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

ictp
(proceedings), Approximation Theory, paper, remainder estimation

Abstract

Authors

T. Popoviciu
Institutul de Calcul

Keywords

?

Paper coordinates

T. Popoviciu, Sur la délimitation du reste dans les formules d’approximation linéaires de l’analyse, 1960 Symposium on the numerical treatment of ordinary differential equations, integral and integro-differential equations «Centre international provisoire de Calcul» (Rome, 1960) pp. 441-446 Birkhäuser, Basel (in French)

PDF

1960 c -Popoviciu- Symposium – Sur la delimitation du reste dans les formules d’approximation lineaires de l’analyse

About this paper

Journal
Publisher Name
DOI
Print ISSN
Online ISSN

[MR0128598]

google scholar link

??

Paper (preprint) in HTML form

1960 c -Popoviciu- Symposium - On the delimitation of the remainder in linear approximation formulas
Original text
Rate this translation
Your feedback will be used to help improve Google Translate
Reprinted from the
Symposium, International Provisional Computing Centre, 1960
Birkhäuser Verlag Basel

ON THE DELIMITATION OF THE REMAINDER IN LINEAR APPROXIMATION FORMULAS OF ANALYSIS

by Tiberiu Popoviciu

University of Cluj

Cluj: (Rumania)

  1. In many approximation formulas of analysis the remainder; or complementary term, R [ f ] R [ f ] R[f]R[f]R[f]is a linear functional (additive and homogeneous) 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 I I IIIof the real axis. The usual formulas of interpolation (polynomial or trigonometric), numerical derivation and integration, etc., have a remainder of this form.
In applications it is important to be able to properly delimit the rest R [ f ] R [ f ] R[f]R[f]R[f]. For this purpose, we have sought, at least in specific, well-defined cases, to put the remainder in various suitable forms. For example, in the form of a definite integral or a linear combination of a finite number of values ​​of the derivatives, of various orders, of the function f f fff, etc.
There is a large body of work on the structure of the remainder R [ f ] R [ f ] R[f]R[f]R[f]I will only cite AA Markoff [4], GD Birkhoff [2]: G. Kowalewski [3], Rv Mises [5], J. Radon [12], E. Ya. Remez [13]; A. Sard [14].
I ′ I ′ J^(')J^{\prime}I′obtained, using the theory of higher-order convex functions that I ′ I ′ j^(')j^{\prime}I′I have studied in the past [7,9], a new representation of the remainder, which is more general and better highlights its structure [ 10 , 11 ] [ 10 , 11 ] [10,11][10,11][10,11].
In this communication I will make some remarks on this representation.
We will assume in the following that the function f f fffand the functional R [ f ] R [ f ] R[f]R[f]R[f]are real and that S S SSScontains all polynomials.
2. We say that R [ f ] R [ f ] R[f]R[f]R[f]is of the simple form if there exists an integer n ≥ − 1 n ≥ − 1 n >= -1n \geq-1n≥−1such that we have
R [ f ] = K ⋅ [ ξ 1 , ξ 2 , … , ξ n + 2 ; f ] ; f ∈ S R [ f ] = K ⋅ ξ 1 , ξ 2 , … , ξ n + 2 ; f ; f ∈ S R[f]=K*[xi_(1),xi_(2),dots,xi_(n+2);f];quad f in SR[f]=K \cdot\left[\xi_{1}, \xi_{2}, \ldots, \xi_{n+2}; f\right]; \quad f \in SR[f]=K⋅[ξ1,ξ2,…,ξn+2;f];f∈S
Or 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]East ≠ 0 ≠ 0 !=0\neq 0≠0, independent of the function f f fffand the ξ i ξ i xi_(i)\xi_{i}ξi, i = 1 , 2 , … , n + 2 i = 1 , 2 , … , n + 2 i=1,2,dots,n+2i=1,2, \ldots, n+2i=1,2,…,n+2are n + 2 n + 2 n+2n+2n+2distinct points of the interval I I III(which can generally depend on the function f f fffand even located at 1 n 1 n 1^(n)1^{n}1ninterior of I I III. if n ≧ 0 n ≧ 0 n >= 0n \geqq 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}; f\right][ξ1,ξ2,…,ξn+2;f]denotes the divided difference (of ardré n + 1 n + 1 n+1n+1n+1) of the function f f fffon the knots 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+2.
The number n n nnnis 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 on any polynomial of degree n n nnn, but that we have 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.
Of simple form is, for example, the remainder in Taylor's formula, in Lagrange's interpolation formula, in Gauss's quadrature formula, etc.
Let us recall the following property:
I. The necessary and sufficient condition for R [ f ] R [ f ] R[f]R[f]R[f], assumed from the degree of accuracy n n nnn, either of the simple form is that we have R [ f ] ≠ 0 R [ f ] ≠ 0 R[f]!=0R[f] \neq 0R[f]≠0for everything f ∈ S f ∈ S f in Sf \in Sf∈S, convex of order n n nnn.
In this case it is, moreover, necessary that R [ f ] R [ f ] R[f]R[f]R[f]keep his sign for f f fffconvex of order n n nnn. Noting that the function x n + 1 x n + 1 x^(n+1)x^{n+1}xn+1is indeed convex of order n n nnn, the previous condition can also 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] R[f]>0 \tag{2} \end{equation*}(2)R[xn+1]R[f]>0
Condition (2), for all f ∈ S f ∈ S f in Sf \in Sf∈Sconvex of order n n nnn, is therefore necessary and sufficient for R [ f ] R [ f ] R[f]R[f]R[f]either of the simple form (1). Note that for cola is also necessary (but not sufficient) that L ′ L ′ l^(')l^{\prime}L′we have 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] 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∈S, non-concave of order n n nnn.
Recall that the function f f fffis said to be convex resp, non-concave of order n n nnnon I I IIIif the divided difference of order n + 1 n + 1 n+1n+1n+1on n + 2 n + 2 n+2n+2n+2any distinct points of I I IIIremains constantly positive resp. non-negative.
3. If R [ f ] R [ f ] R[f]R[f]R[f]is of the simple form, we can delimit it by 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| M \tag{4} \end{equation*}(4)|R[f]|≦|R[xn+1]|M
Or
(5) M = sup x i ∈ I | [ x 1 , x 2 , … , x n + 2 ; f ] | (5) M = sup x i ∈ I   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=supxi∈I|[x1,x2,…,xn+2;f]|
Besides, if f f fffhas a derivative of order n + 1 n + 1 n+1n+1n+1(bounded) on I I III, the number (5) is given by the equality
M = 1 ( n + 1 ) ! sup x ∈ I | f ( n + 1 ) ( x ) | . M = 1 ( n + 1 ) ! sup x ∈ I   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)!supx∈I|f(n+1)(x)|.
The delimitation (4) is valid in a more general case than in that of the simplicity of R [ f ] R [ f ] R[f]R[f]R[f]. We have the following property:
II. The delimitation (4) is valid if R [ f ] R [ f ] R[f]R[f]R[f]is of degree of exactness n and if inequality (3) is verified for any function f ∈ S f ∈ S f in Sf \in Sf∈S, non-concave of order n n nnn.
4. To be able to affirm that the rest R [ f ] R [ f ] R[f]R[f]R[f]is of the simple form, it is enough to know the criteria allowing us to affirm that (under 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 (2) is verified for all f ∈ S f ∈ S f in Sf \in Sf∈Sconvex of order n n nnn.
Here we will present a criterion allowing us to affirm that (under 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 order n n nnn, therefore a criterion allowing the application of property II. This criterion is based on the remarkable properties of convergence and the conservation of the convexity characteristics of the function f f fff, by the corresponding polynomials of SN Bernstein [1,8,15].
Let's suppose that I ≡ [ 0 , 1 ] I ≡ [ 0 , 1 ] I-=[0,1]I \equiv[0,1]I≡[0,1]and that the elements of S S SSShave a derivative of order I ( ≧ 0 ) I ( ≧ 0 ) j( >= 0)j(\geqq 0)I(≧0)continue on [ 0 , 1 ] [ 0 , 1 ] [0.1][0.1][0,1](1 has a zeroth-order derivative and is the function itself). Let us then consider the linear functional R [ f ] R [ f ] R[f]R[f]R[f], degree of accuracy n n nnnand which is limited in relation to the norm
(6) ‖ f ‖ = ∑ i = 0 I sup x ∈ [ 0 , 1 ] | f ( i ) ( x ) | (6) ‖ f ‖ = ∑ i = 0 I   sup x ∈ [ 0 , 1 ]   f ( i ) ( x ) {:(6)||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{6} \end{equation*}(6)‖f‖=∑i=0Isupx∈[0,1]|f(i)(x)|

Let's ask

π k , L = ( − 1 ) n + 1 n ! ∫ x 1 ( t − x ) n t k ( 1 − t ) L d t π k , L = ( − 1 ) n + 1 n ! ∫ x 1   ( t − x ) n t k ( 1 − t ) L d t pi_(k,l)=((-1)^(n+1))/(n!)int_(x)^(1)(tx)^(n)t^(k)(1-t)^(l)dt\pi_{k, l}=\frac{(-1)^{n+1}}{n!} \int_{x}^{1}(tx)^{n} t^{k}(1-t)^{l} dtπk,L=(−1)n+1n!∫x1(t−x)ntk(1−t)Ldt
Under the previous assumptions, we have the following property:
III. For inequality (3) to be verified for any function f ∈ S f ∈ S f in Sf \in Sf∈Snon-concave order n n nnnit is sufficient that one has
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,L]≧0
whatever the non-negative integers k k kkkAnd L L LLL.
The demonstration results from the fact that if
B m = ∑ i = 0 m ( m i ) f ( i m ) x i ( 1 − x ) m − i B m = ∑ i = 0 m   ( m i ) f i m x i ( 1 − x ) m − i B_(m)=sum_(i=0)^(m)((m)/(i))f((i)/(m))x^(i)(1-x)^(mi)B_{m}=\sum_{i=0}^{m}\binom{m}{i} f\left(\frac{i}{m}\right) x^{i}(1-x)^{mi}Bm=∑i=0m(mi)f(im)xi(1−x)m−i
( m ≧ n + 1 ) ( m ≧ n + 1 ) (m >= n+1)(m \geqq n+1)(m≧n+1)
are the SN Bernstein polynomials, we have, under the previous assumptions,
R [ B m ] = ( m − 1 ) ! ( n + 1 ) ! m n ( m − n − 1 ) ! ∑ i = 0 m − n − 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 − n − 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-n-1)((m-n-1)/(i))[(i)/(m),(i+1)/(m),dots,(i+n+1)/(m);f]R[pi_(i,m-n-1-i)]R\left[B_{m}\right]=\frac{(m-1)!(n+1)!}{m^{n}(m-n-1)!} \sum_{i=0}^{m-n-1}\binom{m-n-1}{i}\left[\frac{i}{m}, \frac{i+1}{m}, \ldots, \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)!∑i=0m−n−1(m−n−1i)[im,i+1m,…,i+n+1m;f]R[πi,m−n−1−i]
And
lim m → ∞ R [ x n + 1 ] R [ B m ] = R [ x n + 1 ] R [ f ] lim m → ∞   R x n + 1 R B m = R x n + 1 R [ f ] lim_(m rarr oo)R[x^(n+1)]R[B_(m)]=R[x^(n+1)]R[f]\lim _{m \rightarrow \infty} R\left[x^{n+1}\right] R\left[B_{m}\right]=R\left[x^{n+1}\right] R[f]limm→∞R[xn+1]R[Bm]=R[xn+1]R[f]
and where f f fffis a non-concave function of order n n nnnbelonging to S S SSS.
5. To give an application, either R [ f ] R [ f ] R[f]R[f]R[f]the remainder in the numerical quadrature formula of N. Obrechkoff [6].
∫ 0 1 f ( x ) d x = 2 3 f ( 0 ) + 1 5 f ′ ( 0 ) + 1 30 f ′ ′ ( 0 ) + 1 360 f ′ ′ ′ ( 0 ) + ∫ 0 1   f ( x ) d x = 2 3 f ( 0 ) + 1 5 f ′ ( 0 ) + 1 30 f ′ ′ ( 0 ) + 1 360 f ′ ′ ′ ( 0 ) + int_(0)^(1)f(x)dx=(2)/(3)f(0)+(1)/(5)f^(')(0)+(1)/(30)f^('')(0)+(1)/(360)f^(''')(0)+\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)+∫01f(x)dx=23f(0)+15f′(0)+130f′′(0)+1360f′′′(0)+
+ 1 3 f ( 1 ) − 1 30 f ′ ( 1 ) + R [ f ] + 1 3 f ( 1 ) − 1 30 f ′ ( 1 ) + R [ f ] +(1)/(3)f(1)-(1)/(30)f^(')(1)+R[f]+\frac{1}{3} f(1)-\frac{1}{30} f^{\prime}(1)+R[f]+13f(1)−130f′(1)+R[f]
Or f f fffhas a continuous third-order derivative on [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1][0,1].
In this case R [ f ] R [ f ] R[f]R[f]R[f]is of degree of accuracy 5 and is bounded with respect to the norm (6) for. j = 3 j = 3 j=3j=3I=3。
A simple calculation gives us in oe.cas
R [ x 6 ] = 1 105 > 0 , R [ π k , l ] = 1 6 ! ∫ 0 1 t k + 2 ( 1 − t ) l + 4 d t > 0 R x 6 = 1 105 > 0 , R π k , l = 1 6 ! ∫ 0 1   t k + 2 ( 1 − t ) l + 4 d t > 0 R[x^(6)]=(1)/(105) > 0,quad R[pi_(k,l)]=(1)/(6!)int_(0)^(1)t^(k+2)(1-t)^(l+4)dt > 0R\left[x^{6}\right]=\frac{1}{105}>0, \quad R\left[\pi_{k, l}\right]=\frac{1}{6!} \int_{0}^{1} t^{k+2}(1-t)^{l+4} d t>0R[x6]=1105>0,R[πk,L]=16!∫01tk+2(1−t)L+4dt>0
The delimitation (4) is therefore well applicable and we have
| R [ f ] | ≦ 1 105 sup x i ∈ [ 0 , 1 ] | [ x 1 , x 2 , x 3 , x 4 , x 5 , x 8 ; x 7 ; f ] | | R [ f ] | ≦ 1 105 sup x i ∈ [ 0 , 1 ]   x 1 , x 2 , x 3 , x 4 , x 5 , x 8 ; 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_(8);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_{8} ; x_{7} ; f\right]\right||R[f]|≦1105supxi∈[0,1]|[x1,x2,x3,x4,x5,x8;x7;f]|
If the derivative of order f ( 6 ) f ( 6 ) f^((6))f^{(6)}f(6)exists on [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1][0,1]We have
| R [ f ] | ≦ 1 105 ⋅ 1 6 ! sup x ∈ [ 0 , 1 ] | f ( 6 ) ( x ) | | R [ f ] | ≦ 1 105 ⋅ 1 6 ! sup x ∈ [ 0 , 1 ]   f ( 6 ) ( x ) |R[f]| <= (1)/(105)*(1)/(6!)s u p_(x in[0,1])|f^((6))(x)||R[f]| \leqq \frac{1}{105} \cdot \frac{1}{6!} \sup _{x \in[0,1]}\left|f^{(6)}(x)\right||R[f]|≦1105⋅16!supx∈[0,1]|f(6)(x)|

BIBLIOGRAPHY

[1] Bernstein, SN: Hark. Matem. ob-wa, s.2,t. 13, 1-2 (1912)
[2] Birkhoff, GD: Trans. Bitter. Math. Soc., 7, 107-130 (1906)
[3] Kowalewski, G.: Interpolation undgenäherte Quadratur, 1932
[4] Markoff, AA: Differenzenzechnung, 1896
[5] Mises, Rv: Jf die reine und augew. Math., 174, 56-67 (1936)
[6] obrechkoff, N.: Abh. press. Akad. Wiss., 1940, Nc 4, 1-20
[7] Popoviciu, T.: Mathematica, 8, 1-85 (1934)
[8] ": ibid., 10, 49-54 (1934)
[9] ": ibid., 12, 81-92 (1936)
[10] ": Aucrarile Ses. Gen.
Stuntifice
a Acad.
[13] Ramez, E. Ya.: Zbirnik Praci Institutu Matem. Akad. Nauk. URSR, 3, 21-62 (1939)
[14] Sard, A.: Duke Math. J. 15, 333-345 (1948)
[15] Wigert, S.: Arkiv f. Mast. Astr. och Fysik, Bd. 22B, n. 9, 1-4 (1932).
1960

Related Posts