Abstract
Authors
T. Popoviciu
Institutul de Calcul
Keywords
?
Paper coordinates
T. Popoviciu, Vipuklie funkţii viscih poriadkov i ostatocinîi cilen v nekotorîh approksimationîh formulah analyza, Trudî tretievo Vses. Sezda Moskva, IV (1956), 164-167 (in Russian).
About this paper
Journal
Publisher Name
DOI
Print ISSN
Online ISSN
google scholar link
??
Paper (preprint) in HTML form
Original text
Rate this translation
Your feedback will be used to help improve Google Translate
T. Popovich (Ruzhniya)
CONVEX FUNCTIONS OF HIGHER ORDERS AND THE REMAINDER TERM IN SOME APPROXIMATION FORMULAS OF ANALYSIS
- In sonorax aporoxymannonic formulas of analysis, the remainder term is presented in the form of a lifelike functionalR[f]R[f](of a finite and homogeneous type), which in the general case are defined in the space of continuous functions in the influence of only those differentiable a given infinite or finite number of times in a finite and closed interval[a,b][a, b].
In the following, unless otherwise stated, we will assume thatffis a function of a real variable, and the remainder is a linear functional.
2. An important characteristic of the remainder term is the degree of increase. They say that the remainderR[f]R[f](or the corresponding apyroxinmannone formula, in the simple functionRR(f)) gives an approximation to exactly the degreenn, If
2. An important characteristic of the remainder term is the degree of increase. They say that the remainderR[f]R[f](or the corresponding apyroxinmannone formula, in the simple functionRR(f)) gives an approximation to exactly the degreenn, If
{:(1){:R[1]=R[x]=dots=R]x^(n)]=0","quad R[x^(n+1)]!=0.:}\begin{equation*} \left.R[1]=R[x]=\ldots=R] x^{n}\right]=0, \quad R\left[x^{n+1}\right] \neq 0 . \tag{1} \end{equation*}
For simplicity, we assumed thatR[1]=0R[1]=0, Son > 0n>0.
It is usually possible to express the remainder properlyR[f]c]R[f] \mathrm{c}]momentarily(n+1)(n+1)-nnderivativef^((n+1))f^{(n+1)}functionsff, assuming, of course, that this derivative exists. In this way, the problem of the remainder, first in particular cases, and then in more and more general cases, was studied first by A. A. Markov [3], then by G. D. Birkhoff [1], G. Kovalezskikh [2], J. Radov [7], and others. Recently, E. Remez [8] thoroughly studied the problem from a general point of view, consideringR[f]R[f]as a linear functional(n+1)(n+1)-th derivativef^((n+1))f^{(n+1)}.
3. The study of higher-order convex functions [5] leads to a new form of the remainder term, a form which is capable of unifying the results obtained so far, giving some of them a more general form [and refining [the structure of the remainder term.
It is usually possible to express the remainder properlyR[f]c]R[f] \mathrm{c}]momentarily(n+1)(n+1)-nnderivativef^((n+1))f^{(n+1)}functionsff, assuming, of course, that this derivative exists. In this way, the problem of the remainder, first in particular cases, and then in more and more general cases, was studied first by A. A. Markov [3], then by G. D. Birkhoff [1], G. Kovalezskikh [2], J. Radov [7], and others. Recently, E. Remez [8] thoroughly studied the problem from a general point of view, consideringR[f]R[f]as a linear functional(n+1)(n+1)-th derivativef^((n+1))f^{(n+1)}.
3. The study of higher-order convex functions [5] leads to a new form of the remainder term, a form which is capable of unifying the results obtained so far, giving some of them a more general form [and refining [the structure of the remainder term.
Well-known properties of separated comma differences encourage us to seek for a linear functionalR[f]R[f], satisfying the conditions (1), the expression is as follows:
{:(2)R[f]=K[x_(1),x_(2),dots,x_(n+2);f]:}\begin{equation*} R[f]=K\left[x_{1}, x_{2}, \ldots, x_{n+2} ; f\right] \tag{2} \end{equation*}
rae
s)K!=0K \neq 0- a number that does not record the functionff;
6)x_(1),x_(2),dots,x_(n+2)x_{1}, x_{2}, \ldots, x_{n+2}- defined points of the interval[a,b][a, b]generally говл, завнаяце от фукацыйff:
c) symbol[x_(1),x_(2),dots,x_(n+2);t]\left[x_{1}, x_{2}, \ldots, x_{n+2} ; t\right]means divided by finite difference of ordern+1n+1functionsffon the wallsx_(1),x_(2),dots,x_(n+2)x_{1}, x_{2}, \ldots, x_{n+2}These divided differences are determined, for example, by the recurrence formula
s)K!=0K \neq 0- a number that does not record the functionff;
6)x_(1),x_(2),dots,x_(n+2)x_{1}, x_{2}, \ldots, x_{n+2}- defined points of the interval[a,b][a, b]generally говл, завнаяце от фукацыйff:
c) symbol[x_(1),x_(2),dots,x_(n+2);t]\left[x_{1}, x_{2}, \ldots, x_{n+2} ; t\right]means divided by finite difference of ordern+1n+1functionsffon the wallsx_(1),x_(2),dots,x_(n+2)x_{1}, x_{2}, \ldots, x_{n+2}These divided differences are determined, for example, by the recurrence formula
{:(3)[x_(1),x_(2),dots,x_(n+2);n]=([x_(2),x_(3),dots,x_(n+2);n-[x_(1),x_(2),dots,x_(n+1);n])/(x_(n+2)-x_(1))","[x;f]=f(x).:}\begin{equation*}
\left[x_{1}, x_{2}, \ldots, x_{n+2} ; n\right]=\frac{\left[x_{2}, x_{3}, \ldots, x_{n+2} ; n-\left[x_{1}, x_{2}, \ldots, x_{n+1} ; n\right]\right.}{x_{n+2}-x_{1}},[x ; f]=f(x) . \tag{3}
\end{equation*}
If it is possible to find a number n such that equality (2) holds, we say that the functionR[f]R[f]simple form. In this casennis determined uniquely and we have
K=R[x^(n+1)].K=R\left[x^{n+1}\right] .
- Next, we recall that a convex function of order n in the interval [a,ba, b] is called a functionff, defined in this interval, and
{:(4)[x_(1),x_(2),dots,x_(n+2);f] > 0:}\begin{equation*}
\left[x_{1}, x_{2}, \ldots, x_{n+2} ; f\right]>0 \tag{4}
\end{equation*}
for any purposen+2n+2pointsx_(i)in[a,b],i=1,2dots,n+2x_{i} \in[a, b], i=1,2 \ldots, n+2.
Then we have the following theorem. In order for the remainderR[f]R[f]was of simple form, it is necessary and sufficient thatR[f]≁0R[f] \nsim 0for every function///convex ordernn(and which obviously belongs to the definition of the functional).
Then we have the following theorem. In order for the remainderR[f]R[f]was of simple form, it is necessary and sufficient thatR[f]≁0R[f] \nsim 0for every function///convex ordernn(and which obviously belongs to the definition of the functional).
Using various properties of approximations of higher-order functions, criteria are obtained by which, in some cases, one can conclude whether the remainder term is of simple form.
For example, under certain conditions, although not very limited, if we have in the avdu the most commonly used approximate formulas, it is enough that the functionalR[psi_(2)]R\left[\psi_{2}\right]without changing sign, if
{:(5)psi_(lambda)=((x-lambda+|x-lambda|)/(2))^(2)","quad lambda in[a","b].:}\begin{equation*}
\psi_{\lambda}=\left(\frac{x-\lambda+|x-\lambda|}{2}\right)^{2}, \quad \lambda \in[a, b] . \tag{5}
\end{equation*}
Similarly, the property (see [4]) of S. N. Berishtein polynomials to preserve the convexity of different orders of function can also serve to formulate similar criteria.
Many well-known approximation formulas have a remainder of simple form. The remainder terms of the Lagrange interpolation formula (in particular, the Tevlor formula), the numerical differentiation formulas of A. A. Markov, the numerical integration formulas of Cotes and Gauss, many formulas found and used by Sh. E. Mikeladze, etc., were remainder terms of simple form.
5. If conditions (1) are satisfied, then under sufficiently general assumptions one can find two non-negative numbersAAAndBB, depending on the functionff, so that it would be
5. If conditions (1) are satisfied, then under sufficiently general assumptions one can find two non-negative numbersAAAndBB, depending on the functionff, so that it would be
{:(6)R[f]=A[xi_(1),xi_(2),dots,xi_(n+v);quad f]-B[eta_(1),eta_(2),dots,eta_(n+2);f]:}\begin{equation*}
R[f]=A\left[\xi_{1}, \xi_{2}, \ldots, \xi_{n+v} ; \quad f\right]-B\left[\eta_{1}, \eta_{2}, \ldots, \eta_{n+2} ; f\right] \tag{6}
\end{equation*}
and the pointsxi_(i)\xi_{i}, on the double side, andeta_(i)\eta_{i}- on the other hand, they are disrespectful. Izeem, obviously,A-B=R[x^(n+1)]A-B=R\left[x^{n+1}\right]AndR[f]-R[f]-simple form, if you can put itA=0A=0orB=0B=0.
This result occurs, in particular, if the functionR[R[/]]satisfies some continuity properties, for example, if it has the property of uniform continuity of orderk < n-1k<n-1. i.e. if ov is continuous with respect to the norm of the form
sum_(i=0)^(k)max_(x in[a,B])|^((i))(x)|quad(k < n-1).\sum_{i=0}^{k} \max _{x \in[a, B]}\left|{ }^{(i)}(x)\right| \quad(k<n-1) .
Similar conditions are always satisfied in the most commonly used formulas of numerical differentiation and integration.
Formula (6) can be considered as a generalization of the extension of a linear functional to the difference of two linear and positive functionals, given by F. Riessock [9].
The smallest values ​​of the postlineAA.BBare curled out of the fork
A^(**)=ln!A=s u p_(F in(D))R[I,quadB^(**)=ln[B=A^(**)-R[x^(n+1)]:}A^{*}=\ln !A=\sup _{F \in(D)} R\left[I, \quad B^{*}=\ln \left[B=A^{*}-R\left[x^{n+1}\right]\right.\right.
Where (DD) there is a mazhkestvo of fushil(n+1)(n+1)-separated kovechvle razvostii which remain locked between 0 and 1 (we can assumeA > 0A>0, considering if vado,-R[l]-R[l]meecroR[t]R[t].
6. Лево так полт быт хорошо вавестные опения с помоцо (n+1n+1)- th piercing fupsyviff, if this derivative exists.
6. Лево так полт быт хорошо вавестные опения с помоцо (n+1n+1)- th piercing fupsyviff, if this derivative exists.
Izeem
|R[f]| < (A^(**)+B^(**))/((n+1)!)M_(n+1),quadM_(n+1)=s u p_(x in[a,b])|f^((n+1))||R[f]|<\frac{A^{*}+B^{*}}{(n+1)!} M_{n+1}, \quad M_{n+1}=\sup _{x \in[a, b]}\left|f^{(n+1)}\right|
what follows from the formula for the average value of Conn:
{:(7)[x_(1),x_(2),dots,x_(n+2)n=(1)/((n+1)!)f^((n+1))(xi):}:}\begin{equation*}
\left[x_{1}, x_{2}, \ldots, x_{n+2} n=\frac{1}{(n+1)!} f^{(n+1)}(\xi)\right. \tag{7}
\end{equation*}
Wherexi\xiis located inside the most lalogoy iperazl containing pointsx_(j),l=1,2x_{j}, l=1,2, .dots,n+2\ldots, n+2.
We have, however, generalized (7) by a formula that reveals the connection between the properties of differentiability of different orders of the function [6] and the weft, thus, the structure of the remainder term, when the function has a sufficient number of derivatives.
7. Formulas in the properties of separate cometary functions allow us to clarify the structure of the remainder term. I will define one criterion. Let us consider the so-called Hardy formula:
7. Formulas in the properties of separate cometary functions allow us to clarify the structure of the remainder term. I will define one criterion. Let us consider the so-called Hardy formula:
int_(0)^(4)fdx=0,28(f_(0)+f_(0))+1,62(f_(1)+f_(5))+2,2f_(3)+R[f]quad(f_(l)=f(i))\int_{0}^{4} f d x=0,28\left(f_{0}+f_{0}\right)+1,62\left(f_{1}+f_{5}\right)+2,2 f_{3}+R[f] \quad\left(f_{l}=f(i)\right)
The remainder can be written in the formR[f]=-63[0,0,1,1,3,3,5,5;rho]+109,8[0,1,1,3,3,5,5,6;rho]-63[1,1,3,3,5,5,6,6;rho]R[f]=-63[0,0,1,1,3,3,5,5 ; \rho]+109,8[0,1,1,3,3,5,5,6 ; \rho]-63[1,1,3,3,5,5,6,6 ; \rho].
varphi=int_(c)^(x)f(x)dx,quad epsi in(0,6)\varphi=\int_{c}^{x} f(x) d x, \quad \varepsilon \in(0,6)
(The meaning of separating divisions with repeating terms is well known).
The remainder here does not have a simple form. Calculating by transformations, taking into account formula (7), we find:
The remainder here does not have a simple form. Calculating by transformations, taking into account formula (7), we find:
R[f]=(9)/(700){f^((6))(xi)-(5(63+mu))/(9xx72)f^((8))(eta)},quad xi,eta in(0,6).R[f]=\frac{9}{700}\left\{f^{(6)}(\xi)-\frac{5(63+\mu)}{9 \times 72} f^{(8)}(\eta)\right\}, \quad \xi, \eta \in(0,6) .
if there are 8 gya provvodvaya.
Heremu\mu-negative hour< 32,4<32,4. Takingmu=1,8\mu=1,8, the input for the sound coefficient number(1)/(2)\frac{1}{2}, which can be found in all modern manuals that discuss Hardy's formula. However, one can also put it this waymu=0\mu=0, and then for the coefficient at(35)/(72) < (1)/(2)\frac{35}{72}<\frac{1}{2}we findf^((8))(eta)f^{(8)}(\eta).
Heremu\mu-negative hour< 32,4<32,4. Takingmu=1,8\mu=1,8, the input for the sound coefficient number(1)/(2)\frac{1}{2}, which can be found in all modern manuals that discuss Hardy's formula. However, one can also put it this waymu=0\mu=0, and then for the coefficient at(35)/(72) < (1)/(2)\frac{35}{72}<\frac{1}{2}we findf^((8))(eta)f^{(8)}(\eta).
If we assume only that the function / is continuous in [0, 6], the remainder of Hardy's formula can be written as
{:R[f]-(9)/(700){6![xi_(1),xi_(2),dots,xi_(3);f]-(5(63+mu))/(9xx72)81eta_(4),gamma_(2),dots,eta_(a);f]}.\left.R[f]-\frac{9}{700}\left\{6!\left[\xi_{1}, \xi_{2}, \ldots, \xi_{3} ; f\right]-\frac{5(63+\mu)}{9 \times 72} 81 \eta_{4}, \gamma_{2}, \ldots, \eta_{a} ; f\right]\right\} .
where the separated differences are formed at some nodes located inside the interval[0,6][0,6].
Other formulas of partial integration can be subjected to similar consideration.
8. The definition of the degree of approximation using relations (1) can be generalized if we consider the functionalsR[f]R[f], vanishing on a finite set
8. The definition of the degree of approximation using relations (1) can be generalized if we consider the functionalsR[f]R[f], vanishing on a finite set
{:(8)f_(0)","f_(1)","dots","f_(n):}\begin{equation*}
f_{0}, f_{1}, \ldots, f_{n} \tag{8}
\end{equation*}
functions. The above theory, which corresponds to the casef_(i)=x^(i),i=0,1,dots,nf_{i}=x^{i}, i=0,1, \ldots, ngeneralizes, at least under certain conditions, practically quite broadly, if we introduce the concept of hollowness in relation to functions (8).
Lit.: 1. Birkhoff GD, Trans. Amer. Math. Soc., 7 (1906), 107-136. 2. Kowalewski G., Interpolation and genaherte Quadratur. 1932. 3. Markoff A.A., Differenzenrechnung, 1896. 4. Popoviciu R. Mathematica, 10 (1934), 49-54. 5. Popoviciu T., Lucr. ses. gen. Acad. Rpr, (1950) 183-186. 6. Popoviciu T., A Magyar Tud. Acad. III oszt, Közl., IV (1954), 353-356. 7. Radon J., Monatch. f. Math. u. Phys., 42 (1935), 389-396. 8. Remez E.E. Ya., Rec. trav. Inst. math. Acad. Ukraine, No. 3 (1939),21-62;N4(1910),47-82.921-62 ; \mathcal{N} 4(1910), 47-82.9. R lesz F., Congr. Bologna, 3 (1930), 143-148.