Error estimation in the approximation of function by interpolation cubic splines

Costica Mustata
(original), approximation of univariate functions, Approximation Theory, cubic splines, interpolation, paper, remainder estimation, splines

Abstract

Authors

Costica Mustata
“Tiberiu Popoviciu” Institute of Numerical Analysis, Romanian Academy, Romania

C. Iancu

Keywords

Paper coordinates

C. Mustăţa, C. Iancu, Error estimation in the approximation of function by interpolation cubic splines, Mathematica 29 (52) 1 (1987), 33-39.

PDF

https://ictp.acad.ro/wp-content/uploads/2025/07/1987-Mustata-Mathematica-Error-estimation-in-the-approximation-of-function-bu-interpolation.pdf

About this paper

Journal

Mathematica

Publisher Name

Romanian Academy, Romania

DOI
Print ISSN

1222-9016

 

 

Online ISSN

2601-744X

MR # 89f: 41012

google scholar link

[1] C. Iancu, On the cubic spline of interpolation, Seminar of Functional analysis and Numerical Methods, Preprint nr.4 (1981), 52-71
[2] V.I. Miroshnichenko, On the error of approximation by cubic interpolaiton spines (Russian) Metody spline – Funkji, 93 (1982),3-29.
[3] V.I. Miroshnichenko, On the error of approximation by cubic interpolaiton splines II (Russian), Metody spline – funckji v.chisl. analize 98 (1983), 51-66.
[4] E.J. McShane, Extension of range of funcitons, Bull. Amer. Math. Soc., 40 (1934), 837-842.
[5] C. Mustata, Best approximaiton and unique extension of Lipschitz functions, Journal of Approx. Theory 19, 3 (1977), 222-230.
[6] C. Mustata, On the extension problem with prescribed norm, Seminar of Funcitonal analysis and Numerical Methods, Preprint nr.4 (1981), 93-99.

Paper (preprint) in HTML form

1987-Mustata-Mathematica-Error-estimation-in-the-approximation-of-function-bu-interpolation

ERROR ESTIMATION IN THE APPROXTMATION OF FUNCTIONS BY INTERPOLATION CUBIC SPLINES

C. IANCU, C. MUSTĂTA

  1. In this Note we give estimations for the error of approximation of a continuous function f : [ a , b ] → R f : [ a , b ] → R f:[a,b]rarr Rf:[a, b] \rightarrow Rf:[a,b]→R by interpolation cubic splines with respect to a given division Δ x Δ x Delta_(x)\Delta_{x}Δx of the interval [ a , b ] [ a , b ] [a,b][a, b][a,b].
Let f : [ a , b ] → R f : [ a , b ] → R f:[a,b]rarr Rf:[a, b] \rightarrow Rf:[a,b]→R be a function and let
(1) Δ x : a = x 0 < x 1 < … < x n = b (1) Δ x : a = x 0 < x 1 < … < x n = b {:(1)Delta_(x):a=x_(0) < x_(1) < dots < x_(n)=b:}\begin{equation*} \Delta_{x}: \boldsymbol{a}=x_{0}<x_{1}<\ldots<x_{n}=b \tag{1} \end{equation*}(1)Δx:a=x0<x1<…<xn=b
be a division of the interval [ a , b ] [ a , b ] [a,b][a, b][a,b].
Put
(2) f i = f ( x i ) , i = 0 , 1 , 2 , … , n . (2) f i = f x i , i = 0 , 1 , 2 , … , n . {:(2)f_(i)=f(x_(i))","quad i=0","1","2","dots","n.:}\begin{equation*} f_{i}=f\left(x_{i}\right), \quad i=0,1,2, \ldots, n . \tag{2} \end{equation*}(2)fi=f(xi),i=0,1,2,…,n.
and let us denote by Sp ( 3 , Δ x ) Sp 3 , Δ x Sp(3,Delta_(x))\operatorname{Sp}\left(3, \Delta_{x}\right)Sp(3,Δx) the set of all cubic spline s s sss corresponding to the partition Δ x Δ x Delta_(x)\Delta_{x}Δx and having the properties:
(i) the restriction of s s sss to every interval [ x i − 1 , x i ] x i − 1 , x i [x_(i-1),x_(i)]\left[x_{i-1}, x_{i}\right][xi−1,xi] is a polynomial of degree at most 3 , for i = 1 , 2 , … , n i = 1 , 2 , … , n i=1,2,dots,ni=1,2, \ldots, ni=1,2,…,n;
(ii) s ∈ C 2 [ a , b ] s ∈ C 2 [ a , b ] s inC^(2)[a,b]s \in C^{2}[a, b]s∈C2[a,b], i.e. s s sss is continuously two times differentiable on [ a , b ] [ a , b ] [a,b][a, b][a,b];
(iii) s ( x i ) = f i , i = 0 , 1 , 2 , … , n s x i = f i , i = 0 , 1 , 2 , … , n s(x_(i))=f_(i),i=0,1,2,dots,ns\left(x_{i}\right)=f_{i}, i=0,1,2, \ldots, ns(xi)=fi,i=0,1,2,…,n i.e. s s sss interpolates the function f f fff on the knots in Δ x Δ x Delta_(x)\Delta_{x}Δx.
Put also
(3) h i = x i − x i − 1 , i = 1 , 2 , … , n m i = s ′ ( x i ) , i = 0 , 1 , 2 , … , n M i = s ′ ′ ( x i ) , i = 0 , 1 , 2 , … , n (3) h i = x i − x i − 1 , i = 1 , 2 , … , n m i = s ′ x i , i = 0 , 1 , 2 , … , n M i = s ′ ′ x i , i = 0 , 1 , 2 , … , n {:(3){:[h_(i)=x_(i)-x_(i-1)",",i=1","2","dots","n],[m_(i)=s^(')(x_(i)),","quad i=0","1","2","dots","n],[M_(i)=s^('')(x_(i)),","quad i=0","1","2","dots","n]:}:}\begin{array}{ll} h_{i}=x_{i}-x_{i-1}, & i=1,2, \ldots, n \\ m_{i}=s^{\prime}\left(x_{i}\right) & , \quad i=0,1,2, \ldots, n \tag{3}\\ M_{i}=s^{\prime \prime}\left(x_{i}\right) & , \quad i=0,1,2, \ldots, n \end{array}(3)hi=xi−xi−1,i=1,2,…,nmi=s′(xi),i=0,1,2,…,nMi=s′′(xi),i=0,1,2,…,n
For s s sss in Sp ( 3 , Δ x ) Sp 3 , Δ x Sp(3,Delta_(x))\mathrm{Sp}\left(3, \Delta_{x}\right)Sp(3,Δx), the restriction of the second derivative s ′ ′ s ′ ′ s^('')s^{\prime \prime}s′′ of s s sss to the interval [ x i − 1 , x i ] x i − 1 , x i [x_(i-1),x_(i)]\left[x_{i-1}, x_{i}\right][xi−1,xi] is a polynomial of degree at most 1 , so that
(4) s ′ ′ ( x ) = M i − 1 + M i − M i − 1 x i − x i − 1 ( x − x i − 1 ) x ∈ [ x i − 1 , x i ] , i = 1 ― , n (4) s ′ ′ ( x ) = M i − 1 + M i − M i − 1 x i − x i − 1 x − x i − 1 x ∈ x i − 1 , x i , i = 1 ¯ , n {:[(4)s^('')(x)=M_(i-1)+(M_(i)-M_(i-1))/(x_(i)-x_(i-1))(x-x_(i-1))],[x in[x_(i-1),x_(i)]","quad i= bar(1)","n]:}\begin{gather*} s^{\prime \prime}(x)=M_{i-1}+\frac{M_{i}-M_{i-1}}{x_{i}-x_{i-1}}\left(x-x_{i-1}\right) \tag{4}\\ x \in\left[x_{i-1}, x_{i}\right], \quad i=\overline{1}, n \end{gather*}(4)s′′(x)=Mi−1+Mi−Mi−1xi−xi−1(x−xi−1)x∈[xi−1,xi],i=1―,n
Taking into account the conditions
(5) s ( x i − 1 ) = f i − 1 s ′ ( x i − 1 ) = m i − 1 (5) s x i − 1 = f i − 1 s ′ x i − 1 = m i − 1 {:[(5)s(x_(i-1))=f_(i-1)],[s^(')(x_(i-1))=m_(i-1)]:}\begin{align*} s\left(x_{i-1}\right) & =f_{i-1} \tag{5}\\ s^{\prime}\left(x_{i-1}\right) & =m_{i-1} \end{align*}(5)s(xi−1)=fi−1s′(xi−1)=mi−1
the relation (4) gives :
(6) s ( x ) = M i − M i − 1 6 h i ( x − x i − 1 ) 3 + M i − 1 2 ( x − x i − 1 ) 2 + + m i − 1 ( x − x i − 1 ) + f i − 1 (6) s ( x ) = M i − M i − 1 6 h i x − x i − 1 3 + M i − 1 2 x − x i − 1 2 + + m i − 1 x − x i − 1 + f i − 1 {:[(6)s(x)=(M_(i)-M_(i-1))/(6h_(i))(x-x_(i-1))^(3)+(M_(i-1))/(2)(x-x_(i-1))^(2)+],[quad+m_(i-1)(x-x_(i-1))+f_(i-1)]:}\begin{gather*} s(x)=\frac{M_{i}-M_{i-1}}{6 h_{i}}\left(x-x_{i-1}\right)^{3}+\frac{M_{i-1}}{2}\left(x-x_{i-1}\right)^{2}+ \tag{6}\\ \quad+m_{i-1}\left(x-x_{i-1}\right)+f_{i-1} \end{gather*}(6)s(x)=Mi−Mi−16hi(x−xi−1)3+Mi−12(x−xi−1)2++mi−1(x−xi−1)+fi−1
for x ∈ [ x i − 1 , x i ] x ∈ x i − 1 , x i x in[x_(i-1),x_(i)]x \in\left[x_{i-1}, x_{i}\right]x∈[xi−1,xi] and i = 1 , 2 , … , n i = 1 , 2 , … , n i=1,2,dots,ni=1,2, \ldots, ni=1,2,…,n.
Proposition. Every function s ∈ Sp ( 3 , Δ x ) s ∈ Sp 3 , Δ x s inSp(3,Delta_(x))s \in \mathrm{Sp}\left(3, \Delta_{x}\right)s∈Sp(3,Δx), given by formula (6), is uniquely determined by the conditions:
(i) s ( x i ) = f i , i = 1 , 2 , … , n s x i = f i , i = 1 , 2 , … , n s(x_(i))=f_(i),i=1,2,dots,ns\left(x_{i}\right)=f_{i}, i=1,2, \ldots, ns(xi)=fi,i=1,2,…,n
(ii) s ′ ( x i ) = m i , i = 1 , 2 , … , n s ′ x i = m i , i = 1 , 2 , … , n s^(')(x_(i))=m_(i),i=1,2,dots,ns^{\prime}\left(x_{i}\right)=m_{i}, i=1,2, \ldots, ns′(xi)=mi,i=1,2,…,n
(iii) m 0 = p , M 0 = q , p , q − m 0 = p , M 0 = q , p , q − m_(0)=p,M_(0)=q,p,q-m_{0}=p, M_{0}=q, p, q-m0=p,M0=q,p,q− given real numbers.
Proof. Conditions (i) and (ii) in the Proposition can be rewritten in the form
(7) M i = 6 f i − f i − 1 h i 2 − 6 h i ⋅ m i − 1 − 2 M i − 1 m i = 3 f i − f i − 1 h i − 2 m i − 1 − h i 2 M i − 1 , i = 1 , 2 , … , n (7) M i = 6 f i − f i − 1 h i 2 − 6 h i ⋅ m i − 1 − 2 M i − 1 m i = 3 f i − f i − 1 h i − 2 m i − 1 − h i 2 M i − 1 , i = 1 , 2 , … , n {:[(7)M_(i)=6(f_(i)-f_(i-1))/(h_(i)^(2))-(6)/(h_(i))*m_(i-1)-2M_(i-1)],[m_(i)=3(f_(i)-f_(i-1))/(h_(i))-2m_(i-1)-(h_(i))/(2)M_(i-1)","quad i=1","2","dots","n]:}\begin{align*} M_{i} & =6 \frac{f_{i}-f_{i-1}}{h_{i}^{2}}-\frac{6}{h_{i}} \cdot m_{i-1}-2 M_{i-1} \tag{7}\\ m_{i} & =3 \frac{f_{i}-f_{i-1}}{h_{i}}-2 m_{i-1}-\frac{h_{i}}{2} M_{i-1}, \quad i=1,2, \ldots, n \end{align*}(7)Mi=6fi−fi−1hi2−6hi⋅mi−1−2Mi−1mi=3fi−fi−1hi−2mi−1−hi2Mi−1,i=1,2,…,n
By condition (iii) system (7) is compatible and has a unique solution m 1 , m 2 , … , m n ; M 1 , M 2 , … , M n m 1 , m 2 , … , m n ; M 1 , M 2 , … , M n m_(1),m_(2),dots,m_(n);M_(1),M_(2),dots,M_(n)m_{1}, m_{2}, \ldots, m_{n} ; M_{1}, M_{2}, \ldots, M_{n}m1,m2,…,mn;M1,M2,…,Mn. System (7) can be recursively solved starting from the condition (iii) m 0 = p , M 0 = q m 0 = p , M 0 = q m_(0)=p,M_(0)=qm_{0}=p, M_{0}=qm0=p,M0=q.
2. Estimation of the approximation error. In some papers (see e.g. [2], [3] and the papers quoted there) are given evaluations of the uniform norms ‖ s − f ‖ ‖ s − f ‖ ||s-f||\|s-f\|‖s−f‖ and ‖ s ′ − f ′ ‖ s ′ − f ′ ||s^(')-f^(')||\left\|s^{\prime}-f^{\prime}\right\|‖s′−f′‖ for f f fff satisfying some sufficiently restrictive conditions.
(a) In the following we shall evaluate the uniform norm
(8) ‖ s − f ‖ , (8) ‖ s − f ‖ , {:(8)||s-f||",":}\begin{equation*} \|s-f\|, \tag{8} \end{equation*}(8)‖s−f‖,
supposing that f f fff is a Lipschitz function on [ a , b ] [ a , b ] [a,b][a, b][a,b], i.e. there exists a number K ⩾ 0 K ⩾ 0 K >= 0K \geqslant 0K⩾0 (called a Lipschitz constant) such that
(9) | f ( x ) − f ( y ) | ⩽ K | x − y | , (9) | f ( x ) − f ( y ) | ⩽ K | x − y | , {:(9)|f(x)-f(y)| <= K|x-y|",":}\begin{equation*} |f(x)-f(y)| \leqslant K|x-y|, \tag{9} \end{equation*}(9)|f(x)−f(y)|⩽K|x−y|,
for all x , y ∈ [ a , b ] x , y ∈ [ a , b ] x,y in[a,b]x, y \in[a, b]x,y∈[a,b].
The number
(10) ‖ f ‖ L = sup { | f ( x ) − f ( y ) | | | x − y ∣: x , y ∈ [ a , b ] , x ≠ y } (10) ‖ f ‖ L = sup { | f ( x ) − f ( y ) | | | x − y ∣: x , y ∈ [ a , b ] , x ≠ y } {:(10)||f||_(L)=s u p{|f(x)-f(y)|||x-y∣:x","y in[a","b]","x!=y}:}\begin{equation*} \|f\|_{L}=\sup \{|f(x)-f(y)|| | x-y \mid: x, y \in[a, b], x \neq y\} \tag{10} \end{equation*}(10)‖f‖L=sup{|f(x)−f(y)|||x−y∣:x,y∈[a,b],x≠y}
is the smallest Lipschitz constant for f f fff and is called the Lipschitz norm of f f fff on the interval [ a , b ] [ a , b ] [a,b][a, b][a,b].
The space of all Lipschitz function on [ a , b ] [ a , b ] [a,b][a, b][a,b] is denoted by Lip [ a , b ] [ a , b ] [a,b][a, b][a,b]. The Lipschitz norm of the restriction of f f fff to the division Δ x Δ x Delta_(x)\Delta_{x}Δx is given by
(11) ‖ f | Δ x ‖ L = max { | [ x i − 1 , x i ; f ] | : i = 1 , 2 , … , n } (11) f Δ x L = max x i − 1 , x i ; f : i = 1 , 2 , … , n {:(11)||f|_(Delta x)||_(L)=max{|[x_(i-1),x_(i);f]|:i=1,2,dots,n}:}\begin{equation*} \left\|\left.f\right|_{\Delta x}\right\|_{L}=\max \left\{\left|\left[x_{i-1}, x_{i} ; f\right]\right|: i=1,2, \ldots, n\right\} \tag{11} \end{equation*}(11)‖f|Δx‖L=max{|[xi−1,xi;f]|:i=1,2,…,n}
where [ x i − 1 , x i ; f ] = ( f ( x i ) − f ( x i − 1 ) ) / ( x i − x i − 1 ) x i − 1 , x i ; f = f x i − f x i − 1 / x i − x i − 1 [x_(i-1),x_(i);f]=(f(x_(i))-f(x_(i-1)))//(x_(i)-x_(i-1))\left[x_{i-1}, x_{i} ; f\right]=\left(f\left(x_{i}\right)-f\left(x_{i-1}\right)\right) /\left(x_{i}-x_{i-1}\right)[xi−1,xi;f]=(f(xi)−f(xi−1))/(xi−xi−1) is the divided difference of the function f f fff on the knots x i − 1 , x i x i − 1 , x i x_(i-1),x_(i)x_{i-1}, x_{i}xi−1,xi.
In the sequel we shall need the following extension result of McShane [4]: Let X X XXX be a matric space, Y Y YYY a subset of X X XXX and let f : Y → R f : Y → R f:Yrarr Rf: \mathrm{Y} \rightarrow Rf:Y→R be a Lipschitz function. Then there exists a Lipschitz function F : X → R F : X → R F:X rarr RF: X \rightarrow RF:X→R such that F | Y = f F Y = f F|_(Y)=f\left.F\right|_{Y}=fF|Y=f and ‖ F ‖ L = ‖ f ‖ L ‖ F ‖ L = ‖ f ‖ L ||F||_(L)=||f||_(L)\|F\|_{L}=\|f\|_{L}‖F‖L=‖f‖L. In [6] it was proved that for every f ∈ Lip Y ¯ f ∈ Lip Y ¯ f in Lip bar(Y)f \in \operatorname{Lip} \bar{Y}f∈LipY¯ and every K ⩾ ‖ f ‖ L K ⩾ ‖ f ‖ L K >= ||f||_(L)K \geqslant\|f\|_{L}K⩾‖f‖L there exists an extension F : X → R F : X → R F:X rarr RF: X \rightarrow RF:X→R of f f fff. such that ‖ F ‖ L = K ‖ F ‖ L = K ||F||_(L)=K\|F\|_{L}=K‖F‖L=K.
By this result, if f ∈ Lip [ a , b ] f ∈ Lip [ a , b ] f in Lip[a,b]f \in \operatorname{Lip}[a, b]f∈Lip[a,b], then the restriction f | Δ x f Δ x f|_(Delta x)\left.f\right|_{\Delta x}f|Δx of f f fff to Δ x Δ x Delta_(x)\Delta_{x}Δx has at least one extension F ∈ Lip [ a , b ] F ∈ Lip [ a , b ] F in Lip[a,b]F \in \operatorname{Lip}[a, b]F∈Lip[a,b] such that ‖ F ‖ L = ‖ f ‖ L ‖ F ‖ L = ‖ f ‖ L ||F||_(L)=||f||_(L)\|F\|_{L}=\|f\|_{L}‖F‖L=‖f‖L. It is obvious that such an extension is f f fff itself, but the following two functions
F 1 ( x ) = sup { f ( x k ) − ‖ f ‖ L ⋅ | x − x k | : k = 0 , 1 , 2 , … , n } (12) F 2 ( x ) = inf { f ( x k ) + ‖ f ‖ L ⋅ | x − x k | : k = 0 , 1 , … , n } F 1 ( x ) = sup f x k − ‖ f ‖ L ⋅ x − x k : k = 0 , 1 , 2 , … , n (12) F 2 ( x ) = inf f x k + ‖ f ‖ L ⋅ x − x k : k = 0 , 1 , … , n {:[F_(1)(x)=s u p{f(x_(k))-||f||_(L)*|x-x_(k)|:k=0,1,2,dots,n}],[(12)F_(2)(x)=i n f{f(x_(k))+||f||_(L)*|x-x_(k)|:k=0,1,dots,n}]:}\begin{align*} & F_{1}(x)=\sup \left\{f\left(x_{k}\right)-\|f\|_{L} \cdot\left|x-x_{k}\right|: k=0,1,2, \ldots, n\right\} \\ & F_{2}(x)=\inf \left\{f\left(x_{k}\right)+\|f\|_{L} \cdot\left|x-x_{k}\right|: k=0,1, \ldots, n\right\} \tag{12} \end{align*}F1(x)=sup{f(xk)−‖f‖L⋅|x−xk|:k=0,1,2,…,n}(12)F2(x)=inf{f(xk)+‖f‖L⋅|x−xk|:k=0,1,…,n}
are also extensions of f f fff with norm ‖ f ‖ L ‖ f ‖ L ||f||_(L)\|f\|_{L}‖f‖L, i.e.
(13) ‖ F 1 ‖ L = ‖ F 2 ‖ L = ‖ f ‖ L and F 1 | Δ x = F 2 | Δ x = f | Δ x (13) F 1 L = F 2 L = ‖ f ‖ L  and  F 1 Δ x = F 2 Δ x = f Δ x {:(13)||F_(1)||_(L)=||F_(2)||_(L)=||f||_(L)" and "F_(1)|_(Delta x)=F_(2)|_(Delta x)=f|_(Delta x):}\begin{equation*} \left\|F_{1}\right\|_{L}=\left\|F_{2}\right\|_{L}=\|f\|_{L} \text { and }\left.F_{1}\right|_{\Delta x}=\left.F_{2}\right|_{\Delta x}=\left.f\right|_{\Delta x} \tag{13} \end{equation*}(13)‖F1‖L=‖F2‖L=‖f‖L and F1|Δx=F2|Δx=f|Δx
and every extension F F FFF of f | Δ x f Δ x f|_(Delta x)\left.f\right|_{\Delta x}f|Δx such that ‖ F ‖ L = ‖ f ‖ L ‖ F ‖ L = ‖ f ‖ L ||F||_(L)=||f||_(L)\|F\|_{L}=\|f\|_{L}‖F‖L=‖f‖L verifies F 1 ⩽ F ′ ⩽ F 2 F 1 ⩽ F ′ ⩽ F 2 F_(1) <= F^(') <= F_(2)F_{1} \leqslant F^{\prime} \leqslant F_{2}F1⩽F′⩽F2 (see [5]). In particular
(14) F 1 ( x ) ⩽ f ( x ) ⩽ F 2 ( x ) , (14) F 1 ( x ) ⩽ f ( x ) ⩽ F 2 ( x ) , {:(14)F_(1)(x) <= f(x) <= F_(2)(x)",":}\begin{equation*} F_{1}(x) \leqslant f(x) \leqslant F_{2}(x), \tag{14} \end{equation*}(14)F1(x)⩽f(x)⩽F2(x),
for all x ∈ [ a , b ] x ∈ [ a , b ] x in[a,b]x \in[a, b]x∈[a,b].
From (14) it follows
s ( x ) − F 1 ( x ) ⩾ s ( x ) − f ( x ) ⩾ s ( x ) − F 2 ( x ) , x ∈ [ a , b ] s ( x ) − F 1 ( x ) ⩾ s ( x ) − f ( x ) ⩾ s ( x ) − F 2 ( x ) , x ∈ [ a , b ] s(x)-F_(1)(x) >= s(x)-f(x) >= s(x)-F_(2)(x),quad x in[a,b]s(x)-F_{1}(x) \geqslant s(x)-f(x) \geqslant s(x)-F_{2}(x), \quad x \in[a, b]s(x)−F1(x)⩾s(x)−f(x)⩾s(x)−F2(x),x∈[a,b]
so that
(15) min { ‖ s − F 1 ‖ , ‖ s − F 2 ‖ } ⩽ ‖ s − f ‖ ⩽ max { ‖ s − F 1 ‖ , ‖ s − F 2 ‖ } (15) min s − F 1 , s − F 2 ⩽ ‖ s − f ‖ ⩽ max s − F 1 , s − F 2 {:(15)min{||s-F_(1)||,||s-F_(2)||} <= ||s-f|| <= max{||s-F_(1)||,||s-F_(2)||}:}\begin{equation*} \min \left\{\left\|s-F_{1}\right\|,\left\|s-F_{2}\right\|\right\} \leqslant\|s-f\| \leqslant \max \left\{\left\|s-F_{1}\right\|,\left\|s-F_{2}\right\|\right\} \tag{15} \end{equation*}(15)min{‖s−F1‖,‖s−F2‖}⩽‖s−f‖⩽max{‖s−F1‖,‖s−F2‖}
Taking into account the fact that the functions F 1 F 1 F_(1)F_{1}F1 and F 2 F 2 F_(2)F_{2}F2 given by (12) are piecewise linear, the calculation of the norms ‖ s − F 1 ‖ s − F 1 ||s-F_(1)||\left\|s-F_{1}\right\|‖s−F1‖ and ‖ s − F 2 ‖ s − F 2 ||s-F_(2)||\left\|s-F_{2}\right\|‖s−F2‖ reduces to the calculation of the norm of third degree polynomials on compact subintervals of [ a , b ] [ a , b ] [a,b][a, b][a,b].
If
a i = ‖ ( s − F 1 ) | [ x i − 1 , x i ] ‖ and (16) b i = ‖ ( s − F 2 ) | [ x i − 1 , x i ] ‖ a i = s − F 1 x i − 1 , x i  and  (16) b i = s − F 2 x i − 1 , x i {:[a_(i)=||(s-F_(1))|_([x_(i-1),x_(i)])||" and "],[(16)b_(i)=||(s-F_(2))|_([x_(i-1),x_(i)])||]:}\begin{align*} & a_{i}=\left\|\left.\left(s-F_{1}\right)\right|_{\left[x_{i-1}, x_{i}\right]}\right\| \text { and } \\ & b_{i}=\left\|\left.\left(s-F_{2}\right)\right|_{\left[x_{i-1}, x_{i}\right]}\right\| \tag{16} \end{align*}ai=‖(s−F1)|[xi−1,xi]‖ and (16)bi=‖(s−F2)|[xi−1,xi]‖
for i = 1 , 2 , … , n i = 1 , 2 , … , n i=1,2,dots,ni=1,2, \ldots, ni=1,2,…,n, then
‖ s − F 1 ′ ‖ = max { a i : i = 1 , 2 , … , n } s − F 1 ′ = max a i : i = 1 , 2 , … , n ||s-F_(1)^(')||=max{a_(i):i=1,2,dots,n}\left\|s-F_{1}^{\prime}\right\|=\max \left\{a_{i}: i=1,2, \ldots, n\right\}‖s−F1′‖=max{ai:i=1,2,…,n}
and
‖ s − F 2 ‖ = max { b i : i = 1 , 2 , … , n } s − F 2 = max b i : i = 1 , 2 , … , n ||s-F_(2)||=max{b_(i):i=1,2,dots,n}\left\|s-F_{2}\right\|=\max \left\{b_{i}: i=1,2, \ldots, n\right\}‖s−F2‖=max{bi:i=1,2,…,n}
In order to calculate the numbers a i , b i , i = 1 , 2 , … , n a i , b i , i = 1 , 2 , … , n a_(i),b_(i),i=1,2,dots,na_{i}, b_{i}, i=1,2, \ldots, nai,bi,i=1,2,…,n, we have to distinct three cases :

Case 1.: f i − 1 < f i f i − 1 < f i f_(i-1) < f_(i)f_{i-1}<f_{i}fi−1<fi.

In this case, for x ∈ [ x i − 1 , x i ] x ∈ x i − 1 , x i x in[x_(i-1),x_(i)]x \in\left[x_{i-1}, x_{i}\right]x∈[xi−1,xi] we have
F 1 ( x ) = { f i − 1 − ‖ f ‖ L ( x − x i − 1 ) , x ∈ [ x i − 1 , x ― ] f i + ‖ f ‖ L ( x − x i ) , x ∈ ( x ― , x i ] F 1 ( x ) = f i − 1 − ‖ f ‖ L x − x i − 1 , x ∈ x i − 1 , x _ f i + ‖ f ‖ L x − x i , x ∈ x _ , x i F_(1)(x)={[f_(i-1)-||f||_(L)(x-x_(i-1))","quad x in[x_(i-1),x_]],[f_(i)+||f||_(L)(x-x_(i))","quad x in(x_,x_(i)]]:}F_{1}(x)=\left\{\begin{array}{l} f_{i-1}-\|f\|_{L}\left(x-x_{i-1}\right), \quad x \in\left[x_{i-1}, \underline{x}\right] \\ f_{i}+\|f\|_{L}\left(x-x_{i}\right), \quad x \in\left(\underline{x}, x_{i}\right] \end{array}\right.F1(x)={fi−1−‖f‖L(x−xi−1),x∈[xi−1,x―]fi+‖f‖L(x−xi),x∈(x―,xi]
where
x ― = x i + x i − 1 2 + f i − 1 − f i 2 ‖ f ‖ L x _ = x i + x i − 1 2 + f i − 1 − f i 2 ‖ f ‖ L x_=(x_(i)+x_(i-1))/(2)+(f_(i-1)-f_(i))/(2||f||_(L))\underline{x}=\frac{x_{i}+x_{i-1}}{2}+\frac{f_{i-1}-f_{i}}{2\|f\|_{L}}x―=xi+xi−12+fi−1−fi2‖f‖L
and
F 2 ( x ) = { f i − 1 + ‖ f ‖ L ( x − x i − 1 ) , x ∈ [ x i − 1 , x ¯ ] f i − ‖ f ‖ L ( x − x i ) , x ∈ ( x ¯ , x i ] F 2 ( x ) = f i − 1 + ‖ f ‖ L x − x i − 1 ,      x ∈ x i − 1 , x ¯ f i − ‖ f ‖ L x − x i ,      x ∈ x ¯ , x i F_(2)(x)={[f_(i-1)+||f||_(L)(x-x_(i-1))",",x in[x_(i-1),( bar(x))]],[f_(i)-||f||_(L)(x-x_(i))",",x in(( bar(x)),x_(i)]]:}F_{2}(x)= \begin{cases}f_{i-1}+\|f\|_{L}\left(x-x_{i-1}\right), & x \in\left[x_{i-1}, \bar{x}\right] \\ f_{i}-\|f\|_{L}\left(x-x_{i}\right), & x \in\left(\bar{x}, x_{i}\right]\end{cases}F2(x)={fi−1+‖f‖L(x−xi−1),x∈[xi−1,x¯]fi−‖f‖L(x−xi),x∈(x¯,xi]
where
x ¯ = x i + x i − 1 2 − f i − 1 − f i 2 ‖ f ‖ L x ¯ = x i + x i − 1 2 − f i − 1 − f i 2 ‖ f ‖ L bar(x)=(x_(i)+x_(i-1))/(2)-(f_(i-1)-f_(i))/(2||f||_(L))\bar{x}=\frac{x_{i}+x_{i-1}}{2}-\frac{f_{i-1}-f_{i}}{2\|f\|_{L}}x¯=xi+xi−12−fi−1−fi2‖f‖L
Since
x i − 1 < x ― < x ¯ < x i x i − 1 < x _ < x ¯ < x i x_(i-1) < x_ < bar(x) < x_(i)x_{i-1}<\underline{x}<\bar{x}<x_{i}xi−1<x―<x¯<xi
we have
a i = max { ‖ ( s − F 1 ) | [ x i − 1 , x i ] ‖ , ‖ ( s − F 1 ) | [ x ― , [ x ¯ ] ‖ , ‖ ( s − F 1 ) | [ x ¯ , x i ] ‖ } a i = max s − F 1 x i − 1 , x i , s − F 1 [ x _ , [ x ¯ ] , s − F 1 x ¯ , x i a_(i)=max{||(s-F_(1))|_([x_(i-1),x_(i)])||,||(s-F_(1))|_([x_,[ bar(x)])||,||(s-F_(1))|_([( bar(x)),x_(i)])||}a_{i}=\max \left\{\left\|\left.\left(s-F_{1}\right)\right|_{\left[x_{i-1}, x_{i}\right]}\right\|,\left\|\left.\left(s-F_{1}\right)\right|_{[\underline{x},[\bar{x}]}\right\|,\left\|\left.\left(s-F_{1}\right)\right|_{\left[\bar{x}, x_{i}\right]}\right\|\right\}ai=max{‖(s−F1)|[xi−1,xi]‖,‖(s−F1)|[x―,[x¯]‖,‖(s−F1)|[x¯,xi]‖}
and
b i = max { ‖ ( s − F 2 ) | [ x i − 1 , x ] ‖ , ‖ ( s − F 2 ) | [ x ― , x ¯ ] ‖ , ‖ ( s − F 2 ) | [ x ¯ , x i ‖ } b i = max s − F 2 x i − 1 , x , s − F 2 [ x _ , x ¯ ] , s − F 2 x ¯ , x i b_(i)=max{||(s-F_(2))|_([x_(i-1),x])||,||(s-F_(2))|_([x_, bar(x)])||,||(s-F_(2))|_([( bar(x)),x_(i):})||}b_{i}=\max \left\{\left\|\left.\left(s-F_{2}\right)\right|_{\left[x_{i-1}, x\right]}\right\|,\left\|\left.\left(s-F_{2}\right)\right|_{[\underline{x}, \bar{x}]}\right\|,\left\|\left.\left(s-\mathbb{F}_{2}\right)\right|_{\left[\bar{x}, x_{i}\right.}\right\|\right\}bi=max{‖(s−F2)|[xi−1,x]‖,‖(s−F2)|[x―,x¯]‖,‖(s−F2)|[x¯,xi‖}
Case 2. f i − 1 > f i f i − 1 > f i f_(i-1) > f_(i)f_{i-1}>f_{i}fi−1>fi.
In this case x i − 1 < x ¯ < x ― < x i x i − 1 < x ¯ < x _ < x i x_(i-1) < bar(x) < x_ < x_(i)x_{i-1}<\bar{x}<\underline{x}<x_{i}xi−1<x¯<x―<xi and therefore the norms of s − F 1 s − F 1 s-F_(1)s-\boldsymbol{F}_{1}s−F1 and s − F 2 s − F 2 s-F_(2)s-F_{2}s−F2 are calculated on the intervals [ x i − 1 , x ¯ ] , [ x ¯ , x ― ] , [ x ― , x i ] x i − 1 , x ¯ , [ x ¯ , x _ ] , x _ , x i [x_(i-1),( bar(x))],[ bar(x),x_],[x_,x_(i)]\left[x_{i-1}, \bar{x}\right],[\bar{x}, \underline{x}],\left[\underline{x}, x_{i}\right][xi−1,x¯],[x¯,x―],[x―,xi].
Case 3. f i − 1 = f i f i − 1 = f i f_(i-1)=f_(i)f_{i-1}=f_{i}fi−1=fi.
In this case x ― = x ¯ = ( x i − 1 + x i ) / 2 x _ = x ¯ = x i − 1 + x i / 2 x_= bar(x)=(x_(i-1)+x_(i))//2\underline{x}=\bar{x}=\left(x_{i-1}+x_{i}\right) / 2x―=x¯=(xi−1+xi)/2 and the norms of s − F 1 s − F 1 s-F_(1)s-F_{1}s−F1 and s − F 2 s − F 2 s-F_(2)s-F_{2}s−F2 are calculated on the intervals [ x i − 1 , ( x i + x i − 1 ) / 2 ] , [ ( x i + x i − 1 ) / 2 , x i ] x i − 1 , x i + x i − 1 / 2 , x i + x i − 1 / 2 , x i [x_(i-1),(x_(i)+x_(i-1))//2],[(x_(i)+x_(i-1))//2,x_(i)]\left[x_{i-1},\left(x_{i}+x_{i-1}\right) / 2\right],\left[\left(x_{i}+x_{i-1}\right) / 2, x_{i}\right][xi−1,(xi+xi−1)/2],[(xi+xi−1)/2,xi].
In concrete situations, the numbers a i a i a_(i)a_{i}ai and b i b i b_(i)b_{i}bi can be easily calculated. We do not enter into details, but let us mention that, in general, can be obtained evaluations from above of the norms occurring in the expressins of a i a i a_(i)a_{i}ai and b i b i b_(i)b_{i}bi, depending only on m i , m i − 1 , M i , M i − 1 , h i m i , m i − 1 , M i , M i − 1 , h i m_(i),m_(i-1),M_(i),M_(i-1),h_(i)m_{i}, m_{i-1}, M_{i}, M_{i-1}, h_{i}mi,mi−1,Mi,Mi−1,hi and ‖ f ‖ L ‖ f ‖ L ||f||_(L)\|f\|_{L}‖f‖L.
Concerning the exactity of the evaluations (15) we show that in the set of all real valued Lipschitz functions g g ggg on [ a , b ] [ a , b ] [a,b][a, b][a,b] with norm ‖ g ‖ L == ‖ f ‖ L ‖ g ‖ L == ‖ f ‖ L ||g||_(L)==||f||_(L)\|g\|_{L}= =\|f\|_{L}‖g‖L==‖f‖L and such that g ( x i ) = f i , i = 0 , 1 , 2 , … , n g x i = f i , i = 0 , 1 , 2 , … , n g(x_(i))=f_(i),i=0,1,2,dots,ng\left(x_{i}\right)=f_{i}, i=0,1,2, \ldots, ng(xi)=fi,i=0,1,2,…,n, there exists two functions f ¯ f ¯ bar(f)\bar{f}f¯ and f ― f _ f_\underline{f}f― such that the evaluations (15) are the best possible in this set.
Let
(17) E ( f | Δ x ; ⌊ a , b ⌋ ) = { g ∈ Lip ⌞ a , b ] : g ( x i ) = f ( x i ) , i = 0 , 1 , 2 , … , n ‖ g ‖ L = ‖ f ‖ L } (17) E f Δ x ; ⌊ a , b ⌋ = g ∈ Lip ⌞ a , b : g x i = f x i , i = 0 , 1 , 2 , … , n ‖ g ‖ L = ‖ f ‖ L {:[(17)E(f|_(Delta_(x));|__ a,b __|)={g inLip_(llcorner)a,b]:g(x_(i))=f(x_(i))","quad i=0","1","2","dots","n],[{:||g||_(L)=||f||_(L)}]:}\begin{gather*} E\left(\left.f\right|_{\Delta_{x}} ;\lfloor a, b\rfloor\right)=\left\{g \in \operatorname{Lip}_{\llcorner } a, b\right]: g\left(x_{i}\right)=f\left(x_{i}\right), \quad i=0,1,2, \ldots, n \tag{17}\\ \left.\|g\|_{L}=\|f\|_{L}\right\} \end{gather*}(17)E(f|Δx;⌊a,b⌋)={g∈Lip⌞a,b]:g(xi)=f(xi),i=0,1,2,…,n‖g‖L=‖f‖L}
Obviously, the functions F 1 F 1 F_(1)F_{1}F1 and F 2 F 2 F_(2)F_{2}F2 defined by (12) belong to E ( f | Δ g ; [ a , b ] ) E f Δ g ; [ a , b ] E(f|_(Delta_(g));[a,b])E\left(\left.f\right|_{\Delta_{g}} ;[a, b]\right)E(f|Δg;[a,b]) and if g ∈ E ( f | Δ x ; [ a , b ] ) g ∈ E f Δ x ; [ a , b ] g in E(f|_(Delta_(x));[a,b])g \in E\left(\left.f\right|_{\Delta_{x}} ;[a, b]\right)g∈E(f|Δx;[a,b]) then
F 1 ( x ) ⩽ g ( x ) ⩽ F 2 ( x ) , x ∈ [ a , b ] . F 1 ( x ) ⩽ g ( x ) ⩽ F 2 ( x ) , x ∈ [ a , b ] . F_(1)(x) <= g(x) <= F_(2)(x),quad x in[a,b].\boldsymbol{F}_{\mathbf{1}}(x) \leqslant \boldsymbol{g}(x) \leqslant \boldsymbol{F}_{\mathbf{2}}(x), \quad x \in[\boldsymbol{a}, \boldsymbol{b}] .F1(x)⩽g(x)⩽F2(x),x∈[a,b].
For every interval [ x i − 1 , x i ] , i = 1 , 2 , … , n x i − 1 , x i , i = 1 , 2 , … , n [x_(i-1),x_(i)],i=1,2,dots,n\left[x_{i-1}, x_{i}\right], i=1,2, \ldots, n[xi−1,xi],i=1,2,…,n, let us define the function f ¯ i f ¯ i bar(f)_(i)\bar{f}_{i}f¯i in the following way:
(18) f ¯ i = { F 1 | ∣ x i − 1 , x i ] if a i = max { a i , b i } F 2 | | x i − 1 , x i | if b i = max { a i , b i } (18) f ¯ i = F 1 ∣ x i − 1 , x i  if  a i = max a i , b i F 2 x i − 1 , x i  if  b i = max a i , b i {:(18) bar(f)_(i)={[F_(1)|_({:∣x_(i-1),x_(i)])" if "a_(i)=max{a_(i),b_(i)}],[F_(2)|_(|x_(i-1),x_(i)|)" if "b_(i)=max{a_(i),b_(i)}]:}:}\bar{f}_{i}=\left\{\begin{array}{l} \left.F_{1}\right|_{\left.\mid x_{i-1}, x_{i}\right]} \text { if } a_{i}=\max \left\{a_{i}, b_{i}\right\} \tag{18}\\ \left.F_{2}\right|_{\left|x_{i-1}, x_{i}\right|} \text { if } b_{i}=\max \left\{a_{i}, b_{i}\right\} \end{array}\right.(18)f¯i={F1|∣xi−1,xi] if ai=max{ai,bi}F2||xi−1,xi| if bi=max{ai,bi}
Let the function f ¯ : [ a , b ] → R f ¯ : [ a , b ] → R bar(f):[a,b]rarr R\bar{f}:[a, b] \rightarrow Rf¯:[a,b]→R be defined by
(19) f ¯ | [ x i − 1 , x i ] = f ¯ i , i = 1 , 2 , … , n . (19) f ¯ x i − 1 , x i = f ¯ i , i = 1 , 2 , … , n . {:(19)( bar(f))|_([x_(i-1),x_(i)])= bar(f)_(i)","quad i=1","2","dots","n.:}\begin{equation*} \left.\bar{f}\right|_{\left[x_{i-1}, x_{i}\right]}=\bar{f}_{i}, \quad i=1,2, \ldots, n . \tag{19} \end{equation*}(19)f¯|[xi−1,xi]=f¯i,i=1,2,…,n.
Then f ¯ ∈ Lip [ a , b ] f ¯ ∈ Lip [ a , b ] bar(f)in Lip[a,b]\bar{f} \in \operatorname{Lip}[\boldsymbol{a}, \boldsymbol{b}]f¯∈Lip[a,b] and, as can be easily seen from the definition of the function f → f → vec(f)\vec{f}f→,
(20) ‖ f ¯ − s ‖ ⩽ ‖ g − s ‖ (20) ‖ f ¯ − s ‖ ⩽ ‖ g − s ‖ {:(20)|| bar(f)-s|| <= ||g-s||:}\begin{equation*} \|\bar{f}-s\| \leqslant\|g-s\| \tag{20} \end{equation*}(20)‖f¯−s‖⩽‖g−s‖
for every function g ∈ E ( f ∣ Δ x ; [ a , b ] ) g ∈ E f ∣ Δ x ; [ a , b ] g inE(f∣Delta_(x);[a,b])g \in \mathbb{E}\left(f \mid \Delta_{x} ;[a, b]\right)g∈E(f∣Δx;[a,b]).
Similarly, the function f ― : [ a , b ] → R f _ : [ a , b ] → R f_:[a,b]rarr R\underline{f}:[a, b] \rightarrow Rf―:[a,b]→R defined by
(21) f ― ∣ x i − 1 , x i ] = f ― i , i = 1 , 2 , … , n (21) f _ ∣ x i − 1 , x i = f _ i , i = 1 , 2 , … , n {:(21)f__({:∣x_(i-1),x_(i)])=f__(i)","quad i=1","2","dots","n:}\begin{equation*} \underline{f}_{\left.\mid x_{i-1}, x_{i}\right]}=\underline{f}_{i}, \quad i=1,2, \ldots, n \tag{21} \end{equation*}(21)f―∣xi−1,xi]=f―i,i=1,2,…,n
where
(22) f ― i = { F 1 | [ x i − 1 , x i ] if a i = min { a i , b i } F 2 | [ x i − 1 , x i ] if b i = min { a i , b i } (22) f _ i = F 1 x i − 1 , x i  if  a i = min a i , b i F 2 x i − 1 , x i  if  b i = min a i , b i {:(22)f__(i)={[F_(1)|_([x_(i-1),x_(i)])" if "a_(i)=min{a_(i),b_(i)}],[F_(2)|_([x_(i-1),x_(i)])" if "b_(i)=min{a_(i),b_(i)}]:}:}\underline{f}_{i}=\left\{\begin{array}{l} \left.F_{1}\right|_{\left[x_{i-1}, x_{i}\right]} \text { if } a_{i}=\min \left\{a_{i}, b_{i}\right\} \tag{22}\\ \left.F_{2}\right|_{\left[x_{i-1}, x_{i}\right]} \text { if } b_{i}=\min \left\{a_{i}, b_{i}\right\} \end{array}\right.(22)f―i={F1|[xi−1,xi] if ai=min{ai,bi}F2|[xi−1,xi] if bi=min{ai,bi}
for i = 1 , 2 , … , n i = 1 , 2 , … , n i=1,2,dots,ni=1,2, \ldots, ni=1,2,…,n, verifies the inequality
(23) ‖ g − s ‖ ⩽ ‖ f − s ‖ (23) ‖ g − s ‖ ⩽ ‖ f − s ‖ {:(23)||g-s|| <= ||f-s||:}\begin{equation*} \|g-s\| \leqslant\|f-s\| \tag{23} \end{equation*}(23)‖g−s‖⩽‖f−s‖
for every function g ∈ E ( f ∣ Δ x ; [ a , b ] ) g ∈ E f ∣ Δ x ; [ a , b ] g in E(f∣Delta_(x);[a,b])g \in E\left(f \mid \Delta_{x} ;[a, b]\right)g∈E(f∣Δx;[a,b]).
(b) Evaluation of the norm ‖ f ′ − s ′ ‖ f ′ − s ′ ||f^(')-s^(')||\left\|f^{\prime}-s^{\prime}\right\|‖f′−s′‖.
In the following we shall suppose f ∈ C 1 [ a , b ] f ∈ C 1 [ a , b ] f inC^(1)[a,b]f \in C^{1}[a, b]f∈C1[a,b]. In this case f ∈ Lip [ a , b ] f ∈ Lip [ a , b ] f in Lip[a,b]f \in \operatorname{Lip}[a, b]f∈Lip[a,b] and
(24) ‖ f ‖ L = max { | f ′ ( x ) | : x ∈ [ a , b ] } . (24) ‖ f ‖ L = max f ′ ( x ) : x ∈ [ a , b ] . {:(24)||f||_(L)=max{|f^(')(x)|:x in[a,b]}.:}\begin{equation*} \|f\|_{L}=\max \left\{\left|f^{\prime}(x)\right|: x \in[a, b]\right\} . \tag{24} \end{equation*}(24)‖f‖L=max{|f′(x)|:x∈[a,b]}.
The formulae (12) become
F 1 ( x ) = sup { f ( x i ) − max x ∈ [ a , b ] | f ′ ( x ) | ⋅ | x − x i | : i = 0 , 1 , … , n } F 1 ( x ) = sup f x i − max x ∈ [ a , b ]   f ′ ( x ) ⋅ x − x i : i = 0 , 1 , … , n F_(1)(x)=s u p{f(x_(i))-max_(x in[a,b])|f^(')(x)|*|x-x_(i)|:i=0,1,dots,n}F_{1}(x)=\sup \left\{f\left(x_{i}\right)-\max _{x \in[a, b]}\left|f^{\prime}(x)\right| \cdot\left|x-x_{i}\right|: i=0,1, \ldots, n\right\}F1(x)=sup{f(xi)−maxx∈[a,b]|f′(x)|⋅|x−xi|:i=0,1,…,n}
and
F 2 ( x ) = inf { f ( x i ) + max x ∈ [ a , b ] | f ′ ( x ) | ⋅ | x − x i | : i = 0 , 1 , … , n } F 2 ( x ) = inf f x i + max x ∈ [ a , b ]   f ′ ( x ) ⋅ x − x i : i = 0 , 1 , … , n F_(2)(x)=i n f{f(x_(i))+max_(x in[a,b])|f^(')(x)|*|x-x_(i)|:i=0,1,dots,n}F_{2}(x)=\inf \left\{f\left(x_{i}\right)+\max _{x \in[a, b]}\left|f^{\prime}(x)\right| \cdot\left|x-x_{i}\right|: i=0,1, \ldots, n\right\}F2(x)=inf{f(xi)+maxx∈[a,b]|f′(x)|⋅|x−xi|:i=0,1,…,n}
These functions are in Lip [ a , b ] Lip [ a , b ] Lip[a,b]\operatorname{Lip}[a, b]Lip[a,b] but, in general, they do not belong to C 1 [ a , b ] C 1 [ a , b ] C^(1)[a,b]C^{1}[a, b]C1[a,b]. They are differentiable on ( a , b a , b a,ba, ba,b ) excepting (eventually) the points in Δ x Δ x Delta_(x)\Delta_{x}Δx and the points of the form
x = x i + x i − 1 2 + f i − 1 − f i 2 ‖ f ‖ L , x = x i + x i − 1 2 + f i − 1 − f i 2 ‖ f ‖ L x = x i + x i − 1 2 + f i − 1 − f i 2 ‖ f ‖ L , x = x i + x i − 1 2 + f i − 1 − f i 2 ‖ f ‖ L x=(x_(i)+x_(i-1))/(2)+(f_(i-1)-f_(i))/(2||f||_(L)),x=(x_(i)+x_(i-1))/(2)+(f_(i-1)-f_(i))/(2||f||_(L))x=\frac{x_{i}+x_{i-1}}{2}+\frac{f_{i-1}-f_{i}}{2\|f\|_{L}}, x=\frac{x_{i}+x_{i-1}}{2}+\frac{f_{i-1}-f_{i}}{2\|f\|_{L}}x=xi+xi−12+fi−1−fi2‖f‖L,x=xi+xi−12+fi−1−fi2‖f‖L
If f i − 1 < f i f i − 1 < f i f_(i-1) < f_(i)f_{i-1}<f_{i}fi−1<fi, then the functions s − F 1 s − F 1 s-F_(1)s-F_{1}s−F1 and s − F 2 s − F 2 s-F_(2)s-F_{2}s−F2 are continuously differentiable on every interval ( x i − 1 , x ― x i − 1 , x _ x_(i-1),x_x_{i-1}, \underline{x}xi−1,x― ), ( x ― , x ¯ x _ , x ¯ x_, bar(x)\underline{x}, \bar{x}x―,x¯ ), ( x ¯ , x i x ¯ , x i bar(x),x_(i)\bar{x}, x_{i}x¯,xi ). We have
s ′ ( x ) − f ′ ( x ) = M i − M i − 1 2 h i ( x − x i − 1 ) 2 + M i − 1 ⋅ ( x − x i − 1 ) + m i − 1 − f ′ ( x ) s ′ ( x ) − f ′ ( x ) = M i − M i − 1 2 h i x − x i − 1 2 + M i − 1 ⋅ x − x i − 1 + m i − 1 − f ′ ( x ) s^(')(x)-f^(')(x)=(M_(i)-M_(i-1))/(2h_(i))(x-x_(i-1))^(2)+M_(i-1)*(x-x_(i-1))+m_(i-1)-f^(')(x)s^{\prime}(x)-f^{\prime}(x)=\frac{M_{i}-M_{i-1}}{2 h_{i}}\left(x-x_{i-1}\right)^{2}+M_{i-1} \cdot\left(x-x_{i-1}\right)+m_{i-1}-f^{\prime}(x)s′(x)−f′(x)=Mi−Mi−12hi(x−xi−1)2+Mi−1⋅(x−xi−1)+mi−1−f′(x)
(25)
for all x ∈ [ x i − 1 , x i ] x ∈ x i − 1 , x i x in[x_(i-1),x_(i)]x \in\left[x_{i-1}, x_{i}\right]x∈[xi−1,xi].
Since
− ‖ f ‖ Σ ⩽ − f ′ ( x ) ⩽ ‖ f ‖ L , x ∈ [ x i − 1 , x i , ] − ‖ f ‖ Σ ⩽ − f ′ ( x ) ⩽ ‖ f ‖ L , x ∈ x i − 1 , x i , -||f||_(Sigma) <= -f^(')(x) <= ||f||_(L),quad x in[x_(i-1),x_(i),]-\|f\|_{\Sigma} \leqslant-f^{\prime}(x) \leqslant\|f\|_{L}, \quad x \in\left[x_{i-1}, x_{i},\right]−‖f‖Σ⩽−f′(x)⩽‖f‖L,x∈[xi−1,xi,]
it follows
s ′ ( x ) − ‖ f ‖ L ⩽ s ′ ( x ) − f ′ ( x ) ⩽ s ′ ( x ) + ‖ f ‖ L , x ∈ [ x i − 1 , x i ] , s ′ ( x ) − ‖ f ‖ L ⩽ s ′ ( x ) − f ′ ( x ) ⩽ s ′ ( x ) + ‖ f ‖ L , x ∈ x i − 1 , x i , s^(')(x)-||f||_(L) <= s^(')(x)-f^(')(x) <= s^(')(x)+||f||_(L),x in[x_(i-1),x_(i)],s^{\prime}(x)-\|f\|_{L} \leqslant s^{\prime}(x)-f^{\prime}(x) \leqslant s^{\prime}(x)+\|f\|_{L}, x \in\left[x_{i-1}, x_{i}\right],s′(x)−‖f‖L⩽s′(x)−f′(x)⩽s′(x)+‖f‖L,x∈[xi−1,xi],
so that
∣ s ′ ( x ) − f ′ ( x ) ⩽ max { ‖ s ′ + ‖ f ‖ L ‖ , ‖ s ′ − ‖ f ‖ L ‖ } ∣ s ′ ( x ) − f ′ ( x ) ⩽ max s ′ + f L , s ′ − f L } ∣s^(')(x)-f^(')(x) <= max{||s^(')+||f||_(L)||,||s^(')-||f||_(L)||_(}):}\mid s^{\prime}(x)-f^{\prime}(x) \leqslant \max \left\{\left\|s^{\prime}+\right\| f\left\|_{L}\right\|,\left\|s^{\prime}-\right\| f\left\|_{L}\right\|_{\}}\right.∣s′(x)−f′(x)⩽max{‖s′+‖f‖L‖,‖s′−‖f‖L‖}
for every x ∈ [ x i − 1 , x i ] x ∈ x i − 1 , x i x in[x_(i-1),x_(i)]x \in\left[x_{i-1}, x_{i}\right]x∈[xi−1,xi], where the norms occurring in theright member of the above inequality are calculated on the interval [ x i − 1 , , x i ] , i = 1 , 2 , … n x i − 1 , , x i , i = 1 , 2 , … n [x_(i-1,),x_(i)],i=1,2,dots n\left[x_{i-1,}, x_{i}\right], i=1,2, \ldots n[xi−1,,xi],i=1,2,…n.
Denoting
(26) c i = ‖ s ′ + ‖ f ‖ L ‖ d i = ‖ s ′ − ‖ f ‖ L ‖ (26) c i = s ′ + f L d i = s ′ − f L {:[(26)c_(i)=||s^(')+||f||_(L)||],[d_(i)=||s^(')-||f||_(L)||]:}\begin{align*} c_{i} & =\left\|s^{\prime}+\right\| f\left\|_{L}\right\| \tag{26}\\ d_{i} & =\left\|s^{\prime}-\right\| f\left\|_{L}\right\| \end{align*}(26)ci=‖s′+‖f‖L‖di=‖s′−‖f‖L‖
where the norms are again calculated on [ x i − 1 , x i ] , i = 1 , 2 , … , n x i − 1 , x i , i = 1 , 2 , … , n [x_(i-1),x_(i)],i=1,2,dots,n\left[x_{i-1}, x_{i}\right], i=1,2, \ldots, n[xi−1,xi],i=1,2,…,n, we find that the inequalities
(27) min i = 1 , n ― { c i , d i } ⩽ ‖ s ′ − f ′ ‖ ⩽ max i = 1 , n ― { c i , d i } (27) min i = 1 , n ¯   c i , d i ⩽ s ′ − f ′ ⩽ max i = 1 , n ¯   c i , d i {:(27)min_(i= bar(1,n)){c_(i),d_(i)} <= ||s^(')-f^(')|| <= max_(i= bar(1,n)){c_(i),d_(i)}:}\begin{equation*} \min _{i=\overline{1, n}}\left\{c_{i}, d_{i}\right\} \leqslant\left\|s^{\prime}-f^{\prime}\right\| \leqslant \max _{i=\overline{1, n}}\left\{c_{i}, d_{i}\right\} \tag{27} \end{equation*}(27)mini=1,n―{ci,di}⩽‖s′−f′‖⩽maxi=1,n―{ci,di}
hold true on the interval [ a , b ] [ a , b ] [a,b][a, b][a,b].
Denoting by
(28) x 0 = x i M i − 2 x i M i − 1 + x i − 1 M i − 1 M i − M i − 1 (28) x 0 = x i M i − 2 x i M i − 1 + x i − 1 M i − 1 M i − M i − 1 {:(28)x_(0)=(x_(i)M_(i)-2x_(i)M_(i-1)+x_(i-1)M_(i-1))/(M_(i)-M_(i-1)):}\begin{equation*} x_{0}=\frac{x_{i} M_{i}-2 x_{i} M_{i-1}+x_{i-1} M_{i-1}}{M_{i}-M_{i-1}} \tag{28} \end{equation*}(28)x0=xiMi−2xiMi−1+xi−1Mi−1Mi−Mi−1
the root of the equation s ′ ′ ( x ) = 0 s ′ ′ ( x ) = 0 s^('')(x)=0s^{\prime \prime}(x)=0s′′(x)=0 in the interval [ x i − 1 , x i x i − 1 , x i x_(i-1),x_(i)x_{i-1}, x_{i}xi−1,xi ] one gets
(29) c i = { max { | s ′ ( x 0 ) + ‖ f ‖ L | , | m i − 1 + ‖ f ‖ L | , | h i 2 ( M i + M i − 1 ) + m i − 1 + ‖ f ‖ L | } , if x 0 ∈ ( x i − 1 , x i ) max { | m i − 1 + ‖ f ‖ L | , | h i 2 ( M i + M i − 1 ) + m i − 1 + ‖ f ‖ L | } if x 0 ∉ [ x i − 1 , x i ] c i = max s ′ x 0 + ‖ f ‖ L , m i − 1 + ‖ f ‖ L , h i 2 M i + M i − 1 + m i − 1 + ‖ f ‖ L ,  if  x 0 ∈ x i − 1 , x i max m i − 1 + ‖ f ‖ L , h i 2 M i + M i − 1 + m i − 1 + ‖ f ‖ L  if  x 0 ∉ x i − 1 , x i c_(i)={[max{|s^(')(x_(0))+||f||_(L)|,|m_(i-1)+||f||_(L)|,|(h_(i))/(2)(M_(i)+M_(i-1))+m_(i-1)+||f||_(L)|}","],[" if "x_(0)in(x_(i-1),x_(i))],[max{|m_(i-1)+||f||_(L)|,|(h_(i))/(2)(M_(i)+M_(i-1))+m_(i-1)+||f||_(L)|}],[" if "x_(0)!in[x_(i-1),x_(i)]]:}c_{i}=\left\{\begin{array}{r}\max \left\{\left|s^{\prime}\left(x_{0}\right)+\|f\|_{L}\right|,\left|m_{i-1}+\|f\|_{L}\right|,\left|\frac{h_{i}}{2}\left(M_{i}+M_{i-1}\right)+m_{i-1}+\|f\|_{L}\right|\right\}, \\ \text { if } x_{0} \in\left(x_{i-1}, x_{i}\right) \\ \max \left\{\left|m_{i-1}+\|f\|_{L}\right|,\left|\frac{h_{i}}{2}\left(M_{i}+M_{i-1}\right)+m_{i-1}+\|f\|_{L}\right|\right\} \\ \text { if } x_{0} \notin\left[x_{i-1}, x_{i}\right]\end{array}\right.ci={max{|s′(x0)+‖f‖L|,|mi−1+‖f‖L|,|hi2(Mi+Mi−1)+mi−1+‖f‖L|}, if x0∈(xi−1,xi)max{|mi−1+‖f‖L|,|hi2(Mi+Mi−1)+mi−1+‖f‖L|} if x0∉[xi−1,xi] :
and, respectively,
(30) d i = { max { | s ′ ( x 0 ) − ‖ f ‖ L | , | m i − 1 − ‖ f ‖ L | , | h i 2 ( M i + M i − 1 ) + m i − 1 − ‖ f ‖ L | } if x 0 ∈ ( x i − 1 , x i ) max { | m i − 1 − ‖ f ‖ L | , | h i 2 ( M i + M i − 1 ) + m i − 1 − ‖ f ‖ L | } ∴ if x 0 ∉ [ x i − 1 , x i ] d i = max s ′ x 0 − ‖ f ‖ L , m i − 1 − ‖ f ‖ L , h i 2 M i + M i − 1 + m i − 1 − ‖ f ‖ L  if  x 0 ∈ x i − 1 , x i max m i − 1 − ‖ f ‖ L , h i 2 M i + M i − 1 + m i − 1 − ‖ f ‖ L ∴  if  x 0 ∉ x i − 1 , x i d_(i)={[max{|s^(')(x_(0))-||f||_(L)|,|m_(i-1)-||f||_(L)|,|(h_(i))/(2)(M_(i)+M_(i-1))+m_(i-1)-||f||_(L)|}],[" if "x_(0)in(x_(i-1),x_(i))],[max{|m_(i-1)-||f||_(L)|,|(h_(i))/(2)(M_(i)+M_(i-1))+m_(i-1)-||f||_(L)|}],[:." if "x_(0)!in[x_(i-1),x_(i)]]:}d_{i}=\left\{\begin{array}{c}\max \left\{\left|s^{\prime}\left(x_{0}\right)-\|f\|_{L}\right|,\left|m_{i-1}-\|f\|_{L}\right|,\left|\frac{h_{i}}{2}\left(\boldsymbol{M}_{i}+\boldsymbol{M}_{i-1}\right)+m_{i-1}-\|f\|_{L}\right|\right\} \\ \text { if } x_{0} \in\left(x_{i-1}, x_{i}\right) \\ \max \left\{\left|m_{i-1}-\|f\|_{L}\right|,\left|\frac{h_{i}}{2}\left(\boldsymbol{M}_{i}+\boldsymbol{M}_{i-1}\right)+m_{i-1}-\|f\|_{L}\right|\right\} \\ \therefore \text { if } x_{0} \notin\left[x_{i-1}, x_{i}\right]\end{array}\right.di={max{|s′(x0)−‖f‖L|,|mi−1−‖f‖L|,|hi2(Mi+Mi−1)+mi−1−‖f‖L|} if x0∈(xi−1,xi)max{|mi−1−‖f‖L|,|hi2(Mi+Mi−1)+mi−1−‖f‖L|}∴ if x0∉[xi−1,xi]

REFERENCES

  1. C. I a n c u, On the cubic spline of inlerpolation, Seminar of Functional Analysis and Numerical Methods, Preprint N. 4 (1981), 52-71.
  2. V. L. Miroshnichenko, On the error of approximation by cubic interpolation splines (Russian), Metody spline - funkji, 93 (1982), 3-29.
  3. V. L. Miroshnichenko, On the error of approximation by cubbic interpolation splines II (Russian), Metody spline-funckji v chisl. analize 98 (1983), 51-66.
  4. E. J. McShane, Extension of range of functions, Bull. Amer. Math. Soc., 40 (1934), 837-842.
  5. C. Mustă ţa, Best approximation and unique extension of Lipschilz functions, Journal of Approx. Theory 19, 3 (1977), 222-230.
  6. C. M u s t ă t a, On the extension problem with prescribed norm, Seminar of Functional Analysis and Numerical Methods, Preprint N. 4 (1981), 93-99
1987

Related Posts