On the delimitation of the remainder in certain linear approximation formulas of analysis

Tiberiu Popoviciu
(original), Approximation Theory, paper, remainder estimation

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T. Popoviciu
Institutul de Calcul

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T. Popoviciu, Asupra delimitării restului în unele formule de aproximare liniară ale analizei, Stud. Cerc. Mat. (Cluj), 11 (1960) no. 2, pp. 357-362 (in Romanian)

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Studii si Cercetari Matematice

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Academia Republicii S.R.

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https://ictp.acad.ro/scm/journal/article/view/79

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1960 a1-Popoviciu- Stud. Cerc. Mat. (Cluj) - On the delimitation of the remainder in some approximation formulas
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ON THE DELIMITATION OF THE REMAINDER IN SOME LINEAR APPROXIMATION FORMULAS OF ANALYSIS*)

OFTIBERIU POPOVICIUCorresponding member of the RPR Academy(Cluj)

  1. Let's assume that the rest R [ f ] R [ f ] R[f]R[f]R[f]of a linear approximation formula is a linear functional defined on a vector space S S SSS, formed by functions f = f ( x ) f = f ( x ) f=f(x)f=f(x)f=f(x), defined and continuous on an interval and and andandandFunctions f f fffand the linear functional R [ f ] R [ f ] R[f]R[f]R[f]are real and S S SSScontains all polynomials.
We say that R [ f ] R [ f ] R[f]R[f]R[f]is of simple form, if there exists an integer n ≧ − 1 n ≧ − 1 n >= -1n \geqq-1n≧−1, so that equality occurs
(1) R [ f ] = K [ ξ 1 , ξ 2 , … , ξ n + 2 ; f ] , f ∈ S , (1) R [ f ] = K ξ 1 , ξ 2 , … , ξ n + 2 ; f , f ∈ S , {:(1)R[f]=K[xi_(1),xi_(2),dots,xi_(n+2);f]","quad f in S",":}\begin{equation*} R[f]=K\left[\xi_{1}, \xi_{2}, \ldots, \xi_{n+2} ; f\right], \quad f \in S, \tag{1} \end{equation*}(1)R[f]=K[ξ1,ξ2,…,ξn+2;f],f∈S,
where K = R [ x n + 1 ] K = R x n + 1 K=R[x^(n+1)]K=R\left[x^{n+1}\right]K=R[xn+1]is ≠ 0 ≠ 0 !=0\neq 0≠0, independent of function f f fff, and ξ and ξ and xi_(i)\xi_{i}ξand, and = 1 , 2 , … , n + 2 and = 1 , 2 , … , n + 2 i=1,2,dots,n+2i=1,2, \ldots, n+2and=1,2,…,n+2SYNTHESIS n + 2 n + 2 n+2n+2n+2distinct points of the interval and and andandand(which may generally depend on the function f f fffand which are located inside the interval, if n ≥ 0 n ≥ 0 n >= 0n \geq 0n≥0). The notation [ ξ 1 , ξ 2 , … , ξ n + 2 ; f ] ξ 1 , ξ 2 , … , ξ n + 2 ; f [xi_(1),xi_(2),dots,xi_(n+2);f]\left[\xi_{1}, \xi_{2}, \ldots, \xi_{n+2} ; right][ξ1,ξ2,…,ξn+2;f]represents the divided difference of the function f f fffon the nodes ξ 1 , ξ 2 , … , ξ n + 2 ξ 1 , ξ 2 , … , ξ n + 2 xi_(1),xi_(2),dots,xi_(n+2)\xi_{1}, \xi_{2}, \ldots, \xi_{n+2}ξ1,ξ2,…,ξn+2For these notions and for the few properties that will follow, we ask the reader to consult our previous works, in particular, our paper [3] in the previous volume of this journal.
In this case, n n nnnrepresents the degree of accuracy of the remainder and enjoys the property (characteristic) that R [ f ] R [ f ] R[f]R[f]R[f]is zero for any polynomial of degree n n nnn, but R [ x n + 1 ] ≠ 0 R x n + 1 ≠ 0 R[x^(n+1)]!=0R\left[x^{n+1}\right] \neq 0R[xn+1]≠0.
We recall that because the functional R [ f ] R [ f ] R[f]R[f]R[f]having the degree of accuracy n n nnn, to be of simple form, it is necessary and sufficient that R [ f ] ≠ 0 R [ f ] ≠ 0 R[f]!=0R[f] \neq 0R[f]≠0for any function f f fff ∈ S ∈ S in S\in S∈Sconvex of the order n n nnn(on and and andandand). In this case it is also necessary that R [ f ] R [ f ] R[f]R[f]R[f]to keep a constant sign for any convex function of order n n nnn. Noting that the function x n + 1 x n + 1 x^(n+1)x^{n+1}xn+1is convex of order n n nnn, the previous condition can be written
(2) R [ x n + 1 ] ⋅ R [ f ] > 0 . (2) R x n + 1 ⋅ R [ f ] > 0 . {:(2)R[x^(n+1)]*R[f] > 0.:}\begin{equation*} R\left[x^{n+1}\right] \cdot R[f]>0 . \tag{2} \end{equation*}(2)R[xn+1]⋅R[f]>0.
Condition (2) for any function f ∈ S f ∈ S f in Sf \in Sf∈Sconvex of the order n n nnn, is therefore necessary and sufficient for R [ f ] R [ f ] R[f]R[f]R[f]to be of the simple form (1). We note that for this it is also necessary (but not sufficient) that R [ x n + 1 ] ≠ 0 R x n + 1 ≠ 0 R[x^(n+1)]!=0R\left[x^{n+1}\right] \neq 0R[xn+1]≠0and
(3) R [ x n + 1 ] ⋅ R [ f ] ≧ 0 (3) R x n + 1 ⋅ R [ f ] ≧ 0 {:(3)R[x^(n+1)]*R[f] >= 0:}\begin{equation*} R\left[x^{n+1}\right] \cdot R[f] \geqq 0 \tag{3} \end{equation*}(3)R[xn+1]⋅R[f]≧0
for any function f ∈ S f ∈ S f in Sf \in Sf∈Snon-concave of the order n n nnn2.
If the functional R [ f ] R [ f ] R[f]R[f]R[f]is of simple form (1), then we can delimit it with the formula
(4) | R [ f ] | ≦ | R [ x n + 1 ] | ⋅ M , (4) | R [ f ] | ≦ R x n + 1 ⋅ M , {:(4)|R[f]| <= |R[x^(n+1)]|*M",":}\begin{equation*} |R[f]| \leqq\left|R\left[x^{n+1}\right]\right| \cdot M, \tag{4} \end{equation*}(4)|R[f]|≦|R[xn+1]|⋅M,
where
(5) M = soup x and ∈ and | [ x 1 , x 2 , … , x n + 2 ; f ] | (5) M = soup x and ∈ and   x 1 , x 2 , … , x n + 2 ; f {:(5)M=su p_(x_(i)in I)|[x_(1),x_(2),dots,x_(n+2);f]|:}\begin{equation*} M=\sup _{x_{i} \in I}\left|\left[x_{1}, x_{2}, \ldots, x_{n+2} ; f\right]\right| \tag{5} \end{equation*}(5)M=soupxand∈and|[x1,x2,…,xn+2;f]|
Moreover, if f f fffadmits a derivative of order n + 1 n + 1 n+1n+1n+1(bordered) on and and andandand, the number (5) is given by the equality
M = 1 ( n + 1 ) ! soup x ∈ and | f ( n + 1 ) ( x ) | M = 1 ( n + 1 ) ! soup x ∈ and   f ( n + 1 ) ( x ) M=(1)/((n+1)!)su p_(x in I)|f^((n+1))(x)|M=\frac{1}{(n+1)!} \sup _{x \in I}\left|f^{(n+1)}(x)\right|M=1(n+1)!soupx∈and|f(n+1)(x)|
But the delimitation (4) is valid in a more general case. Namely, we will prove that:
Delimitation (4) is valid if R [ f ] R [ f ] R[f]R[f]R[f]has the degree of accuracy n n nnnand if inequality (3) is verified for any function f ∈ S f ∈ S f in Sf \in Sf∈Snon-concave of the order n n nnn.
we R [ x n + 1 ] ≠ 0 R x n + 1 ≠ 0 R[x^(n+1)]!=0R\left[x^{n+1}\right] \neq 0R[xn+1]≠0and for demonstration we can assume R [ x n + 1 ] > 0 R x n + 1 > 0 R[x^(n+1)] > 0R\left[x^{n+1}\right]>0R[xn+1]>0. We then consider the linear functional (defined on S S SSS)
(6) R 1 [ f ] = R [ f ] + ε [ x 1 , x 2 , … , x n + 2 ; f ] (6) R 1 [ f ] = R [ f ] + ε x 1 , x 2 , … , x n + 2 ; f {:(6)R_(1)[f]=R[f]+epsi[x_(1),x_(2),dots,x_(n+2);f]:}\begin{equation*} R_{1}[f]=R[f]+\varepsilon\left[x_{1}, x_{2}, \ldots, x_{n+2} ; f\right] \tag{6} \end{equation*}(6)R1[f]=R[f]+ε[x1,x2,…,xn+2;f]
where x 1 , x 2 , … , x n + 2 x 1 , x 2 , … , x n + 2 x_(1),x_(2),dots,x_(n+2)x_{1}, x_{2}, \ldots, x_{n+2}x1,x2,…,xn+2SYNTHESIS n + 2 n + 2 n+2n+2n+2distinct fixed points (independent of function f f fff) in the interval and and andandand, and ε ε epsilon\varepsilonεis an arbitrary positive number. We will show that R 1 [ f ] R 1 [ f ] R_(1)[f]R_{1}[f]R1[f]is of simple form (1). Indeed, if we take into account the fact that the difference divided by n + 2 n + 2 n+2n+2n+2nodes (not all confused) of a convex function of order n n nnnis, by definition, positive, we deduce that R 1 [ f ] > 0 R 1 [ f ] > 0 R_(1)[f] > 0R_{1}[f]>0R1[f]>0for any function f ∈ S f ∈ S f in Sf \in Sf∈Sconvex ordinal n n nnn. The property of the demonetrah is treated. Taking into account (5) and (6) and also writing the corresponding delimitation (4) for R 1 [ f ] R 1 [ f ] R_(1)[f]R_{1}[f]R1[f], we obtain
| R [ f ] | ≦ ( R [ x n + 1 ] + 2 ε ) M | R [ f ] | ≦ R x n + 1 + 2 ε M |R[f]| <= (R[x^(n+1)]+2epsi)M|R[f]| \leqq\left(R\left[x^{n+1}\right]+2 \varepsilon\right) M|R[f]|≦(R[xn+1]+2ε)M
This inequality being true whatever the positive number is ε ε epsilon\varepsilonε, the delimitation (4) results and the property in question is proven. If we have R [ x n + 1 ] < 0 R x n + 1 < 0 R[x^(n+1)] < 0R\left[x^{n+1}\right]<0R[xn+1]<0, the proof is analogous. We then take in (6) for ε ε epsilon\varepsilonεan arbitrary negative number.
3. To apply the previous property it is sufficient to know criteria that allow us to state that (in the hypothesis R [ x n + 1 ] ≠ 0 R x n + 1 ≠ 0 R[x^(n+1)]!=0R\left[x^{n+1}\right] \neq 0R[xn+1]≠0) inequality (3) is verified for any function f ∈ S f ∈ S f in Sf \in Sf∈S, non-concave of the order n n nnnWe will
present here such a criterion that results from the remarkable property of SN Bernstein's approximation polynomials, to preserve the convexity character of functions [2].
We assume that and = [ 0 , 1 ] and = [ 0 , 1 ] I=[0,1]I=[0,1]and=[0,1]and that the functions of space S S SSSadmit derivatives of the order j ( ≥ 0 ) j ( ≥ 0 ) j( >= 0)j(\geq 0)j(≥0)contained on [ 0 , 1 ] [ 0 , 1 ] [0.1][0.1][0,1]We consider the linear functional R [ f ] R [ f ] R[f]R[f]R[f], having the degree of accuracy n n nnnand which is bounded in the norm
(7) ‖ f ‖ = ∑ and = 0 j soup x ∈ [ 0 , 1 ] | f ( and ) ( x ) | (7) ‖ f ‖ = ∑ and = 0 j   soup x ∈ [ 0 , 1 ]   f ( and ) ( x ) {:(7)||f||=sum_(i=0)^(j)su p_(x in[0,1])|f^((i))(x)|:}\begin{equation*} \|f\|=\sum_{i=0}^{j} \sup _{x \in[0,1]}\left|f^{(i)}(x)\right| \tag{7} \end{equation*}(7)‖f‖=∑and=0jsoupx∈[0,1]|f(and)(x)|
note
(8) π k , it = ( − 1 ) n + 1 n ! ∫ x 1 ( t − x ) n t k ( 1 − t ) it d t (8) π k , it = ( − 1 ) n + 1 n ! ∫ x 1   ( t − x ) n t k ( 1 − t ) it d t {:(8)pi_(k,l)=((-1)^(n+1))/(n!)int_(x)^(1)(tx)^(n)t^(k)(1-t)^(l)dt:}\begin{equation*} \pi_{k, l}=\frac{(-1)^{n+1}}{n!} \int_{x}^{1}(tx)^{n} t^{k}(1-t)^{l} dt \tag{8} \end{equation*}(8)πk,it=(−1)n+1n!∫x1(t−x)ntk(1−t)itdt
In the hypotheses formulated previously, the following property occurs:
In order for inequality (3) to be verified for any function f ∈ S f ∈ S f in Sf \in Sf∈S, non-concave of the order n n nnn, it is (necessary and) sufficient for the inequality to hold R [ x n + 1 ] ⋅ R [ π k , 1 ] ≧ 0 R x n + 1 ⋅ R π k , 1 ≧ 0 R[x^(n+1)]*R[pi_(k),1] >= 0R\left[x^{n+1}\right] \cdot R\left[\pi_{k}, 1\right] \geqq 0R[xn+1]⋅R[πk,1]≧0, whatever they are negative integers k k kkkand it it ititit.
We note that π k , and ( n + 1 ) = x k ( 1 − x ) it π k , and ( n + 1 ) = x k ( 1 − x ) it pi_(k,i)^((n+1))=x^(k)(1-x)^(l)\pi_{k, i}^{(n+1)}=x^{k}(1-x)^{l}πk,and(n+1)=xk(1−x)itIf
B m = B m ( x ; f ) = ∑ and = 0 m ( m and ) f ( and m ) x and ( 1 − x ) m − and B m = B m ( x ; f ) = ∑ and = 0 m   ( m and ) f and m x and ( 1 − x ) m − and B_(m)=B_(m)(x;f)=sum_(i=0)^(m)((m)/(i))f((i)/(m))x^(i)(1-x)^(mi)B_{m}=B_{m}(x ; f)=\sum_{i=0}^{m}\binomial{m}{i} f\left(\frac{i}{m}\right) x^{i}(1-x)^{mi}Bm=Bm(x;f)=∑and=0m(mand)f(andm)xand(1−x)m−and
is the SN Bernstein polynomial of degree m m mmm, then its derivative of the order n + 1 ( m ≧ n + 1 ) n + 1 ( m ≧ n + 1 ) n+1(m >= n+1)n+1(m \geqq n+1)n+1(m≧n+1), we have
B m ( n + 1 ) = ( m − 1 ) ! ( n + 1 ) ! m n ( m − n − 1 ) ! ∑ and = 0 m − n − 1 | and m − n − 1 and ) [ and m , and + 1 m , ⋯ , and + n + 1 m ; f ] π and , m − n − 1 − and ( n + 1 ) B m ( n + 1 ) = ( m − 1 ) ! ( n + 1 ) ! m n ( m − n − 1 ) ! ∑ and = 0 m − n − 1   and m − n − 1 and and m , and + 1 m , ⋯ , and + n + 1 m ; f π and , m − n − 1 − and ( n + 1 ) {:B_(m)^((n+1))=((m-1)!(n+1)!)/(m^(n)(m-n-1)!)sum_(i=0)^(m-n-1)|_(i)^(m-n-1)_(i))[(i)/(m),(i+1)/(m),cdots,(i+n+1)/(m);f]pi_(i,m-n-1-i)^((n+1))\left.B_{m}^{(n+1)}=\left.\frac{(m-1)!(n+1)!}{m^{n}(m-n-1)!} \sum_{i=0}^{m-n-1}\right|_{i} ^{m-n-1}{ }_{i}\right)\left[\frac{i}{m}, \frac{i+1}{m}, \cdots, \frac{i+n+1}{m} ; f\right] \pi_{i, m-n-1-i}^{(n+1)}Bm(n+1)=(m−1)!(n+1)!mn(m−n−1)!∑and=0m−n−1|andm−n−1and)[andm,and+1m,⋯,and+n+1m;f]πand,m−n−1−and(n+1)
from where
B m = ( m − 1 ) ! ( n + 1 ) ! m n ( m − n − 1 ) ! ∑ i = 0 n − n − 1 ( m − n − 1 i ) [ i m , i + 1 m , ⋯ , i + n + 1 m ; f ] π i , m − n − 1 − i + β m B m = ( m − 1 ) ! ( n + 1 ) ! m n ( m − n − 1 ) ! ∑ i = 0 n − n − 1   ( m − n − 1 i ) i m , i + 1 m , ⋯ , i + n + 1 m ; f π i , m − n − 1 − i + β m B_(m)=((m-1)!(n+1)!)/(m^(n)(m-n-1)!)sum_(i=0)^(n-n-1)((m-n-1)/(i))[(i)/(m),(i+1)/(m),cdots,(i+n+1)/(m);f]pi_(i,m-n-1-i)+beta_(m)B_{m}=\frac{(m-1)!(n+1)!}{m^{n}(m-n-1)!} \sum_{i=0}^{n-n-1}\binom{m-n-1}{i}\left[\frac{i}{m}, \frac{i+1}{m}, \cdots, \frac{i+n+1}{m} ; f\right] \pi_{i, m-n-1-i}+\beta_{m}Bm=(m−1)!(n+1)!mn(m−n−1)!∑and=0n−n−1(m−n−1and)[andm,and+1m,⋯,and+n+1m;f]πand,m−n−1−and+βm
where β m β m beta_(m)\beta_{m}βmis a polynomial of degree n n nnn
As SN Bernstein [1] and S. Wigert [5] have shown, if the derivative f ( i ) f ( i ) f^((i))f^{(i)}f(and)of order i ( ≧ 0 ) i ( ≧ 0 ) i( >= 0)i(\geqq 0)and(≧0)of the function f f fffexists and is continuous on [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1][0,1], the string { B m ( i ) } B m ( i ) {B_(m)^((i))}\left\{B_{m}^{(i)}\right\}{Bm(and)}tend to m → ∞ m → ∞ m rarr oom \rightarrow \inftym→∞, evenly on [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1][0,1]by f ( i ) f ( i ) f^((i))f^{(i)}f(and)It follows from this that R [ B m ] → R [ f ] R B m → R [ f ] R[B_(m)]rarr R[f]R\left[B_{m}\right] \rightarrow R[f]R[Bm]→R[f]for m → ∞ m → ∞ m rarr oom \rightarrow \inftym→∞and therefore
(9) lim m → ∞ R [ x n + 1 ] ⋅ R [ B m ] = R [ x n + 1 ] ⋅ R [ f ] . (9) lim m → ∞   R x n + 1 ⋅ R B m = R x n + 1 ⋅ R [ f ] . {:(9)lim_(m rarr oo)R[x^(n+1)]*R[B_(m)]=R[x^(n+1)]*R[f].:}\begin{equation*} \lim _{m \rightarrow \infty} R\left[x^{n+1}\right] \cdot R\left[B_{m}\right]=R\left[x^{n+1}\right] \cdot R[f] . \tag{9} \end{equation*}(9)limm→∞R[xn+1]⋅R[Bm]=R[xn+1]⋅R[f].
If we notice that
R [ B m ] = ( m − 1 ) ! ( n + 1 ) ! m n ( m − n − 1 ) ! ∑ i = 0 m − 1 ( m − n − 1 i ) | i m , i + 1 m , ⋯ , i + n + 1 m ; f ] R [ π i , m − n − 1 − i ] R B m = ( m − 1 ) ! ( n + 1 ) ! m n ( m − n − 1 ) ! ∑ i = 0 m − 1   ( m − n − 1 i ) i m , i + 1 m , ⋯ , i + n + 1 m ; f R π i , m − n − 1 − i {:R[B_(m)]=((m-1)!(n+1)!)/(m^(n)(m-n-1)!)sum_(i=0)^(m-1)((m-n-1)/(i))|(i)/(m),(i+1)/(m),cdots,(i+n+1)/(m);f]R[pi_(i,m-n-1-i)]\left.\left.R\left[B_{m}\right]=\frac{(m-1)!(n+1)!}{m^{n}(m-n-1)!} \sum_{i=0}^{m-1}\binom{m-n-1}{i} \right\rvert\, \frac{i}{m}, \frac{i+1}{m}, \cdots, \frac{i+n+1}{m} ; f\right] R\left[\pi_{i, m-n-1-i}\right]R[Bm]=(m−1)!(n+1)!mn(m−n−1)!∑and=0m−1(m−n−1and)|andm,and+1m,⋯,and+n+1m;f]R[πand,m−n−1−and]
and that the differences divided by n + 2 n + 2 n+2n+2n+2nodes of a non-concave function of order n n nnnare nonnegative, it follows that R [ x n + 1 ] ⋅ R [ B n ] ≧ 0 R x n + 1 ⋅ R B n ≧ 0 R[x^(n+1)]*R[B_(n)] >= 0R\left[x^{n+1}\right] \cdot R\left[B_{n}\right] \geqq 0R[xn+1]⋅R[Bn]≧0for any non-concave function of order n n nnn. Taking into account ( 9 ), the property in question results.
4. To give an application, either R [ f ] R [ f ] R[f]R[f]R[f]remainder in the numerical quadrature formula
(10) ∫ 0 1 f ( x ) d x = 2 3 f ( 0 ) + 1 5 f ′ ( 0 ) + 1 30 f ′ ′ ( 0 ) + 1 360 f ′ ′ ′ ( 0 ) + + 1 3 f ( 1 ) − 1 30 f ′ ( 1 ) + R [ f ] (10) ∫ 0 1   f ( x ) d x = 2 3 f ( 0 ) + 1 5 f ′ ( 0 ) + 1 30 f ′ ′ ( 0 ) + 1 360 f ′ ′ ′ ( 0 ) + + 1 3 f ( 1 ) − 1 30 f ′ ( 1 ) + R [ f ] {:[(10)int_(0)^(1)f(x)dx=(2)/(3)f(0)+(1)/(5)f^(')(0)+(1)/(30)f^('')(0)+(1)/(360)f^(''')(0)+],[+(1)/(3)f(1)-(1)/(30)f^(')(1)+R[f]]:}\begin{align*} \int_{0}^{1} f(x) d x= & \frac{2}{3} f(0)+\frac{1}{5} f^{\prime}(0)+\frac{1}{30} f^{\prime \prime}(0)+\frac{1}{360} f^{\prime \prime \prime}(0)+ \tag{10}\\ & +\frac{1}{3} f(1)-\frac{1}{30} f^{\prime}(1)+R[f] \end{align*}(10)∫01f(x)dx=23f(0)+15f′(0)+130f′′(0)+1360f′′′(0)++13f(1)−130f′(1)+R[f]
where f f fffadmits a derivative of order 3, continues on [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1][0,1]
In this case the functional R [ f ] R [ f ] R[f]R[f]R[f]has the degree of accuracy n = 5 n = 5 n=5n=5n=5and is bounded with respect to the norm ( 7 ), for j = 3 j = 3 j=3j=3j=3, We have
τ k , l = 1 5 ! ∫ x 1 l ( t − x ) 5 t k ( 1 − t ) l d t τ k , l = 1 5 ! ∫ x 1 l   ( t − x ) 5 t k ( 1 − t ) l d t tau_(k,l)=(1)/(5!)int_(x)^((1)/(l))(t-x)^(5)t^(k)(1-t)^(l)dt\tau_{k, l}=\frac{1}{5!} \int_{x}^{\frac{1}{l}}(t-x)^{5} t^{k}(1-t)^{l} d tτk,it=15!∫x1it(t−x)5tk(1−t)itdt
deduce
R [ x 6 ] = 1 105 > 0 , ∫ 0 1 π k , l d x = 1 6 ∫ 0 1 t 6 + k ( 1 − t ) l d t R x 6 = 1 105 > 0 , ∫ 0 1   π k , l d x = 1 6 ∫ 0 1   t 6 + k ( 1 − t ) l d t R[x^(6)]=(1)/(105) > 0,quadint_(0)^(1)pi_(k,l)dx=(1)/(6)int_(0)^(1)t^(6+k)(1-t)^(l)dtR\left[x^{6}\right]=\frac{1}{105}>0, \quad \int_{0}^{1} \pi_{k, l} d x=\frac{1}{6} \int_{0}^{1} t^{6+k}(1-t)^{l} d tR[x6]=1105>0,∫01πk,itdx=16∫01t6+k(1−t)itdt
and a simple calculation gives us
R [ π k , l ] = 1 6 ∫ 0 1 t k + 2 ( 1 − t ) l + 4 d t > 0 R π k , l = 1 6 ∫ 0 1   t k + 2 ( 1 − t ) l + 4 d t > 0 R[pi_(k,l)]=(1)/(6)int_(0)^(1)t^(k+2)(1-t)^(l+4)dt > 0R\left[\pi_{k, l}\right]=\frac{1}{6} \int_{0}^{1} t^{k+2}(1-t)^{l+4} d t>0R[πk,it]=16∫01tk+2(1−t)it+4dt>0
Therefore, in this case, the delimitation (4) can be applied and we have
R [ f ] ≦ 1 105 sup x i ∈ [ 0 , 1 ] | [ x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 ; f ] | R [ f ] ≦ 1 105 sup x i ∈ [ 0 , 1 ]   x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 ; f R[f] <= (1)/(105)s u p_(x_(i)in[0,1])|[x_(1),x_(2),x_(3),x_(4),x_(5),x_(6),x_(7);f]|R[f] \leqq \frac{1}{105} \sup _{x_{i} \in[0,1]}\left|\left[x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7} ; f\right]\right|R[f]≦1105soupxand∈[0,1]|[x1,x2,x3,x4,x5,x6,x7;f]|
If the derivative of the order 6 , f ( 6 ) 6 , f ( 6 ) 6,f^((6))6, f^{(6)}6,f(6), exists on [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1][0,1], we have
| R [ f ] | ≤ 1 150 ⋅ 1 6 ! sup x ∈ [ 0 , 1 ] | f ( 6 ) ( x ) | | R [ f ] | ≤ 1 150 ⋅ 1 6 ! sup x ∈ [ 0 , 1 ]   f ( 6 ) ( x ) |R[f]| <= (1)/(150)*(1)/(6!)s u p_(x in[0,1])|f^((6))(x)||R[f]| \leq \frac{1}{150} \cdot \frac{1}{6!} \sup _{x \in[0,1]}\left|f^{(6)}(x)\right||R[f]|≤1150⋅16!soupx∈[0,1]|f(6)(x)|

ABOUT THE EVALUATION OF THE RESIDUAL MEMBER OF SOME LINEAR FORMULAS OF APPROXIMATE MATHEMATICAL ANALYSIS

KPATKOE CONTENTS

It is assumed that there is a remainder R [ f ] R [ f ] R[f]R[f]R[f]some linear approximation formula is a linear functional defined on a vector space S S SSS, educated by fuctions f = f ( x ) f = f ( x ) f=f(x)f=f(x)f=f(x), defined and continuous on the interval I I IIand. Functions f f fffand functional k [ f ] k [ f ] k[f]k[f]k[f]they are real, and space S S SSS, contains all polynomials. Based on some previous results [3], the following property is proved in this paper:
Оты имело место оценка (4), где М дается формулой (5), работано очты K [ f ] K [ f ] K[f]K[f]K[f]he had order of accuracy n n nnnи очные неравенство (3) was satisfied для люфой функции f ∈ S f ∈ S f in Sf \in Sf∈Snon-concave order n n nnn.
It is in order of functional accuracy K [ f ] K [ f ] K[f]K[f]K[f], the number is understood n n nnnwith that property, что R [ f ] R [ f ] R[f]R[f]R[f]equals zero for any polynomial n n nnn-of the degree, but R [ x n + 1 ] ≠ 0 R x n + 1 ≠ 0 R[x^(n+1)]!=0R\left[x^{n+1}\right] \neq 0R[xn+1]≠0.
In the continuation, there is an indication that gives the opportunity to learn (under the assumption R [ x n + 1 ] ≠ 0 R x n + 1 ≠ 0 R[x^(n+1)]!=0R\left[x^{n+1}\right] \neq 0R[xn+1]≠0) is satisfied if inequality (3) is satisfied for any function f ∈ S f ∈ S f in Sf \in Sf∈Snon-concave degree n n nnn. This sign is based on the application of properties of the polynomial approximation С. N. Bernstein preserves the convexity character [2].
With this purpose, it is assumed that I = [ 0 , 1 ] I = [ 0 , 1 ] I=[0,1]I=[0,1]and=[0,1]и что элементы прострация S S SSShave derivative orders j ( ≥ 0 ) j ( ≥ 0 ) j( >= 0)j(\geq 0)j(≥0)continuous on [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1][0,1]. It is also assumed that the linear functional R [ f ] R [ f ] R[f]R[f]R[f]limited relative to the norm (7). Under these assumptions, the property is proved:
Для того, что неравенство (3) was satisfied f ∈ S f ∈ S f in Sf \in Sf∈Snon-concave order n n nnn, (necessary and) sufficient to have inequality R [ x n + 1 ] ⋅ R [ π k , l ] ≥ 0 R x n + 1 ⋅ R π k , l ≥ 0 R[x^(n+1)]*R[pi_(k,l)] >= 0R\left[x^{n+1}\right] \cdot R\left[\pi_{k, l}\right] \geq 0R[xn+1]⋅R[πk,it]≥0, какие ни были неотрицательные целе рядом к и l l llit.
The above results are applied to limit the rest of the quadratic formulas (10).

SUR LA DÉLIMITATION DU RESTE DANS CERTAINES FORMULES D'APPROXIMATION LINÉAIRES DE L'ANALYSF,

SUMMARY

We assume that the rest R [ f ] R [ f ] R[f]R[f]R[f]A linear approximation formula is a linear function defined on a vector space S S SSS, formed by the functions f = f ( x ) f = f ( x ) f=f(x)f=f(x)f=f(x), defined et continues sur un intervalle l l llit, The functions f f fffand the functional R ( f ] R ( f ] R(f]R(f]R(f]are real, and space S S SSScontains all the polynomials. Starting from some previous results [3], we demonstrate in the present work the following property:
Pour que la délimitation (4) ait lieu, où M M MMMest donné par (5), il suffit que R [ f ] R [ f ] R[f]R[f]R[f]has the degree of accuracy n n nnnet que l'inégalité (3) soit verified pour toule fonction f ∈ S f ∈ S f in Sf \in Sf∈S, non-concave order n n nnn.
Nous entendons here par degree d'exactitude d'une fonctionnelle R [ f ] R [ f ] R[f]R[f]R[f]a number n n nnnhaving the property that R [ f ] R [ f ] R[f]R[f]R[f]est nul pour tout polynome de degree n n nnn, corn R [ x n + 1 ] ≠ 0 R x n + 1 ≠ 0 R[x^(n+1)]!=0R\left[x^{n+1}\right] \neq 0R[xn+1]≠0.
On donne ensuite un criterio qui permet de connaître si (dans 1'hypothèse R [ x n + 1 ] ≠ 0 R x n + 1 ≠ 0 R[x^(n+1)]!=0R\left[x^{n+1}\right] \neq 0R[xn+1]≠0) l'inégalité (3) is verified pour toute fonction f ∈ S f ∈ S f in Sf \in Sf∈S, non-concave order n n nnn. What criteria are based on the utilization of the property that SN Bernstein's approximation polynomials preserve the convexity characteristics of functions [2].
It is assumed to this end that I = [ 0 , 1 ] I = [ 0 , 1 ] I=[0,1]I=[0,1]and=[0,1]et que les éléments de l'espace S S SSShave derivatives of order j ( ≥ 0 ) j ( ≥ 0 ) j( >= 0)j(\geq 0)j(≥0)keep dancing [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1][0,1]. On suppose aussi que la fonctionnelle linear R [ f ] R [ f ] R[f]R[f]R[f]soit bornée par rapport à la norme (7). Dans cette hypothèse we demonstrate the following property:
Pour que l'inégalité (3) soit verificatie quelles que soit la fonotion f ∈ S f ∈ S f in Sf \in Sf∈Snon-cencave d'ordre n, il est (nécessaire et) suffisant qu'ait lieu l'inégalité R [ x n + 1 ] . R [ π k , l ] ≧ 0 R x n + 1 . R π k , l ≧ 0 R[x^(n+1)].R[pi_(k),l] >= 0R\left[x^{n+1}\right] . R\left[\pi_{k}, l\right] \geqq 0R[xn+1].R[πk,it]≧0, quels que scient les entiers non-négatifs k k kkkand l l llit.
Les résultats ci-dessus s'applique à la délimitation du reste de la formula de quadrature (10).

BIBLIOGRAPHY

  1. SN Bernstein, Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités. Soob. Харьк. Math. Ob-va, series 2, 13, 1-2(1912).
  2. T. Popoviciu, Sur l'approximation des fonctions convexes d'ordre supérieur. Mathematica, 10, 49-54 (1934).
    • On the remainder in some linear approximation formulas of analysis. Studii si Cercetari de Matematica (Cluj), X, 2, 337-389 (1959).
    • Sur le reste dans certaines formulaes linéaires d'approximation de l'analyse. Mathematica, 1(24), 95-142 (1960).
  5. S. Wigert, Sur l'approximation par polynomes des fonctions continues. Arkiv för Mat Astr., och Fysik, 22 B, No. 9, 1-4 (1932).
    Received November 29, 1060.
  1. *) This work is also published in French in the journal "Mathematica" vol. 2(25), issue 1.
1960

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