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.
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
ERROR ESTIMATION IN THE APPROXTMATION OF FUNCTIONS BY INTERPOLATION CUBIC SPLINES
- In this Note we give estimations for the error of approximation of a continuous function f:[a,b]rarr Rf:[a, b] \rightarrow R by interpolation cubic splines with respect to a given division Delta_(x)\Delta_{x} of the interval [a,b][a, b].
Let f:[a,b]rarr Rf:[a, b] \rightarrow R be a function and let
{:(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*}
be a division of the interval [a,b][a, b].
Put
Put
{:(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*}
and let us denote by Sp(3,Delta_(x))\operatorname{Sp}\left(3, \Delta_{x}\right) the set of all cubic spline ss corresponding to the partition Delta_(x)\Delta_{x} and having the properties:
(i) the restriction of ss to every interval [x_(i-1),x_(i)]\left[x_{i-1}, x_{i}\right] is a polynomial of degree at most 3 , for i=1,2,dots,ni=1,2, \ldots, n;
(ii) s inC^(2)[a,b]s \in C^{2}[a, b], i.e. ss is continuously two times differentiable on [a,b][a, b];
(iii) s(x_(i))=f_(i),i=0,1,2,dots,ns\left(x_{i}\right)=f_{i}, i=0,1,2, \ldots, n i.e. ss interpolates the function ff on the knots in Delta_(x)\Delta_{x}.
(i) the restriction of ss to every interval [x_(i-1),x_(i)]\left[x_{i-1}, x_{i}\right] is a polynomial of degree at most 3 , for i=1,2,dots,ni=1,2, \ldots, n;
(ii) s inC^(2)[a,b]s \in C^{2}[a, b], i.e. ss is continuously two times differentiable on [a,b][a, b];
(iii) s(x_(i))=f_(i),i=0,1,2,dots,ns\left(x_{i}\right)=f_{i}, i=0,1,2, \ldots, n i.e. ss interpolates the function ff on the knots in Delta_(x)\Delta_{x}.
Put also
{:(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}
For ss in Sp(3,Delta_(x))\mathrm{Sp}\left(3, \Delta_{x}\right), the restriction of the second derivative s^('')s^{\prime \prime} of ss to the interval [x_(i-1),x_(i)]\left[x_{i-1}, x_{i}\right] 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 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*}
Taking into account the conditions
{:[(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*}
the relation (4) gives :
{:[(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*}
for x in[x_(i-1),x_(i)]x \in\left[x_{i-1}, x_{i}\right] and i=1,2,dots,ni=1,2, \ldots, n.
Proposition. Every function s inSp(3,Delta_(x))s \in \mathrm{Sp}\left(3, \Delta_{x}\right), given by formula (6), is uniquely determined by the conditions:
(i) s(x_(i))=f_(i),i=1,2,dots,ns\left(x_{i}\right)=f_{i}, i=1,2, \ldots, n
(ii) s^(')(x_(i))=m_(i),i=1,2,dots,ns^{\prime}\left(x_{i}\right)=m_{i}, i=1,2, \ldots, n
(iii) m_(0)=p,M_(0)=q,p,q-m_{0}=p, M_{0}=q, p, q- given real numbers.
Proposition. Every function s inSp(3,Delta_(x))s \in \mathrm{Sp}\left(3, \Delta_{x}\right), given by formula (6), is uniquely determined by the conditions:
(i) s(x_(i))=f_(i),i=1,2,dots,ns\left(x_{i}\right)=f_{i}, i=1,2, \ldots, n
(ii) s^(')(x_(i))=m_(i),i=1,2,dots,ns^{\prime}\left(x_{i}\right)=m_{i}, i=1,2, \ldots, n
(iii) m_(0)=p,M_(0)=q,p,q-m_{0}=p, M_{0}=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)-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*}
By condition (iii) system (7) is compatible and has a unique solution 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}. System (7) can be recursively solved starting from the condition (iii) m_(0)=p,M_(0)=qm_{0}=p, M_{0}=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\| and ||s^(')-f^(')||\left\|s^{\prime}-f^{\prime}\right\| for ff satisfying some sufficiently restrictive conditions.
(a) In the following we shall evaluate the uniform norm
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\| and ||s^(')-f^(')||\left\|s^{\prime}-f^{\prime}\right\| for ff satisfying some sufficiently restrictive conditions.
(a) In the following we shall evaluate the uniform norm
{:(8)||s-f||",":}\begin{equation*}
\|s-f\|, \tag{8}
\end{equation*}
supposing that ff is a Lipschitz function on [a,b][a, b], i.e. there exists a number K >= 0K \geqslant 0 (called a Lipschitz constant) such that
{:(9)|f(x)-f(y)| <= K|x-y|",":}\begin{equation*}
|f(x)-f(y)| \leqslant K|x-y|, \tag{9}
\end{equation*}
for all x,y in[a,b]x, y \in[a, b].
The number
The number
{:(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*}
is the smallest Lipschitz constant for ff and is called the Lipschitz norm of ff on the interval [a,b][a, b].
The space of all Lipschitz function on [a,b][a, b] is denoted by Lip [a,b][a, b]. The Lipschitz norm of the restriction of ff to the division Delta_(x)\Delta_{x} is given by
{:(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*}
where [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) is the divided difference of the function ff on the knots x_(i-1),x_(i)x_{i-1}, x_{i}.
In the sequel we shall need the following extension result of McShane [4]: Let XX be a matric space, YY a subset of XX and let f:Yrarr Rf: \mathrm{Y} \rightarrow R be a Lipschitz function. Then there exists a Lipschitz function F:X rarr RF: X \rightarrow R such that F|_(Y)=f\left.F\right|_{Y}=f and ||F||_(L)=||f||_(L)\|F\|_{L}=\|f\|_{L}. In [6] it was proved that for every f in Lip bar(Y)f \in \operatorname{Lip} \bar{Y} and every K >= ||f||_(L)K \geqslant\|f\|_{L} there exists an extension F:X rarr RF: X \rightarrow R of ff. such that ||F||_(L)=K\|F\|_{L}=K.
By this result, if f in Lip[a,b]f \in \operatorname{Lip}[a, b], then the restriction f|_(Delta x)\left.f\right|_{\Delta x} of ff to Delta_(x)\Delta_{x} has at least one extension F in Lip[a,b]F \in \operatorname{Lip}[a, b] such that ||F||_(L)=||f||_(L)\|F\|_{L}=\|f\|_{L}. It is obvious that such an extension is ff itself, but the following two functions
{:[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*}
are also extensions of ff with norm ||f||_(L)\|f\|_{L}, i.e.
{:(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*}
and every extension FF of f|_(Delta x)\left.f\right|_{\Delta x} such that ||F||_(L)=||f||_(L)\|F\|_{L}=\|f\|_{L} verifies F_(1) <= F^(') <= F_(2)F_{1} \leqslant F^{\prime} \leqslant F_{2} (see [5]). In particular
{:(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*}
for all x in[a,b]x \in[a, b].
From (14) it follows
From (14) it follows
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]
so that
{:(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*}
Taking into account the fact that the functions F_(1)F_{1} and F_(2)F_{2} given by (12) are piecewise linear, the calculation of the norms ||s-F_(1)||\left\|s-F_{1}\right\| and ||s-F_(2)||\left\|s-F_{2}\right\| reduces to the calculation of the norm of third degree polynomials on compact subintervals of [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)])||]:}\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*}
for i=1,2,dots,ni=1,2, \ldots, n, then
||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\}
and
||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\}
In order to calculate the numbers a_(i),b_(i),i=1,2,dots,na_{i}, b_{i}, i=1,2, \ldots, n, we have to distinct three cases :
Case 1.: f_(i-1) < f_(i)f_{i-1}<f_{i}.
In this case, for x in[x_(i-1),x_(i)]x \in\left[x_{i-1}, x_{i}\right] we have
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.
where
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}}
and
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}
where
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}}
Since
x_(i-1) < x_ < bar(x) < x_(i)x_{i-1}<\underline{x}<\bar{x}<x_{i}
we have
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\}
and
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\}
Case 2. f_(i-1) > f_(i)f_{i-1}>f_{i}.
In this case x_(i-1) < bar(x) < x_ < x_(i)x_{i-1}<\bar{x}<\underline{x}<x_{i} and therefore the norms of s-F_(1)s-\boldsymbol{F}_{1} and s-F_(2)s-F_{2} are calculated on the intervals [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].
In this case x_(i-1) < bar(x) < x_ < x_(i)x_{i-1}<\bar{x}<\underline{x}<x_{i} and therefore the norms of s-F_(1)s-\boldsymbol{F}_{1} and s-F_(2)s-F_{2} are calculated on the intervals [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].
Case 3. f_(i-1)=f_(i)f_{i-1}=f_{i}.
In this case x_= bar(x)=(x_(i-1)+x_(i))//2\underline{x}=\bar{x}=\left(x_{i-1}+x_{i}\right) / 2 and the norms of s-F_(1)s-F_{1} and s-F_(2)s-F_{2} are calculated on the intervals [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].
In this case x_= bar(x)=(x_(i-1)+x_(i))//2\underline{x}=\bar{x}=\left(x_{i-1}+x_{i}\right) / 2 and the norms of s-F_(1)s-F_{1} and s-F_(2)s-F_{2} are calculated on the intervals [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].
In concrete situations, the numbers a_(i)a_{i} and b_(i)b_{i} 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} and b_(i)b_{i}, 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} and ||f||_(L)\|f\|_{L}.
Concerning the exactity of the evaluations (15) we show that in the set of all real valued Lipschitz functions gg on [a,b][a, b] with norm ||g||_(L)==||f||_(L)\|g\|_{L}= =\|f\|_{L} and such that g(x_(i))=f_(i),i=0,1,2,dots,ng\left(x_{i}\right)=f_{i}, i=0,1,2, \ldots, n, there exists two functions bar(f)\bar{f} and f_\underline{f} such that the evaluations (15) are the best possible in this set.
Let
{:[(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*}
Obviously, the functions F_(1)F_{1} and F_(2)F_{2} defined by (12) belong to E(f|_(Delta_(g));[a,b])E\left(\left.f\right|_{\Delta_{g}} ;[a, b]\right) and if g in E(f|_(Delta_(x));[a,b])g \in E\left(\left.f\right|_{\Delta_{x}} ;[a, b]\right) then
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}] .
For every interval [x_(i-1),x_(i)],i=1,2,dots,n\left[x_{i-1}, x_{i}\right], i=1,2, \ldots, n, let us define the function bar(f)_(i)\bar{f}_{i} in the following way:
{:(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.
Let the function bar(f):[a,b]rarr R\bar{f}:[a, b] \rightarrow R be defined by
{:(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*}
Then bar(f)in Lip[a,b]\bar{f} \in \operatorname{Lip}[\boldsymbol{a}, \boldsymbol{b}] and, as can be easily seen from the definition of the function vec(f)\vec{f},
{:(20)|| bar(f)-s|| <= ||g-s||:}\begin{equation*}
\|\bar{f}-s\| \leqslant\|g-s\| \tag{20}
\end{equation*}
for every function g inE(f∣Delta_(x);[a,b])g \in \mathbb{E}\left(f \mid \Delta_{x} ;[a, b]\right).
Similarly, the function f_:[a,b]rarr R\underline{f}:[a, b] \rightarrow R defined by
Similarly, the function f_:[a,b]rarr R\underline{f}:[a, b] \rightarrow R defined by
{:(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*}
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)}]:}:}\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.
for i=1,2,dots,ni=1,2, \ldots, n, verifies the inequality
{:(23)||g-s|| <= ||f-s||:}\begin{equation*}
\|g-s\| \leqslant\|f-s\| \tag{23}
\end{equation*}
for every function g in E(f∣Delta_(x);[a,b])g \in E\left(f \mid \Delta_{x} ;[a, b]\right).
(b) Evaluation of the norm ||f^(')-s^(')||\left\|f^{\prime}-s^{\prime}\right\|.
(b) Evaluation of the norm ||f^(')-s^(')||\left\|f^{\prime}-s^{\prime}\right\|.
In the following we shall suppose f inC^(1)[a,b]f \in C^{1}[a, b]. In this case f in Lip[a,b]f \in \operatorname{Lip}[a, b] and
{:(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*}
The formulae (12) become
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\}
and
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\}
These functions are in Lip[a,b]\operatorname{Lip}[a, b] but, in general, they do not belong to C^(1)[a,b]C^{1}[a, b]. They are differentiable on ( a,ba, b ) excepting (eventually) the points in Delta_(x)\Delta_{x} 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=\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}}
If f_(i-1) < f_(i)f_{i-1}<f_{i}, then the functions s-F_(1)s-F_{1} and s-F_(2)s-F_{2} are continuously differentiable on every interval ( x_(i-1),x_x_{i-1}, \underline{x} ), ( x_, bar(x)\underline{x}, \bar{x} ), ( bar(x),x_(i)\bar{x}, x_{i} ). We have
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)
(25)
for all x in[x_(i-1),x_(i)]x \in\left[x_{i-1}, x_{i}\right].
Since
for all x in[x_(i-1),x_(i)]x \in\left[x_{i-1}, x_{i}\right].
Since
-||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]
it follows
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],
so that
∣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.
for every x in[x_(i-1),x_(i)]x \in\left[x_{i-1}, x_{i}\right], where the norms occurring in theright member of the above inequality are calculated on the interval [x_(i-1,),x_(i)],i=1,2,dots n\left[x_{i-1,}, x_{i}\right], i=1,2, \ldots n.
Denoting
{:[(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*}
where the norms are again calculated on [x_(i-1),x_(i)],i=1,2,dots,n\left[x_{i-1}, x_{i}\right], i=1,2, \ldots, n, we find that the inequalities
{:(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*}
hold true on the interval [a,b][a, b].
Denoting by
Denoting by
{:(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*}
the root of the equation s^('')(x)=0s^{\prime \prime}(x)=0 in the interval [ x_(i-1),x_(i)x_{i-1}, x_{i} ] 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)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. :
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)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.
(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)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. :
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)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.
REFERENCES
- 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.
- V. L. Miroshnichenko, On the error of approximation by cubic interpolation splines (Russian), Metody spline - funkji, 93 (1982), 3-29.
- V. L. Miroshnichenko, On the error of approximation by cubbic interpolation splines II (Russian), Metody spline-funckji v chisl. analize 98 (1983), 51-66.
- E. J. McShane, Extension of range of functions, Bull. Amer. Math. Soc., 40 (1934), 837-842.
- C. Mustă ţa, Best approximation and unique extension of Lipschilz functions, Journal of Approx. Theory 19, 3 (1977), 222-230.
- 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