Estimates of the error of interval quadrature formulas on some classes of di erentiable functions

âèçíà÷à1 ïîõèáêó iíòåðâàëüíî ̈ êâàäðàòóðíî ̈ ôîðìóëè íà êëàñi ôóíêöié H. Iíòåðâàëüíi êâàäðàòóðíi ôîðìóëè ðîçãëÿäàëèñü, íàïðèêëàä, â ðîáîòàõ Êóçüìiíî ̈ À.Ë., Øàðiïîâà Ð.Í., Áàáåíêî Â.Ô., Ìîòîðíîãî Â.Ï., Áîðîäà÷åâà Ñ.Â. òà iíøèõ ìàòåìàòèêiâ. ßêùî ïåðåéòè â (0.1) äî ãðàíèöi, êîëè h→ 0, òî îäåðæèìî çâè÷àéíó êâàäðàòóðíó ôîðìóëó. Çàäà÷à ïðî îïòèìiçàöiþ iíòåðâàëüíèõ êâàäðàòóðíèõ ôîðìóë (0.1) ñêëàäà1òüñÿ ó çíàõîäæåííi âåëè÷èíè

sntervl qudrture formuls were onsideredD for exmpleD in I E VF prom the point of view of pplitionD intervl qudrture formuls re more nturl thn ordiE nry qudrture formulsD euse the results of mesuring physil quntityD in some sesD due to the instlltion of mesuring instrumentsD is the verging of the funtion tht hrterizes the mesured vlueF sf we tke limit in @IA when h → 0, then we otin the usul qudrture formulF he prolem of optimiztion of intervl qudrture formuls @IA onsists in (nding the vlue whih is lled the error of the optiml qudrture formul in the lss H, nd the sets of oe0ients c 0 = {c 0 k } n k=1 nd nodes x 0 = {x 0 k } n k=1 , for whih the ext lower limit @PA is rehedF he prolem of optimiztion of intervl qudrture formuls ws onsideredD for exmpleD in Q E VF st turned out tht in mny ses for lsses of 2π−periodi ontinuous funtions the optiml intervl qudrture formul is the formul where n − re the nodesD h ∈ (0, π/n). udrture formul @QA is ext for ny onstntD iFeF the integrl over the segment [0, 2π] is equl to the qudrture sumF o we my ssume tht the funtion f in @QA is equel to zero in the verge iFeF vet f (t) e 2π−periodi funtion integrle on the periodD let us denote y f h (t ) the teklov funtionD nd let S h f e the teklov opertorF tht is he qudrture sum in @QA my e represented s 2π n tht isD the intervl qudrture formul @QA oinides with the retngles formul for the funtion f h (t). e introdue the following lssesX H ω : ontinuous funtions whose moduli of ontinuity stisfy ω(f ; t) ≤ ω(t), where ω(t) is given onvex modulus of ontinuityF W r H ω (r = 1, 2, ...)− the lss of 2π− periodi funtionsD for whih rEth derivtive pinllyD denote y W r H ω n the sulss of 2π/n− periodi funtions from W r H ω . sn S the optimlity of the intervl qudrture formul @QA on the lss W r 1 is provedD nd in the pper T the optimlity of the intervl qudrture formul @QA on the lss W r ∞ is provedF sn the present pperD we otin the ext estimte of the error of the intervl qudrture formul @QA on the lss W r H ω .
Theorem 1. The equality holds, where H and H n are the classes of functions dened in above.
Proof. ine H n is suset of the set H, the left prt does not exeed the right oneF yn the other hndD due to the properties of the lss H @symmetryD shift invrineAD for ny funtion f ∈ HD the funtion hs the following properties ine the integrls nd qudrti sums for the funtions f (t) nd ψ f (t) oinideD then the errors of qudrture formul @PA for funtions f (t) nd ψ f (t) s well D nd this implies equlity @RAF Corollary 1. Due to equations (3 -5) we have sneedD sine the funtion f (t) is 2π/n− periodiD the terms in the rightEhnd side of eqution @TA oinideD therefore henote y I r (r Enturl numerA the integrtion opertorD iFeF where D r (t) Efernoulli kernelX he rightEhnd side of eqution @UA my e represented y the integrtion opertorX If ω is a given convex modulus of continuity, then the equality holds, where the function ρ(x) is dened by the equation and ρ −1 (x) is a function inverse to ρ(x).

Corollary 2. If the function ψ(t) satises the conditions of Theorem 2 and
xote tht the onditions of heorem P for the funtion ψ(t) hold for fernoulli funtions D r (t). ellyD if r = 2ν is evenD then on segments [0, π] nd [π, 2π] the onditions of the theorem stisfy for the funtions D 2ν (t) nd if r = 2ν +1− is oddD then for the funtions D 2ν+1 (t) on the segments [−π/2, π/2] nd [π/2, 3π/2], in ddition to the onditions of the heoremD @IIA holdsF st is ovious tht the onditions of heorem P nd ondition @IIA hold for the orresponding segments nd for the funtions D r (nt) sn the pper W the funtions were onsidered s wellD nd it ws oserved tht funtions R 2ν (h, t) hve the sme properties s funtions D 2ν+1 (h, t)D nd funtions R 2ν+1 (h, t) ehves similrly to funtions D 2ν (h, t)F sn prtiulrD R 2ν (h, t) is n odd funtionD nd R 2ν+1 (h, t)− is evenD nd heorem P holds for themF st is esy to verify tht similr properties hve funtions Lemma 1. The functions K r,n (h, t), if h ∈ [0, π/2n], have the following properties: 1. K r,n (h, t) are equal to zero on the average. 2. K 2ν,n (h, t) is odd, K 2ν+1,n (h, t) is even. In particular, the functions K 2ν,n (h, t) are equal to zero at the points kπ/n, k ∈ Z.
Proof. roposition I follows from the de(nition of the funtions K r,n (h, t).
e prove the oddness of the funtion K 2ν,n (h, t). o this end we use the periodiity nd evenness of the funtion D 2 nu (t).
imilrly we proveD the evenness of the funtion K 2ν+1,n (h, t).
roposition Q follows from the de(nition of the funtions K r,n (h, t). roposition R follows from the xewtonEveinitz formul nd ropositions Q nd PF o prove S we hve to hek S for the funtions K 1,n (h, t), nd then to ssume tht for some positive integer r > 1 the ttement S does not hold nd to get ontrditionF roposition T follows from the previous oneF vemm is provedF Lemma 2. If h ∈ [0, π/2n], then for the functions K 2ν,n (h, t) the conditions of Theorem 2 and condition (11) are satised on segments [−π/2n, π/2n] and [π/2n, 3π/2n].
Theorem 3. Any 2π/n periodic function, equal to zero on the average, whose (r − 1)-st derivative is absolutely continuous, may be represented in the form Proof.pirst onside the se r = 1. epply the formul of integrtion y prts he (rst term on the right is equl to f (x), nd the seond is equl to zero euse y the ondition of the theoremD the funtion f (x) is equl to zero on the vergeF uppose tht heorem holds for nturl numer r nd show tht the sttement holds for nturl numer r + 1. sing the eqution d dt B r+1 (t) = B r (t) nd integrting the prts we otin Theorem 4. If r is an even natural number, then for any h ∈ [0, π/2n] the equality holds.