Taylor series of functions of several variables and remainder terms in certain approximation formulas of analysis

Tiberiu Popoviciu
(original), Approximation Theory, paper

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T. Popoviciu
Institutul de Calcul

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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).

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1965 b -Popoviciu- Vipuklie funktii viscih poriadkov
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T. Popovich (Ruzhniya)

CONVEX FUNCTIONS OF HIGHER ORDERS AND THE REMAINDER TERM IN SOME APPROXIMATION FORMULAS OF ANALYSIS

  1. In sonorax aporoxymannonic formulas of analysis, the remainder term is presented in the form of a lifelike functional R [ f ] R [ f ] R[f]R[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 ] [a,b][a, b][a,b].
In the following, unless otherwise stated, we will assume that f f fffis 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 remainder R [ f ] R [ f ] R[f]R[f]R[f](or the corresponding apyroxinmannone formula, in the simple function R R RRR(f)) gives an approximation to exactly the degree n n nnn, If
(1) R [ 1 ] = R [ x ] = … = R ] x n ] = 0 , R [ x n + 1 ] ≠ 0 . (1) R [ 1 ] = R [ x ] = … = R ] x n = 0 , R x n + 1 ≠ 0 . {:(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*}(1)R[1]=R[x]=…=R]xn]=0,R[xn+1]≠0.
For simplicity, we assumed that R [ 1 ] = 0 R [ 1 ] = 0 R[1]=0R[1]=0R[1]=0, So n > 0 n > 0 n > 0n>0n>0.
It is usually possible to express the remainder properly R [ f ] c ] R [ f ] c ] R[f]c]R[f] \mathrm{c}]R[f]c]momentarily ( n + 1 ) ( n + 1 ) (n+1)(n+1)(n+1)- n n nnnderivative f ( n + 1 ) f ( n + 1 ) f^((n+1))f^{(n+1)}f(n+1)functions f f fff, 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, considering R [ f ] R [ f ] R[f]R[f]R[f]as a linear functional ( n + 1 ) ( n + 1 ) (n+1)(n+1)(n+1)-th derivative f ( n + 1 ) f ( n + 1 ) f^((n+1))f^{(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 functional R [ f ] R [ f ] R[f]R[f]R[f], satisfying the conditions (1), the expression is as follows:
(2) R [ f ] = K [ x 1 , x 2 , … , x n + 2 ; f ] (2) R [ f ] = K x 1 , x 2 , … , x n + 2 ; f {:(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*}(2)R[f]=K[x1,x2,…,xn+2;f]
rae
s) K ≠ 0 K ≠ 0 K!=0K \neq 0K≠0- a number that does not record the function f f fff;
6) 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+2- defined points of the interval [ a , b ] [ a , b ] [a,b][a, b][a,b]generally говл, завнаяце от фукацый f f fff:
c) symbol [ x 1 , x 2 , … , x n + 2 ; t ] x 1 , x 2 , … , x n + 2 ; t [x_(1),x_(2),dots,x_(n+2);t]\left[x_{1}, x_{2}, \ldots, x_{n+2} ; t\right][x1,x2,…,xn+2;t]means divided by finite difference of order n + 1 n + 1 n+1n+1n+1functions f f fffon the walls 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+2These divided differences are determined, for example, by the recurrence formula
(3) [ x 1 , x 2 , … , x n + 2 ; n ] = [ x 2 , x 3 , … , x n + 2 ; n - [ x 1 , x 2 , … , x n + 1 ; n ] x n + 2 - x 1 , [ x ; f ] = f ( x ) . (3) x 1 , x 2 , … , x n + 2 ; n = x 2 , x 3 , … , x n + 2 ; n - x 1 , x 2 , … , x n + 1 ; n x n + 2 - x 1 , [ x ; f ] = f ( x ) . {:(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*}(3)[x1,x2,…,xn+2;n]=[x2,x3,…,xn+2;n-[x1,x2,…,xn+1;n]xn+2-x1,[x;f]=f(x).
If it is possible to find a number n such that equality (2) holds, we say that the function R [ f ] R [ f ] R[f]R[f]R[f]simple form. In this case n n nnnis determined uniquely and we have
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].
  1. Next, we recall that a convex function of order n in the interval [ a , b a , b a,ba, ba,b] is called a function f f fff, defined in this interval, and
(4) [ x 1 , x 2 , … , x n + 2 ; f ] > 0 (4) x 1 , x 2 , … , x n + 2 ; f > 0 {:(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*}(4)[x1,x2,…,xn+2;f]>0
for any purpose n + 2 n + 2 n+2n+2n+2points x i ∈ [ a , b ] , i = 1 , 2 … , n + 2 x i ∈ [ a , b ] , i = 1 , 2 … , n + 2 x_(i)in[a,b],i=1,2dots,n+2x_{i} \in[a, b], i=1,2 \ldots, n+2xi∈[a,b],i=1,2…,n+2.
Then we have the following theorem. In order for the remainder R [ f ] R [ f ] R[f]R[f]R[f]was of simple form, it is necessary and sufficient that R [ f ] ≁ 0 R [ f ] ≁ 0 R[f]≁0R[f] \nsim 0R[f]≁0for every function / / ////convex order n n nnn(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 functional R [ ψ 2 ] R ψ 2 R[psi_(2)]R\left[\psi_{2}\right]R[ψ2]without changing sign, if
(5) ψ λ = ( x − λ + | x − λ | 2 ) 2 , λ ∈ [ a , b ] . (5) ψ λ = x − λ + | x − λ | 2 2 , λ ∈ [ a , b ] . {:(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*}(5)ψλ=(x-λ+|x-λ|2)2,λ∈[a,b].
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 numbers A A AAAAnd B B BBB, depending on the function f f fff, so that it would be
(6) R [ f ] = A [ ξ 1 , ξ 2 , … , ξ n + v ; f ] − B [ η 1 , η 2 , … , η n + 2 ; f ] (6) R [ f ] = A ξ 1 , ξ 2 , … , ξ n + v ; f − B η 1 , η 2 , … , η n + 2 ; f {:(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*}(6)R[f]=A[ξ1,ξ2,…,ξn+v;f]-B[η1,η2,…,ηn+2;f]
and the points ξ i ξ i xi_(i)\xi_{i}ξi, on the double side, and η i η i eta_(i)\eta_{i}ηi- on the other hand, they are disrespectful. Izeem, obviously, A − B = R [ x n + 1 ] A − B = R x n + 1 A-B=R[x^(n+1)]A-B=R\left[x^{n+1}\right]A-B=R[xn+1]And R [ f ] − R [ f ] − R[f]-R[f]-R[f]-simple form, if you can put it A = 0 A = 0 A=0A=0A=0or B = 0 B = 0 B=0B=0B=0.
This result occurs, in particular, if the function R [ R [ R[R[R[/ ] ] ]]]satisfies some continuity properties, for example, if it has the property of uniform continuity of order k < n − 1 k < n − 1 k < n-1k<n-1k<n-1. i.e. if ov is continuous with respect to the norm of the form
∑ i = 0 k max x ∈ [ a , B ] | ( i ) ( x ) | ( k < n − 1 ) . ∑ i = 0 k   max x ∈ [ a , B ]   ( i ) ( x ) ( k < n − 1 ) . 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) .∑i=0kmaxx∈[a,B]|(i)(x)|(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 postline A A AAA. B B BBBare curled out of the fork
A ∗ = ln ! A = sup F ∈ ( D ) R [ I , B ∗ = ln ⁡ [ B = A ∗ − R [ x n + 1 ] A ∗ = ln ! A = sup F ∈ ( D )   R I , B ∗ = ln ⁡ B = A ∗ − R x n + 1 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.A∗=ln!A=supF∈(D)R[I,B∗=ln⁡[B=A∗-R[xn+1]
Where ( D D DDD) there is a mazhkestvo of fushil ( n + 1 ) ( n + 1 ) (n+1)(n+1)(n+1)-separated kovechvle razvostii which remain locked between 0 and 1 (we can assume A > 0 A > 0 A > 0A>0A>0, considering if vado, − R [ l ] − R [ l ] -R[l]-R[l]-R[l]meecro R [ t ] R [ t ] R[t]R[t]R[t].
6. Лево так полт быт хорошо вавестные опения с помоцо ( n + 1 n + 1 n+1n+1n+1)- th piercing fupsyvi f f fff, if this derivative exists.
Izeem
| R [ f ] | < A ∗ + B ∗ ( n + 1 ) ! M n + 1 , M n + 1 = sup x ∈ [ a , b ] | f ( n + 1 ) | | R [ f ] | < A ∗ + B ∗ ( n + 1 ) ! M n + 1 , M n + 1 = sup x ∈ [ a , b ]   f ( n + 1 ) |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||R[f]|<A∗+B∗(n+1)!Mn+1,Mn+1=supx∈[a,b]|f(n+1)|
what follows from the formula for the average value of Conn:
(7) [ x 1 , x 2 , … , x n + 2 n = 1 ( n + 1 ) ! f ( n + 1 ) ( ξ ) (7) x 1 , x 2 , … , x n + 2 n = 1 ( n + 1 ) ! f ( n + 1 ) ( ξ ) {:(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*}(7)[x1,x2,…,xn+2n=1(n+1)!f(n+1)(ξ)
Where ξ ξ xi\xiξis located inside the most lalogoy iperazl containing points x j , l = 1 , 2 x j , l = 1 , 2 x_(j),l=1,2x_{j}, l=1,2xj,l=1,2, . … , n + 2 … , n + 2 dots,n+2\ldots, n+2…,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:
∫ 0 4 f d x = 0 , 28 ( f 0 + f 0 ) + 1 , 62 ( f 1 + f 5 ) + 2 , 2 f 3 + R [ f ] ( f l = f ( i ) ) ∫ 0 4   f d x = 0 , 28 f 0 + f 0 + 1 , 62 f 1 + f 5 + 2 , 2 f 3 + R [ f ] f l = f ( i ) 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)∫04fdx=0,28(f0+f0)+1,62(f1+f5)+2,2f3+R[f](fl=f(i))
The remainder can be written in the form R [ f ] = − 63 [ 0 , 0 , 1 , 1 , 3 , 3 , 5 , 5 ; ρ ] + 109 , 8 [ 0 , 1 , 1 , 3 , 3 , 5 , 5 , 6 ; ρ ] − 63 [ 1 , 1 , 3 , 3 , 5 , 5 , 6 , 6 ; ρ ] R [ f ] = − 63 [ 0 , 0 , 1 , 1 , 3 , 3 , 5 , 5 ; ρ ] + 109 , 8 [ 0 , 1 , 1 , 3 , 3 , 5 , 5 , 6 ; ρ ] − 63 [ 1 , 1 , 3 , 3 , 5 , 5 , 6 , 6 ; ρ ] 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]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]R[f]=-63[0,0,1,1,3,3,5,5;ρ]+109,8[0,1,1,3,3,5,5,6;ρ]-63[1,1,3,3,5,5,6,6;ρ].
φ = ∫ c x f ( x ) d x , ε ∈ ( 0 , 6 ) φ = ∫ c x   f ( x ) d x , ε ∈ ( 0 , 6 ) 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)φ=∫cxf(x)dx,ε∈(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:
R [ f ] = 9 700 { f ( 6 ) ( ξ ) − 5 ( 63 + μ ) 9 × 72 f ( 8 ) ( η ) } , ξ , η ∈ ( 0 , 6 ) . R [ f ] = 9 700 f ( 6 ) ( ξ ) − 5 ( 63 + μ ) 9 × 72 f ( 8 ) ( η ) , ξ , η ∈ ( 0 , 6 ) . 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) .R[f]=9700{f(6)(ξ)-5(63+μ)9×72f(8)(η)},ξ,η∈(0,6).
if there are 8 gya provvodvaya.
Here μ μ mu\muμ-negative hour < 32 , 4 < 32 , 4 < 32,4<32,4<32,4. Taking μ = 1 , 8 μ = 1 , 8 mu=1,8\mu=1,8μ=1,8, the input for the sound coefficient number 1 2 1 2 (1)/(2)\frac{1}{2}12, which can be found in all modern manuals that discuss Hardy's formula. However, one can also put it this way μ = 0 μ = 0 mu=0\mu=0μ=0, and then for the coefficient at 35 72 < 1 2 35 72 < 1 2 (35)/(72) < (1)/(2)\frac{35}{72}<\frac{1}{2}3572<12we find f ( 8 ) ( η ) f ( 8 ) ( η ) f^((8))(eta)f^{(8)}(\eta)f(8)(η).
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 ! [ ξ 1 , ξ 2 , … , ξ 3 ; f ] − 5 ( 63 + μ ) 9 × 72 81 η 4 , γ 2 , … , η a ; f ] } . R [ f ] − 9 700 6 ! ξ 1 , ξ 2 , … , ξ 3 ; f − 5 ( 63 + μ ) 9 × 72 81 η 4 , γ 2 , … , η a ; f . {: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\} .R[f]-9700{6![ξ1,ξ2,…,ξ3;f]-5(63+μ)9×7281η4,γ2,…,ηa;f]}.
where the separated differences are formed at some nodes located inside the interval [ 0 , 6 ] [ 0 , 6 ] [0,6][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 functionals R [ f ] R [ f ] R[f]R[f]R[f], vanishing on a finite set
(8) f 0 , f 1 , … , f n (8) f 0 , f 1 , … , f n {:(8)f_(0)","f_(1)","dots","f_(n):}\begin{equation*} f_{0}, f_{1}, \ldots, f_{n} \tag{8} \end{equation*}(8)f0,f1,…,fn
functions. The above theory, which corresponds to the case f i = x i , i = 0 , 1 , … , n f i = x i , i = 0 , 1 , … , n f_(i)=x^(i),i=0,1,dots,nf_{i}=x^{i}, i=0,1, \ldots, nfi=xi,i=0,1,…,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 ; N 4 ( 1910 ) , 47 − 82.9 21 − 62 ; N 4 ( 1910 ) , 47 − 82.9 21-62;N4(1910),47-82.921-62 ; \mathcal{N} 4(1910), 47-82.921-62;N4(1910),47-82.9. R lesz F., Congr. Bologna, 3 (1930), 143-148.
1956

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