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T. Popoviciu
Institutul de Calcul
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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)
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Symposium, International Provisional Computing Centre, 1960
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)
- In many approximation formulas of analysis the remainder; or complementary term,R[f]R[f]is a linear functional (additive and homogeneous) defined on a vector spaceSS, formed by functionsf=f(x)f=f(x), defined and continuous on an intervalIIof 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 restR[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 functionff, etc.
There is a large body of work on the structure of the remainderR[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].
J^(')J^{\prime}obtained, using the theory of higher-order convex functions thatj^(')j^{\prime}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].
J^(')J^{\prime}obtained, using the theory of higher-order convex functions thatj^(')j^{\prime}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].
In this communication I will make some remarks on this representation.
We will assume in the following that the functionffand the functionalR[f]R[f]are real and thatSScontains all polynomials.
2. We say thatR[f]R[f]is of the simple form if there exists an integern >= -1n \geq-1such that we have
2. We say thatR[f]R[f]is of the simple form if there exists an integern >= -1n \geq-1such that we have
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 S
OrK=R[x^(n+1)]K=R\left[x^{n+1}\right]East!=0\neq 0, independent of the functionffand thexi_(i)\xi_{i},i=1,2,dots,n+2i=1,2, \ldots, n+2aren+2n+2distinct points of the intervalII(which can generally depend on the functionffand even located at1^(n)1^{n}interior ofII. ifn >= 0n \geqq 0). The notation[xi_(1),xi_(2),dots,xi_(n+2);f]\left[\xi_{1}, \xi_{2}, \ldots, \xi_{n+2}; f\right]denotes the divided difference (of ardrén+1n+1) of the functionffon the knotsxi_(1),xi_(2),dots,xi_(n+2)\xi_{1}, \xi_{2}, \ldots, \xi_{n+2}.
The numbernnis the degree of accuracy of the remainder and enjoys the property (characteristic) thatR[f]R[f]is zero on any polynomial of degreenn, but that we haveR[x^(n+1)]!=0R\left[x^{n+1}\right] \neq 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 forR[f]R[f], assumed from the degree of accuracynn, either of the simple form is that we haveR[f]!=0R[f] \neq 0for everythingf in Sf \in S, convex of ordernn.
I. The necessary and sufficient condition forR[f]R[f], assumed from the degree of accuracynn, either of the simple form is that we haveR[f]!=0R[f] \neq 0for everythingf in Sf \in S, convex of ordernn.
In this case it is, moreover, necessary thatR[f]R[f]keep his sign forffconvex of ordernn. Noting that the functionx^(n+1)x^{n+1}is indeed convex of ordernn, the previous condition can also be written
{:(2)R[x^(n+1)]R[f] > 0:}\begin{equation*} R\left[x^{n+1}\right] R[f]>0 \tag{2} \end{equation*}
Condition (2), for allf in Sf \in Sconvex of ordernn, is therefore necessary and sufficient forR[f]R[f]either of the simple form (1). Note that for cola is also necessary (but not sufficient) thatl^(')l^{\prime}we haveR[x^(n+1)]!=0R\left[x^{n+1}\right] \neq 0And
{:(3)R[x^(n+1)]R[f] >= 0:}\begin{equation*} R\left[x^{n+1}\right] R[f] \geqq 0 \tag{3} \end{equation*}
for any functionf in Sf \in S, non-concave of ordernn.
Recall that the functionffis said to be convex resp, non-concave of ordernnonIIif the divided difference of ordern+1n+1onn+2n+2any distinct points ofIIremains constantly positive resp. non-negative.
3. IfR[f]R[f]is of the simple form, we can delimit it by the formula
Recall that the functionffis said to be convex resp, non-concave of ordernnonIIif the divided difference of ordern+1n+1onn+2n+2any distinct points ofIIremains constantly positive resp. non-negative.
3. IfR[f]R[f]is of the simple form, we can delimit it by the formula
{:(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*}
Or
{:(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*}
Besides, ifffhas a derivative of ordern+1n+1(bounded) onII, the number (5) is given by the equality
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| .
The delimitation (4) is valid in a more general case than in that of the simplicity ofR[f]R[f]. We have the following property:
II. The delimitation (4) is valid ifR[f]R[f]is of degree of exactness n and if inequality (3) is verified for any functionf in Sf \in S, non-concave of ordernn.
4. To be able to affirm that the restR[f]R[f]is of the simple form, it is enough to know the criteria allowing us to affirm that (under the hypothesisR[x^(n+1)]!=0R\left[x^{n+1}\right] \neq 0) inequality (2) is verified for allf in Sf \in Sconvex of ordernn.
II. The delimitation (4) is valid ifR[f]R[f]is of degree of exactness n and if inequality (3) is verified for any functionf in Sf \in S, non-concave of ordernn.
4. To be able to affirm that the restR[f]R[f]is of the simple form, it is enough to know the criteria allowing us to affirm that (under the hypothesisR[x^(n+1)]!=0R\left[x^{n+1}\right] \neq 0) inequality (2) is verified for allf in Sf \in Sconvex of ordernn.
Here we will present a criterion allowing us to affirm that (under the hypothesisR[x^(n+1)]!=0R\left[x^{n+1}\right] \neq 0) inequality (3) is verified for any functionf in Sf \in S, non-concave of ordernn, 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 functionff, by the corresponding polynomials of SN Bernstein [1,8,15].
Let's suppose thatI-=[0,1]I \equiv[0,1]and that the elements ofSShave a derivative of orderj( >= 0)j(\geqq 0)continue on[0.1][0.1](1 has a zeroth-order derivative and is the function itself). Let us then consider the linear functionalR[f]R[f], degree of accuracynnand which is limited in relation to the norm
{:(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*}
Let's ask
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} dt
Under the previous assumptions, we have the following property:
III. For inequality (3) to be verified for any functionf in Sf \in Snon-concave ordernnit is sufficient that one has
III. For inequality (3) to be verified for any functionf in Sf \in Snon-concave ordernnit is sufficient that one has
R[x^(n+1)]R[pi_(k,l)] >= 0R\left[x^{n+1}\right] R\left[\pi_{k, l}\right] \geqq 0
whatever the non-negative integerskkAndLL.
The demonstration results from the fact that if
The demonstration results from the fact that if
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}
(m >= n+1)(m \geqq n+1)
are the SN Bernstein polynomials, we have, under the previous assumptions,
are the SN Bernstein polynomials, we have, under the previous assumptions,
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]
And
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]
and whereffis a non-concave function of ordernnbelonging toSS.
5. To give an application, eitherR[f]R[f]the remainder in the numerical quadrature formula of N. Obrechkoff [6].
5. To give an application, eitherR[f]R[f]the remainder in the numerical quadrature formula of N. Obrechkoff [6].
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)+
+(1)/(3)f(1)-(1)/(30)f^(')(1)+R[f]+\frac{1}{3} f(1)-\frac{1}{30} f^{\prime}(1)+R[f]
Orffhas a continuous third-order derivative on[0,1][0,1].
In this caseR[f]R[f]is of degree of accuracy 5 and is bounded with respect to the norm (6) for.j=3j=3。
In this caseR[f]R[f]is of degree of accuracy 5 and is bounded with respect to the norm (6) for.j=3j=3。
A simple calculation gives us in oe.cas
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>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>0
The delimitation (4) is therefore well applicable and we have
|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|
If the derivative of orderf^((6))f^{(6)}exists on[0,1][0,1]We have
|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|
If the derivative of orderf^((6))f^{(6)}exists on[0,1][0,1]We have
|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|
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[2] Birkhoff, GD: Trans. Bitter. Math. Soc., 7, 107-130 (1906)
[3] Kowalewski, G.: Interpolation undgenäherte Quadratur, 1932
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[8] ": ibid., 10, 49-54 (1934)
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[9] ": ibid., 12, 81-92 (1936)
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