M , b − aπ n D ( n, x ) · (cid:13)(cid:13) f ( n +1) (cid:13)(cid:13) , if f ( n +1) ∈ L ([ a, b ]) , b − a n k P n − + P n − S ( · , x ) k ∞ · ( M − m ) , if m ≤ f ( n ) ≤ M , b − a n ( M − m ) · (cid:13)(cid:13) f ( n +1) (cid:13)(cid:13) ∞ , if f ( n +1) ∈ L ∞ ([ a, b ]) , holds for all x ∈ (cid:2) a, a + b (cid:3) . Quadrature rules and error bounds
Representations of Quadratures.
In viewing (2.1), the integral R ba f ( y ) dy can be expressed by the general quadrature rule: Z ba f ( y ) dy = Q n ( f, x ) + E n ( f, x )(5.1)where Q n ( f, x ) := b − an f ( x ) + f ( a + b − x )2 + n − X k =1 G k ( x ) ! , (5.2)and E n ( f, x ) := − n ! Z ba ( x − t ) n − k ( t, x ) f ( n ) ( t ) dt. (5.3)for all a ≤ x ≤ a + b .In particular cases, we have: • If x = a , then Z ba f ( y ) dy = Q n ( f, a ) + E n ( f, a )(5.4)such that Q n ( f, a ) := b − an f ( a ) + f ( b )2 + n − X k =1 G ak ! , and E n ( f, a ) := − n ! Z ba ( a − t ) n − k ( t, a ) f ( n ) ( t ) dt, where, G ak = ( n − k ) k ! · (cid:16) − k +1 (cid:17) (cid:18) b − a (cid:19) k f ( k − (cid:18) a + b (cid:19) , and k ( t, a ) = t − a + b , for all t ∈ ( a, b ). • If x = a + b , then Z ba f ( y ) dy = Q n (cid:18) f, a + b (cid:19) + E n (cid:18) f, a + b (cid:19) (5.5)such that Q n (cid:18) f, a + b (cid:19) := b − an f (cid:0) a + b (cid:1) + f (cid:0) a +3 b (cid:1) n − X k =1 G a + b k ! , and E n (cid:18) f, a + b (cid:19) := − n ! Z ba (cid:18) a + b − t (cid:19) n − k (cid:18) t, a + b (cid:19) f ( n ) ( t ) dt, where, G a + b k = ( n − k ) k ! · (cid:18) b − a (cid:19) k nh f ( k − ( a ) + ( − k +1 f ( k − ( b ) i + (cid:16) − k +1 (cid:17) f ( k − (cid:18) a + b (cid:19)(cid:27) , and k (cid:18) t, a + b (cid:19) = t − a, t ∈ (cid:2) a, a + b (cid:3) t − a + b , t ∈ (cid:0) a + b , a +3 b (cid:1) t − b, t ∈ (cid:2) a +3 b , b (cid:3) . • If x = a + b , then Z ba f ( y ) dy = Q n (cid:18) f, a + b (cid:19) + E n (cid:18) f, a + b (cid:19) , (5.6)such that Q n (cid:18) f, a + b (cid:19) := b − an f (cid:18) a + b (cid:19) + n − X k =1 G a + b k ! , and E n (cid:18) f, a + b (cid:19) := − n ! Z ba (cid:18) a + b − t (cid:19) n − k (cid:18) t, a + b (cid:19) f ( n ) ( t ) dt, where, G a + b k = ( n − k ) k ! · (cid:18) b − a (cid:19) k h f ( k − ( a ) + ( − k +1 f ( k − ( b ) i , and k (cid:18) t, a + b (cid:19) = (cid:26) t − a, t ∈ (cid:2) a, a + b (cid:3) t − b, t ∈ (cid:2) a + b , b (cid:3) . A general quadrature rule via harmonic sequence of polynomials can be consid-ered as follows: Z ba f ( y ) dy = Q n ( f, P n , x ) + E n ( f, P n , x ) , ∀ x ∈ (cid:20) a, a + b (cid:21) (5.7) ENERALIZATIONS OF GUESSAB–SCHMEISSER FORMULA 23 where Q n ( f, P n , x ) is the quadrature formula given by Q n ( f, P n , x ) := b − an " f ( x ) + f ( a + b − x )2 + n − X k =1 n T k ( x ) + f F k ( a, b ) o , with error term E n ( f, P n , x ) := − ( − n − n Z ba P n − ( t ) S ( t, x ) f ( n ) ( t ) dt, such that T k ( x ) = ( − k n P k ( x ) f ( k ) ( x ) + P k ( a + b − x ) f ( k ) ( a + b − x ) o and f F k ( a, b ) = ( − k ( n − k ) b − a h P k ( a ) f ( k − ( a ) − P k ( b ) f ( k − ( b ) i . Remark 5.
Identities (5.1) and (5.7) can be considered as Euler–Maclaurin typeformulae for symmetric values.
Remark 6.
As we mentioned at the end of intoduction section, many authors usedsome key or general expansion formulas such as Euler–Maclaurin type formulae andBernoulli polynomials ( cf. [24]) to construct some quadrature rules of Newton–Cotes and Gauss types as done in Franji´c works [30]–[34]. Our expansions, theidentities (5.1) and (5.7) can be considered as general key formuals instaed of thoseused in [30]–[34] to construct several quadrature formulas for an arbitrary n -thdifferentiable real function. The same remark holds for the formuals (1.5), (1.9)and (1.10).5.2. Errors bounds via Chebyshev-Gr¨uss type inequalities.
In what follows,error bounds for the quadrature rules obtained in Section 5 are proved. The proofof these bounds can be deduced from Corollary 2 and Corollary 6.
Proposition 1.
Let I be a real interval, a, b ∈ I ◦ ( a < b ) . Let f : I → R besuch that f is n -times differentiable function such that f ( n +1) ( n ≥ is absolutelycontinuous on [ a, b ] . If f ( n − ( a ) = f ( n − ( b ) , then for all n ≥ we have (5.8) |E n ( f, a ) |≤ n − (cid:0) n − n (cid:1) n − n − n +212 n · ( n !) ( b − a ) n +2 · (cid:13)(cid:13) f ( n +1) (cid:13)(cid:13) ∞ , if f ( n +1) ∈ L ∞ ([ a, b ]) (cid:0) n − n (cid:1) n − n − n +24 n · ( n !) (cid:0) − n − − − n − (cid:1) ( b − a ) n − · ( M − m ) , if m ≤ f ( n ) ≤ M, ( b − a ) n + 32 n − · ( n !) · π B ( n ) · (cid:13)(cid:13) f ( n +1) (cid:13)(cid:13) , if f ( n +1) ∈ L ([ a, b ]) , n − (cid:0) n − n (cid:1) n − n − n +28 n · ( n !) ( b − a ) n +1 · ( M − m ) , if m ≤ f ( n ) ≤ M, (cid:0) n − n (cid:1) n − n − n +28 n · ( n !) (cid:0) − n − − − n − (cid:1) ( b − a ) n +1 · (cid:13)(cid:13) f ( n +1) (cid:13)(cid:13) ∞ , if f ( n +1) ∈ L ∞ ([ a, b ]) , , (5.9) (cid:12)(cid:12)(cid:12)(cid:12) E n (cid:18) f, a + b (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ n − (cid:0) n − n (cid:1) n − n − n +212 n · ( n !) ( b − a ) n +2 · (cid:13)(cid:13) f ( n +1) (cid:13)(cid:13) ∞ , if f ( n +1) ∈ L ∞ ([ a, b ]) (cid:0) n − n (cid:1) n − n − n +24 n · ( n !) (cid:0) − n − − − n − (cid:1) ( b − a ) n − · ( M − m ) , if m ≤ f ( n ) ≤ M, ( b − a ) n + 32 n − · ( n !) · π p A ( n ) + B ( n ) · (cid:13)(cid:13) f ( n +1) (cid:13)(cid:13) , if f ( n +1) ∈ L ([ a, b ]) , ( b − a ) n +1 n − (cid:0) n − n (cid:1) n − n − n +28 n · ( n !) · ( M − m ) , if m ≤ f ( n ) ≤ M, (cid:0) n − n (cid:1) n − n − n +28 n · ( n !) (cid:0) − n − − − n − (cid:1) ( b − a ) n +1 · (cid:13)(cid:13) f ( n +1) (cid:13)(cid:13) ∞ , if f ( n +1) ∈ L ∞ ([ a, b ]) , (5.10) (cid:12)(cid:12)(cid:12)(cid:12) E n (cid:18) f, a + b (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ n − (cid:0) n − n (cid:1) n − n − n +212 n · ( n !) ( b − a ) n +2 · (cid:13)(cid:13) f ( n +1) (cid:13)(cid:13) ∞ , if f ( n +1) ∈ L ∞ ([ a, b ]) (cid:0) n − n (cid:1) n − n − n +24 n · ( n !) (cid:0) − n − − − n − (cid:1) ( b − a ) n − · ( M − m ) , if m ≤ f ( n ) ≤ M, ( b − a ) n + 32 n − · ( n !) · π A ( n ) · (cid:13)(cid:13) f ( n +1) (cid:13)(cid:13) , if f ( n +1) ∈ L ([ a, b ]) , n − (cid:0) n − n (cid:1) n − n − n +28 n · ( n !) ( b − a ) n +1 · ( M − m ) , if m ≤ f ( n ) ≤ M, (cid:0) n − n (cid:1) n − n − n +28 n · ( n !) (cid:0) − n − − − n − (cid:1) ( b − a ) n +1 · (cid:13)(cid:13) f ( n +1) (cid:13)(cid:13) ∞ , if f ( n +1) ∈ L ∞ ([ a, b ]) , , where A ( n ) = 2 ( n − (2 n −
1) (2 n −
2) (2 n − and B ( n ) = 2 n − (2 n −
1) (2 n −
2) + 4 n (2 n −
1) + 2 n (2 n −
1) (2 n −
2) (2 n − , ∀ n ≥ . And finally (5.11) |E n ( f, P n , x ) |≤ ( b − a ) n k P n − + P n − S ( · , x ) k ∞ · (cid:13)(cid:13) f ( n +1) (cid:13)(cid:13) ∞ , if f ( n +1) ∈ L ∞ ([ a, b ]) b − a n ( M − m ) ( M − m ) , if m ≤ f ( n ) ≤ M , ( b − a ) π n D ( n, x ) · (cid:13)(cid:13) f ( n +1) (cid:13)(cid:13) , if f ( n +1) ∈ L ([ a, b ]) , ( b − a ) n k P n − + P n − S ( · , x ) k ∞ · ( M − m ) , if m ≤ f ( n ) ≤ M , ( b − a ) n ( M − m ) · (cid:13)(cid:13) f ( n +1) (cid:13)(cid:13) ∞ , if f ( n +1) ∈ L ∞ ([ a, b ]) , ENERALIZATIONS OF GUESSAB–SCHMEISSER FORMULA 25 where D ( n, x ) and m , M are defined in Theorem 11. Remark 7.
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