On the Preservation of the Shape of Convexity of Functions by Interpolation

Tiberiu Popoviciu
Approximation Theory, convexity, interpolation, Mathematical Analysis

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Tiberiu Popoviciu
Institutul de Calcul

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T. Popoviciu, Sur la conservation de l’allure de convexité des fonctions par interpolation, An. Şti. Univ. “Al. I. Cuza” Iaşi Secţ. I a Mat. (N.S.) 14 (1968), pp. 7-14 (in French). Communication présentée au Congrès International des Mathématiciens, Moscou, 16-26 août 1966.

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1968 d -Popoviciu- An. Sti. Univ. Al. I. Cuza Iasi – Sur la conservation de l’allure de convexite des fonctions par interpolation

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published by the “Alexandru Ioan Cuza” University of Iaşi

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1968 d -Popoviciu- An. Sci. Univ. Al. I. Cuza Iasi - On the conservation of the convexity of
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ON THE PRESERVATION OF THE SHAPE OF CONVEXITY OF FUNCTIONS BY INTERPOLATION

ABOUTTIBERIU POPOVICIUin ClujPaper presented at the International Congress of Mathemoticians, Moscow, August 16-26, 1966

  1. Consider an operator F [ f ∣ x ] F [ f ∣ x ] F[f∣x]F[f \mid x]F[f∣x]which transforms the function f f fff, real of a real variable defined on the set AND AND ANDANDANDof the real axis, into a function of x x xxxreal defined on a set I I IIIof the real axis.
There is no need to specify the nature of the operator at this point. F [ f ∣ x ] F [ f ∣ x ] F[f∣x]F[f \mid x]F[f∣x], its definition set and the structures of sets AND , I AND , I E,IE, IAND,I. In the sequel AND AND ANDANDANDwill generally be an interval and the operator F [ f ∣ x ] F [ f ∣ x ] F[f∣x]F[f \mid x]F[f∣x]a particular linear (additive and homogeneous) operator. The function F [ f ∣ x ] F [ f ∣ x ] F[f∣x]F[f \mid x]F[f∣x]of x x xxxwill always be a polynomial, so I I IIIcan be any set of the real axis.
Definition. We say that the operator F [ t ∣ x ] F [ t ∣ x ] F[t∣x]F[t \mid x]F[t∣x]preserves (on I) the nonconcavity of order n n nnn(of the function f f fff) if the function F [ f ∣ x ] F [ f ∣ x ] F[f∣x]F[f \mid x]F[f∣x]of x x xxxis nonconcave of order n n nnn(on I) for any function f f fffnon-concave order n n nnn(on AND AND ANDANDAND ).
An analogous definition can be given for the conservation of convexity, non-convexity and concavity of order n n nnnof the function f f fffby the operator F [ f ∣ x ] F [ f ∣ x ] F[f∣x]F[f \mid x]F[f∣x]. Note that if the operator F [ f ∣ x ] F [ f ∣ x ] F[f∣x]F[f \mid x]F[f∣x]preserves the non-concavity (convexity) of order n n nnn, the operator − F [ f ∣ x ] − F [ f ∣ x ] -F[f∣x]-F[f \mid x]−F[f∣x]preserves the non-convexity (concavity) of order n n nnnof f f fffon the same set I I IIIand vice versa. It follows that if a linear operator F [ f ∣ x ] F [ f ∣ x ] F[f∣x]F[f \mid x]F[f∣x]preserves the non-concavity (convexity) of order n n nnn, it also preserves the nonconvexity (concavity) of order n n nnnof f f fffand vice versa on the same set I I III.
2. We assume that we know the definitions and properties of non-concave, convex, non-convex and concave functions of order n n nnnThese definitions are obtained by the conservation of the sign of the
divided differences of order n + 1 n + 1 n+1n+1n+1of the function. For these properties one can see, for example, my previous work on higher-order convex functions, the list of which is unnecessary to reproduce here.
We designate by [ x 1 , x 2 , … , x m ; f ] x 1 , x 2 , … , x m ; f [x_(1),x_(2),dots,x_(m);f]\left[x_{1}, x_{2}, \ldots, x_{m} ; f\right][x1,x2,…,xm;f]the divided difference of order m − 1 m − 1 m-1m-1m−1and by L ( x 1 , x 2 , … , x m ; f ∣ x ) L x 1 , x 2 , … , x m ; f ∣ x L(x_(1),x_(2),dots,x_(m);f∣x)L\left(x_{1}, x_{2}, \ldots, x_{m} ; f \mid x\right)L(x1,x2,…,xm;f∣x)the Lagrange-Hermite polynomial of the function f f fffon the knots x a , a = 1 , 2 , … , m x a , a = 1 , 2 , … , m x_(alpha),alpha=1,2,dots,mx_{\alpha}, \alpha=1,2, \ldots, mxa,a=1,2,…,m. The knots x a x a x_(alpha)x_{\alpha}xaare not necessarily distinct, but the divided difference and the Lagrange-Hermite polynomial have well-known definitions apart from the function values f f fffon the nodes, also the values ​​of a certain number of successive derivatives of f f fffon the nodes which have a multiplicity order greater than 1.
3. Let k 1 , k 2 , … , k p k 1 , k 2 , … , k p k_(1),k_(2),dots,k_(p)k_{1}, k_{2}, \ldots, k_{p}k1,k2,…,kpnatural numbers ( p ≧ 1 p ≧ 1 p >= 1p \geqq 1p≧1), of sum equal to m m mmmand either k = max ( k 1 , k 2 , … , k p ) k = max k 1 , k 2 , … , k p k=max(k_(1),k_(2),dots,k_(p))k=\max \left(k_{1}, k_{2}, \ldots, k_{p}\right)k=max(k1,k2,…,kp). Consider the Lagrange-Hermite polynomial L ( x ) = L ( x 1 , x 2 , … , x m ; f ∣ x ) L ( x ) = L x 1 , x 2 , … , x m ; f ∣ x L(x)=L(x_(1),x_(2),dots,x_(m);f∣x)L(x)=L\left(x_{1}, x_{2}, \ldots, x_{m} ; f \mid x\right)L(x)=L(x1,x2,…,xm;f∣x)on the m m mmmknots x 1 , x 2 , … , x m x 1 , x 2 , … , x m x_(1),x_(2),dots,x_(m)x_{1}, x_{2}, \ldots, x_{m}x1,x2,…,xmof which k ^ a k ^ a hat(k)_(alpha)\hat{k}_{\alpha}k^acoincide with and a , a = 1 , 2 , … , p and a , a = 1 , 2 , … , p y_(alpha),alpha=1,2,dots,py_{\alpha}, \alpha=1,2, \ldots, panda,a=1,2,…,p, THE and 1 , and 2 , … , and p and 1 , and 2 , … , and p y_(1),y_(2),dots,y_(p)y_{1}, y_{2}, \ldots, y_{p}and1,and2,…,andpbeing p p pppdistinct points of the real axis.
We know that the polynomial L ( x ) L ( x ) L(x)L(x)L(x)is completely characterized by the fact that it is of degree m − 1 m − 1 m-1m-1m−1and that it verifies the equalities
(1) L ( c ) ( and a ) = f ( c ) ( and a ) , c = 0 , 1 , … , k a − 1 , a = 1 , 2 , … , p L ( c ) and a = f ( c ) and a , c = 0 , 1 , … , k a − 1 , a = 1 , 2 , … , p quadL^((gamma))(y_(alpha))=f^((gamma))(y_(alpha)),quad gamma=0,1,dots,k_(alpha)-1,quad alpha=1,2,dots,p\quad L^{(\gamma)}\left(y_{\alpha}\right)=f^{(\gamma)}\left(y_{\alpha}\right), \quad \gamma=0,1, \ldots, k_{\alpha}-1, \quad \alpha=1,2, \ldots, pL(c)(anda)=f(c)(anda),c=0,1,…,ka−1,a=1,2,…,p
where the accents signify successive derivations ( f ( 0 ) ( x ) = f ( x ) ) f ( 0 ) ( x ) = f ( x ) (f^((0))(x)=f(x))\left(f^{(0)}(x)=f(x)\right)(f(0)(x)=f(x)).
We have
(2) L ( x 1 , x 2 , … , x m ; f ∣ x ) = ∑ β = 0 k − 1 H β [ f ∣ x ] (2) L x 1 , x 2 , … , x m ; f ∣ x = ∑ β = 0 k − 1   H β [ f ∣ x ] {:(2)L(x_(1),x_(2),dots,x_(m);f∣x)=sum_(beta=0)^(k-1)H_(beta)[f∣x]:}\begin{equation*} L\left(x_{1}, x_{2}, \ldots, x_{m} ; f \mid x\right)=\sum_{\beta=0}^{k-1} H_{\beta}[f \mid x] \tag{2} \end{equation*}(2)L(x1,x2,…,xm;f∣x)=∑b=0k−1Hb[f∣x]
H β [ f ∣ x ] = ∑ ( β ) f ( β ) ( y α ) h β , α ( x ) , β = 0 , 1 , … , k − 1 H β [ f ∣ x ] = ∑ ( β )   f ( β ) y α h β , α ( x ) , β = 0 , 1 , … , k − 1 H_(beta)[f∣x]=sum(beta)f^((beta))(y_(alpha))h_(beta,alpha)(x),quad beta=0,1,dots,k-1H_{\beta}[f \mid x]=\sum^{(\beta)} f^{(\beta)}\left(y_{\alpha}\right) h_{\beta, \alpha}(x), \quad \beta=0,1, \ldots, k-1Hb[f∣x]=∑(b)f(b)(anda)hb,a(x),b=0,1,…,k−1, summation ∑ ( β ) ∑ ( β )   sum(beta)\sum^{(\beta)}∑(b)being extended to all values ​​of α α alpha\alphaafor which k α ≥ β + 1 k α ≥ β + 1 k_(alpha) >= beta+1k_{\alpha} \geq \beta+1ka≥b+1. The polynomial h β , α ( x ) h β , α ( x ) h_(beta,alpha)(x)h_{\beta, \alpha}(x)hb,a(x)is of degree m − 1 m − 1 m-1m-1m−1, is equal to L ( x 1 , x 2 , … , x m ; f ∣ x ) L x 1 , x 2 , … , x m ; f ∣ x L(x_(1),x_(2),dots,x_(m);f∣x)L\left(x_{1}, x_{2}, \ldots, x_{m} ; f \mid x\right)L(x1,x2,…,xm;f∣x)for a function f f fffsuitably chosen and verifies the equalities
(4) { h β , α ( γ ) ( y δ ) = 0 h β , α ( β ) ( y α ) = 1 . { γ = 0 , 1 , … , k δ − 1 , δ = 1 , 2 , … , α − 1 , α + 1 , … , p γ = 0 , 1 , … , β − 1 , β + 1 , … , h α − 1 , δ = α (4) h β , α ( γ ) ( y δ ) = 0 h β , α ( β ) ( y α ) = 1 . γ = 0 , 1 , … , k δ − 1 , δ = 1 , 2 , … , α − 1 , α + 1 , … , p γ = 0 , 1 , … , β − 1 , β + 1 , … , h α − 1 , δ = α {:(4){[{:h_(beta,alpha)^((gamma))(y_(delta))=0:}],[{:h_(beta,alpha)^((beta))(y_(alpha))=1.:}]quad{[gamma=0","1","dots","k_(delta)-1","quad delta=1","2","dots","alpha-1","alpha+1","dots","p],[gamma=0","1","dots","beta-1","quad beta+1","dots","h_(alpha)-1","quad delta=alpha]:}:}\left\{\begin{array} { l } { h _ { \beta , \alpha } ^ { ( \gamma ) } ( y _ { \delta } ) = 0 } \tag{4}\\ { h _ { \beta , \alpha } ^ { ( \beta ) } ( y _ { \alpha } ) = 1 . } \end{array} \quad \left\{\begin{array}{l} \gamma=0,1, \ldots, k_{\delta}-1, \quad \delta=1,2, \ldots, \alpha-1, \alpha+1, \ldots, p \\ \gamma=0,1, \ldots, \beta-1, \quad \beta+1, \ldots, h_{\alpha}-1, \quad \delta=\alpha \end{array}\right.\right.(4){hb,a(c)(andd)=0hb,a(b)(anda)=1.{c=0,1,…,kd−1,d=1,2,…,a−1,a+1,…,pc=0,1,…,b−1,b+1,…,ha−1,d=a
  1. Now let's ask
(5) F γ [ f ∣ x ] = ∑ β = 0 γ H β [ f ∣ x ] , γ = 0 , 1 , … , k − 1 . (5) F γ [ f ∣ x ] = ∑ β = 0 γ   H β [ f ∣ x ] , γ = 0 , 1 , … , k − 1 . {:(5)F_(gamma)[f∣x]=sum_(beta=0)^(gamma)H_(beta)[f∣x]","quad gamma=0","1","dots","k-1.:}\begin{equation*} F_{\gamma}[f \mid x]=\sum_{\beta=0}^{\gamma} H_{\beta}[f \mid x], \quad \gamma=0,1, \ldots, k-1 . \tag{5} \end{equation*}(5)Fc[f∣x]=∑b=0cHb[f∣x],c=0,1,…,k−1.
SO F γ [ f ∣ x ] F γ [ f ∣ x ] F_(gamma)[f∣x]F_{\gamma}[f \mid x]Fc[f∣x]is a linear operator that can be assumed, and that we assume, to be defined on the set of functions having a derivative of order γ γ gamma\gammacon an interval E E EEANDcontaining the points y α , α = 1 , 2 , … , p y α , α = 1 , 2 , … , p y_(alpha),alpha=1,2,dots,py_{\alpha}, \alpha=1,2, \ldots, panda,a=1,2,…,p.
We have, in particular, F k − 1 [ f ∣ x ] = L ( x 1 , x 2 , … , x m ; f ∣ x ) F k − 1 [ f ∣ x ] = L x 1 , x 2 , … , x m ; f ∣ x F_(k-1)[f∣x]=L(x_(1),x_(2),dots,x_(m);f∣x)F_{k-1}[f \mid x]=L\left(x_{1}, x_{2}, \ldots, x_{m} ; f \mid x\right)Fk−1[f∣x]=L(x1,x2,…,xm;f∣x).
When all the knots are double, so if k 1 = k 2 = ⋯ = k p = 2 ( m = 2 p ) k 1 = k 2 = ⋯ = k p = 2 ( m = 2 p ) k_(1)=k_(2)=cdots=k_(p)=2(m=2p)k_{1}=k_{2}=\cdots=k_{p}=2(m=2 p)k1=k2=⋯=kp=2(m=2p), F 0 [ f ∣ x ] F 0 [ f ∣ x ] F_(0)[f∣x]F_{0}[f \mid x]F0[f∣x]is the well-known Fejér operator [1].
The problem of preserving non-concavity of a given order n ( ≧ − 1 ≧ − 1 >= -1\geqq-1≧−1) by the operator ( 2 ) or, in general, by the operator ( 5 ), is of some interest. We first have the important results of L. Fejér [1] on the conservation of the sign ( n = − 1 n = − 1 n=-1n=-1n=−1) by the operator F 0 [ f ∣ x ] F 0 [ f ∣ x ] F_(0)[f∣x]F_{0}[f \mid x]F0[f∣x]in the case of all double knots. We have given, among other things, certain results on the conservation of the sign ( n = − 1 n = − 1 n=-1n=-1n=−1) or monotony ( n = 0 n = 0 n=0n=0n=0) by the polynomial ( 2 ) in the case of all simple knots and for the Fejér operator [3, 4, 5].
5. According to (4), the polynomial h β , α ( x ) h β , α ( x ) h_(beta,alpha)(x)h_{\beta, \alpha}(x)hb,a(x)is always divisible by the polynomial ∏ α = 1 P ( x − y α ) min ( β , k α ) ∏ α = 1 P   x − y α min β , k α prod_(alpha=1)^(P)(x-y_(alpha))^(min(beta,k_(alpha)))\prod_{\alpha=1}^{P}\left(x-y_{\alpha}\right)^{\min \left(\beta, k_{\alpha}\right)}∏a=1P(x−anda)min(b,ka). This is also true for the sum ∑ ( β ) h β , α ( x ) ∑ ( β )   h β , α ( x ) sum(beta)h_(beta,alpha)(x)\sum^{(\beta)} h_{\beta, \alpha}(x)∑(b)hb,a(x). This sum is of effective degree at least equal to ∑ α = 1 p min ( β , k α ) ∑ α = 1 p   min β , k α sum_(alpha=1)^(p)min(beta,k_(alpha))\sum_{\alpha=1}^{p} \min \left(\beta, k_{\alpha}\right)∑a=1pmin(b,ka)since its order derivative β β beta\betab, For x = y δ x = y δ x=y_(delta)x=y_{\delta}x=andd, Or y δ ≧ β + 1 y δ ≧ β + 1 y_(delta) >= beta+1y_{\delta} \geqq \beta+1andd≧b+1, is equal to ∑ ( β ) h β , α ( β ) ( y δ ) = h β , δ ( β ) ( y δ ) = 1 ∑ ( β ) h β , α ( β ) y δ = h β , δ ( β ) y δ = 1 sum^((beta))h_(beta,alpha)^((beta))(y_(delta))=h_(beta,delta)^((beta))(y_(delta))=1\sum{ }^{(\beta)} h_{\beta, \alpha}^{(\beta)}\left(y_{\delta}\right)=h_{\beta, \delta}^{(\beta)}\left(y_{\delta}\right)=1∑(b)hb,a(b)(andd)=hb,d(b)(andd)=1, as a result of equalities (4).
We deduce
Theorem 1. If 1 ≦ β ≦ k − 1 , n ≧ β 1 ≦ β ≦ k − 1 , n ≧ β 1 <= beta <= k-1,n >= beta1 \leqq \beta \leqq k-1, n \geqq \beta1≦b≦k−1,n≧band if
(6) n < ∑ α = 1 p min ( β , k α ) , (6) n < ∑ α = 1 p   min β , k α , {:(6)n < sum_(alpha=1)^(p)min(beta,k_(alpha))",":}\begin{equation*} n<\sum_{\alpha=1}^{p} \min \left(\beta, k_{\alpha}\right), \tag{6} \end{equation*}(6)n<∑a=1pmin(b,ka),
the polynomial F β − 1 [ f ∣ x ] F β − 1 [ f ∣ x ] F_(beta-1)[f∣x]F_{\beta-1}[f \mid x]Fb−1[f∣x]does not preserve the non-concavity of order n n nnnon any interval of non-zero length I I III.
Let us suppose, in fact, the opposite. The function x β x β x^(beta)x^{\beta}xbis both nonconcave and nonconvex of order n n nnn(on I I III), SO F β − 1 [ x β ∣ x ] F β − 1 x β ∣ x F_(beta-1)[x^(beta)∣x]F_{\beta-1}\left[x^{\beta} \mid x\right]Fb−1[xb∣x]reduces to a polynomial of degree n n nnn. But if in (2) we pose f ( x ) = x β f ( x ) = x β f(x)=x^(beta)f(x)=x^{\beta}f(x)=xb, we have from (3), x β = F β − 1 [ x β ∣ x ] + β ! ∑ ( β ) h β , α ( x ) x β = F β − 1 x β ∣ x + β ! ∑ ( β )   h β , α ( x ) x^(beta)=F_(beta-1)[x^(beta)∣x]+beta!sum(beta)h_(beta,alpha)(x)x^{\beta}=F_{\beta-1}\left[x^{\beta} \mid x\right]+\beta!\sum^{(\beta)} h_{\beta, \alpha}(x)xb=Fb−1[xb∣x]+b!∑(b)hb,a(x), which according to (6) is impossible.
When the orders of multiplicity k 1 , k 2 , … , k p k 1 , k 2 , … , k p k_(1),k_(2),dots,k_(p)k_{1}, k_{2}, \ldots, k_{p}k1,k2,…,kpnodes are all even and when β = k − 1 β = k − 1 beta=k-1\beta=k-1b=k−1, we can obtain a more precise result by the
Theorem 2. If the orders of multiplicity k 1 , k 2 , … , k p k 1 , k 2 , … , k p k_(1),k_(2),dots,k_(p)k_{1}, k_{2}, \ldots, k_{p}k1,k2,…,kpnodes are all even and if
k = max ( k 1 , k 2 , … , k p ) ≦ n + 1 < ∑ α = 1 p k α = m , k = max k 1 , k 2 , … , k p ≦ n + 1 < ∑ α = 1 p   k α = m , k=max(k_(1),k_(2),dots,k_(p)) <= n+1 < sum_(alpha=1)^(p)k_(alpha)=m,k=\max \left(k_{1}, k_{2}, \ldots, k_{p}\right) \leqq n+1<\sum_{\alpha=1}^{p} k_{\alpha}=m,k=max(k1,k2,…,kp)≦n+1<∑a=1pka=m,
the polynomial F k − 2 [ f ∣ x ] F k − 2 [ f ∣ x ] F_(k-2)[f∣x]F_{k-2}[f \mid x]Fk−2[f∣x]does not preserve the non-concavity of order n n nnnon any interval of non-zero length I I III.
The demonstration is done as for Theorem 1, noting that
h k − 1 , α ( x ) = ( x − y α ) k α − 1 ∏ γ = 1 p ( x − y γ ) k γ ( k α − 1 ) ! ∏ γ = 1 p ( y α − y γ ) k γ h k − 1 , α ( x ) = x − y α k α − 1 ∏ γ = 1 p   x − y γ k γ k α − 1 ! ∏ γ = 1 p   y α − y γ k γ h_(k-1,alpha)(x)=((x-y_(alpha))^(k_(alpha)-1)prod_(gamma=1)^(p)(x-y_(gamma))^(k_(gamma)))/((k_(alpha)-1)!prod_(gamma=1)^(p)(y_(alpha)-y_(gamma))^(k_(gamma)))h_{k-1, \alpha}(x)=\frac{\left(x-y_{\alpha}\right)^{k_{\alpha}-1} \prod_{\gamma=1}^{p}\left(x-y_{\gamma}\right)^{k_{\gamma}}}{\left(k_{\alpha}-1\right)!\prod_{\gamma=1}^{p}\left(y_{\alpha}-y_{\gamma}\right)^{k_{\gamma}}}hk−1,a(x)=(x−anda)ka−1∏c=1p(x−andc)kc(ka−1)!∏c=1p(anda−andc)kc
where in the product ∏ γ = 1 p ( α ) ∏ γ = 1 p   ( α ) prod_(gamma=1)^(p)(alpha)\prod_{\gamma=1}^{p}(\alpha)∏c=1p(a)the value α α alpha\alphaaof γ γ gamma\gammacis excepted. The sum Σ ( k − 1 ) h k − 1 , α ( x ) Σ ( k − 1 ) h k − 1 , α ( x ) Sigma^((k-1))h_(k-1,alpha)(x)\Sigma^{(k-1)} h_{k-1, \alpha}(x)S(k−1)hk−1,a(x)is a polynomial of effective degree m − 1 m − 1 m-1m-1m−1whose first coefficient (that of x m − 1 x m − 1 x^(m-1)x^{m-1}xm−1) East
∑ ( k − 1 ) 1 ( k α − 1 ) ! ∏ γ = 1 p ( y α − y γ ) k γ > 0 ∑ ( k − 1 )   1 k α − 1 ! ∏ γ = 1 p   y α − y γ k γ > 0 sum(k-1)(1)/((k_(alpha)-1)!prod_(gamma=1)^(p)(y_(alpha)-y_(gamma))^(k_(gamma))) > 0\sum^{(k-1)} \frac{1}{\left(k_{\alpha}-1\right)!\prod_{\gamma=1}^{p}\left(y_{\alpha}-y_{\gamma}\right)^{k_{\gamma}}}>0∑(k−1)1(ka−1)!∏c=1p(anda−andc)kc>0
  1. For polynomials (5) it is always sufficient to examine the conservation of non-concavity of order n n nnn, For n n nnnsmall enough. Indeed, if n ≥ m − 1 n ≥ m − 1 n >= m-1n \geq m-1n≥m−1, the generalized interpolation polynomial (5) trivially preserves the non-concavity of order n n nnnof the function on the entire real axis, since any polynomial of degree n n nnnis non-concave of order n n nnneverywhere.
  2. If the nodes are all confused, so if p = 1 , k = m , x 1 == x 2 = … = x m = y 1 = c p = 1 , k = m , x 1 == x 2 = … = x m = y 1 = c p=1,k=m,x_(1)==x_(2)=dots=x_(m)=y_(1)=cp=1, k=m, x_{1}= =x_{2}=\ldots=x_{m}=y_{1}=cp=1,k=m,x1==x2=…=xm=and1=c, we can take for E E EEANDany interval of non-zero length containing the point c c ccc. We then have
    (7) F k − 1 [ f ∣ x ] = L ( x 1 , x 2 , … , x m ; f ( x ) = ∑ β = 0 m − 1 ( x − c ) β β ! f ( β ) ( c ) F k − 1 [ f ∣ x ] = L x 1 , x 2 , … , x m ; f ( x ) = ∑ β = 0 m − 1   ( x − c ) β β ! f ( β ) ( c ) quadF_(k-1)[f∣x]=L(x_(1),x_(2),dots,x_(m);f(x)=sum_(beta=0)^(m-1)((x-c)^(beta))/(beta!)f^((beta))(c):}\quad F_{k-1}[f \mid x]=L\left(x_{1}, x_{2}, \ldots, x_{m} ; f(x)=\sum_{\beta=0}^{m-1} \frac{(x-c)^{\beta}}{\beta!} f^{(\beta)}(c)\right.Fk−1[f∣x]=L(x1,x2,…,xm;f(x)=∑b=0m−1(x−c)bb!f(b)(c)
    and it is the Taylor polynomial of degree m − 1 m − 1 m-1m-1m−1of the function f f fffon the point c c ccc.
We have
Theorem 3. If m ≧ 2 , − 1 ≦ n ≦ m − 3 m ≧ 2 , − 1 ≦ n ≦ m − 3 m >= 2,-1 <= n <= m-3m \geqq 2,-1 \leqq n \leqq m-3m≧2,−1≦n≦m−3, the Taylor polynomial (7) does not preserve the non-concavity of order n n nnnon any non-zero length interval I of the real axis.
Either first n = − 1 n = − 1 n=-1n=-1n=−1. It is then necessary to demonstrate the non-conservation of the sign by the polynomial (7).
Either z z zzWithan interior point of I I IIIand different from c c ccc. The function (polynomial)
(8) φ ( x ) = 1 − x − c z − c + ( x − c z − c ) 2 m = 1 + | x − c z − c | ( | x − c z − c | 2 m − 1 − sgn x − c z − c ) φ ( x ) = 1 − x − c z − c + x − c z − c 2 m = 1 + x − c z − c x − c z − c 2 m − 1 − sgn x − c z − c varphi(x)=1-(x-c)/(z-c)+((x-c)/(z-c))^(2m)=1+|(x-c)/(z-c)|(|(x-c)/(z-c)|^(2m-1)-sgn(x-c)/(z-c))\varphi(x)=1-\frac{x-c}{z-c}+\left(\frac{x-c}{z-c}\right)^{2 m}=1+\left|\frac{x-c}{z-c}\right|\left(\left|\frac{x-c}{z-c}\right|^{2 m-1}-\operatorname{sgn} \frac{x-c}{z-c}\right)f(x)=1−x−cWith−c+(x−cWith−c)2m=1+|x−cWith−c|(|x−cWith−c|2m−1−sgnx−cWith−c)is positive on the real axis, therefore on E E EEAND. If we pose f ( x ) = φ ( x ) f ( x ) = φ ( x ) f(x)=varphi(x)f(x)=\varphi(x)f(x)=f(x), the Taylor polynomial (7) becomes F k − 1 [ φ ∣ x ] = 1 − x − c z − c F k − 1 [ φ ∣ x ] = 1 − x − c z − c F_(k-1)[varphi∣x]=1-(x-c)/(z-c)F_{k-1}[\varphi \mid x]=1-\frac{x-c}{z-c}Fk−1[f∣x]=1−x−cWith−c. It is a
polynomial of effective degree 1 which vanishes on z z zzWithand which therefore also takes negative values ​​on any neighborhood of z z zzWith, so also on I I III.
The theorem is thus demonstrated for n = − 1 n = − 1 n=-1n=-1n=−1.
For n > − 1 n > − 1 n > -1n>-1n>−1the demonstration is analogous. It suffices to take for f ( x ) f ( x ) f(x)f(x)f(x)a function whose order derivative n + 1 n + 1 n+1n+1n+1is equal to the function (8) and to note that the derivative of order n + 1 n + 1 n+1n+1n+1of a non-concave function of order n n nnnmust be non-negative.
And n > m − 3 n > m − 3 n > m-3n>m-3n>m−3the polynomial (7) preserves the non-concavity of order n n nnnon the entire real axis.
8. Theorems 1 , 2 , 3 1 , 2 , 3 1,2,31,2,31,2,3are rather theorems of non-conservation of non-concavity of a certain order n n nnnWe will also give, by some examples, properties of conservation of non-concavity for any order ≧ − 1 ≧ − 1 >= -1\geqq-1≧−1.
Let us first consider the case where we have only two distinct nodes, one of which is simple. So let p = 2 , k 1 = k = m − 1 , k 2 = 1 , x 1 == x 2 = … = x m − 1 = y 1 = a , x m = y 2 = b p = 2 , k 1 = k = m − 1 , k 2 = 1 , x 1 == x 2 = … = x m − 1 = y 1 = a , x m = y 2 = b p=2,k_(1)=k=m-1,k_(2)=1,x_(1)==x_(2)=dots=x_(m-1)=y_(1)=a,x_(m)=y_(2)=bp=2, k_{1}=k=m-1, k_{2}=1, x_{1}= =x_{2}=\ldots=x_{m-1}=y_{1}=a, x_{m}=y_{2}=bp=2,k1=k=m−1,k2=1,x1==x2=…=xm−1=and1=a,xm=and2=band suppose a < b a < b a < ba<ba<b, to fix ideas.
In this case we have
(9) F γ [ f ∣ x ] = ( x − a ) m − 1 ( b − a ) m − 1 f ( b ) + ∑ α = 0 γ 1 α ! [ ( x − a ) α − ( x − a ) m − 1 ( b − a ) m − α − 1 ] f ( α ) ( a ) γ = 0 , 1 , … , m − 2 (9) F γ [ f ∣ x ] = ( x − a ) m − 1 ( b − a ) m − 1 f ( b ) + ∑ α = 0 γ   1 α ! ( x − a ) α − ( x − a ) m − 1 ( b − a ) m − α − 1 f ( α ) ( a ) γ = 0 , 1 , … , m − 2 {:[(9)F_(gamma)[f∣x]=((x-a)^(m-1))/((b-a)^(m-1))f(b)+sum_(alpha=0)^(gamma)(1)/(alpha!)[(x-a)^(alpha)-((x-a)^(m-1))/((b-a)^(m-alpha-1))]f^((alpha))(a)],[gamma=0","1","dots","m-2]:}\begin{gather*} F_{\gamma}[f \mid x]=\frac{(x-a)^{m-1}}{(b-a)^{m-1}} f(b)+\sum_{\alpha=0}^{\gamma} \frac{1}{\alpha!}\left[(x-a)^{\alpha}-\frac{(x-a)^{m-1}}{(b-a)^{m-\alpha-1}}\right] f^{(\alpha)}(a) \tag{9}\\ \gamma=0,1, \ldots, m-2 \end{gather*}(9)Fc[f∣x]=(x−a)m−1(b−a)m−1f(b)+∑a=0c1a![(x−a)a−(x−a)m−1(b−a)m−a−1]f(a)(a)c=0,1,…,m−2
and we can state
Theorem 4. If m ≧ 2 , 0 ≦ γ ≦ m − 2 m ≧ 2 , 0 ≦ γ ≦ m − 2 m >= 2,0 <= gamma <= m-2m \geqq 2,0 \leqq \gamma \leqq m-2m≧2,0≦c≦m−2and if μ γ = 1 ( m − 1 γ ) 1 m − γ − 1 μ γ = 1 ( m − 1 γ ) 1 m − γ − 1 mu_(gamma)=1((m-1)/(gamma))^((1)/(m-gamma-1))\mu_{\gamma}=1\binom{m-1}{\gamma}^{\frac{1}{m-\gamma-1}}mc=1(m−1c)1m−c−1the generalized interpolation polynomial (9) preserves the non-concavity of order γ − 1 γ − 1 gamma-1\gamma-1c−1on the interval [ a , a + μ γ ( b − a ) ] a , a + μ γ ( b − a ) [a,a+mu_(gamma)(b-a)]\left[a, a+\mu_{\gamma}(b-a)\right][a,a+mc(b−a)]and m − γ m − γ m-gammam-\gammam−cis even and on the interval [ a − μ γ ( b − a ) , a + μ γ ( b − a ) ] a − μ γ ( b − a ) , a + μ γ ( b − a ) [a-mu_(gamma)(b-a),a+mu_(gamma)(b-a)]\left[a-\mu_{\gamma}(b-a), a+\mu_{\gamma}(b-a)\right][a−mc(b−a),a+mc(b−a)]and m − γ m − γ m-gammam-\gammam−cis odd.
Similarly, the generalized interpolation polynomial (9) preserves the nonconcavity of order γ γ gamma\gammacon the interval [ a , + ∞ ) [ a , + ∞ ) [a,+oo)[a,+\infty)[a,+∞)and m − γ m − γ m-gammam-\gammam−cis odd and on the entire real axis if m − γ m − γ m-gammam-\gammam−cis even.
The demonstration follows immediately from the formulas
d γ F γ [ f ∣ x ] d x γ = ( m − 1 ) ( m − 2 ) … ( m − γ ) ( x − a b − a ) m − γ − 1 [ b , a , a , … , a ⏟ γ ; f ] + + [ 1 − ( m − 1 γ ) ( x − a b − a ) m − γ − 1 ] f ( γ ) ( a ) d γ + 1 F γ [ f ∣ x ] d x γ + 1 = ( m − 1 ) ( m − 2 ) … ( m − γ − 1 ) ( x − a b − a ) m − γ − 2 = [ b , a , a , … , a ⏟ ⋯ γ ; f ] d γ F γ [ f ∣ x ] d x γ = ( m − 1 ) ( m − 2 ) … ( m − γ ) ( x − a b − a ) m − γ − 1 [ b , a , a , … , a ⏟ γ ; f ] + + 1 − ( m − 1 γ ) ( x − a b − a ) m − γ − 1 f ( γ ) ( a ) d γ + 1 F γ [ f ∣ x ] d x γ + 1 = ( m − 1 ) ( m − 2 ) … ( m − γ − 1 ) ( x − a b − a ) m − γ − 2 = [ b , a , a , … , a ⏟ ⋯ γ ; f ] {:[(d^(gamma)F_(gamma)[f∣x])/(dx^(gamma))=(m-1)(m-2)dots(m-gamma)((x-a)/(b-a))^(m-gamma-1)[b","ubrace(a,a,dots,aubrace)_(gamma);f]+],[+[1-((m-1)/(gamma))((x-a)/(b-a))^(m-gamma-1)]f^((gamma))(a)],[(d^(gamma+1)F_(gamma)[f∣x])/(dx^(gamma+1))=(m-1)(m-2)dots(m-gamma-1)((x-a)/(b-a))^(m-gamma-2)=[b","ubrace(a,a,dots,aubrace)_(cdots gamma);f]]:}\begin{aligned} \frac{d^{\gamma} F_{\gamma}[f \mid x]}{d x^{\gamma}}= & (m-1)(m-2) \ldots(m-\gamma)\binom{x-a}{b-a}^{m-\gamma-1}[b, \underbrace{a, a, \ldots, a}_{\gamma} ; f]+ \\ & +\left[1-\binom{m-1}{\gamma}\binom{x-a}{b-a}^{m-\gamma-1}\right] f^{(\gamma)}(a) \\ \frac{d^{\gamma+1} F_{\gamma}[f \mid x]}{d x^{\gamma+1}}= & (m-1)(m-2) \ldots(m-\gamma-1)\binom{x-a}{b-a}^{m-\gamma-2}=[b, \underbrace{a, a, \ldots, a}_{\cdots \gamma} ; f] \end{aligned}dcFc[f∣x]dxc=(m−1)(m−2)…(m−c)(x−ab−a)m−c−1[b,a,a,…,a⏟c;f]++[1−(m−1c)(x−ab−a)m−c−1]f(c)(a)dc+1Fc[f∣x]dxc+1=(m−1)(m−2)…(m−c−1)(x−ab−a)m−c−2=[b,a,a,…,a⏟⋯c;f]
And b < a b < a b < ab<ab<awe have an analogous property which is obtained in the same way. In this case it is sufficient to replace in the statement of the theorem the intervals [ a , a + μ γ ( b − a ) ] , [ a − μ γ ( b − a ) , a + μ γ ( b − a ) ] a , a + μ γ ( b − a ) , a − μ γ ( b − a ) , a + μ γ ( b − a ) [a,a+mu_(gamma)(b-a)],[a-mu_(gamma)(b-a),a+mu_(gamma)(b-a)]\left[a, a+\mu_{\gamma}(b-a)\right],\left[a-\mu_{\gamma}(b-a), a+\mu_{\gamma}(b-a)\right][a,a+mc(b−a)],[a−mc(b−a),a+mc(b−a)]And [ a , + ∞ ) [ a , + ∞ ) [a,+oo)[a,+\infty)[a,+∞)respectively by [ a − μ γ ( a − b ) , a ] , [ a − μ γ ( a − b ) , a + μ γ ( a − b ) ] a − μ γ ( a − b ) , a , a − μ γ ( a − b ) , a + μ γ ( a − b ) [a-mu_(gamma)(a-b),a],[a-mu_(gamma)(a-b),a+mu_(gamma)(a-b)]\left[a-\mu_{\gamma}(a-b), a\right],\left[a-\mu_{\gamma}(a-b), a+\mu_{\gamma}(a-b)\right][a−mc(a−b),a],[a−mc(a−b),a+mc(a−b)]And ( − ∞ , a ] ( − ∞ , a ] (-oo,a](-\infty, a](−∞,a].
9. Let us further suppose that we have only two distinct nodes, of which k 1 k 1 k_(1)k_{1}k1coincide with a a aaaAnd k 2 k 2 k_(2)k_{2}k2with b b bbb, Or a < b a < b a < ba<ba<b. We have k 1 + k 2 = m k 1 + k 2 = m k_(1)+k_(2)=mk_{1}+k_{2}=mk1+k2=mand we can assume k 1 ≧ 2 , k 2 ≧ 2 k 1 ≧ 2 , k 2 ≧ 2 k_(1) >= 2,k_(2) >= 2k_{1} \geqq 2, k_{2} \geqq 2k1≧2,k2≧2. We can then obtain the polynomials h β , α ( x ) h β , α ( x ) h_(beta,alpha)(x)h_{\beta, \alpha}(x)hb,a(x)in the following form:
h β , 1 ( x ) = ( m − β − 1 ) ! ( x − a ) β β ! ( b − a ) m − β − 1 ( k 1 − β − 1 ) ! ( k 2 − 1 ) ! ⋅ ∫ x b ( t − a ) k 1 β − 1 ( b − t ) k 2 − 1 d t , β = 0 , 1 , … , k 1 − 1 h β , 2 ( x ) = ( − 1 ) β ( m − β − 1 ) ! ( b − x ) β β ! ( b − a ) m − β − 1 ( k 1 − 1 ) ! ( k 2 − β − 1 ) ! ⋅ ∫ a x ( t − a ) k 1 − 1 ( b − t ) k 2 − β − 1 d t , β = 0 , 1 , … , k 2 − 1 h β , 1 ( x ) = ( m − β − 1 ) ! ( x − a ) β β ! ( b − a ) m − β − 1 k 1 − β − 1 ! k 2 − 1 ! ⋅ ∫ x b   ( t − a ) k 1 β − 1 ( b − t ) k 2 − 1 d t , β = 0 , 1 , … , k 1 − 1 h β , 2 ( x ) = ( − 1 ) β ( m − β − 1 ) ! ( b − x ) β β ! ( b − a ) m − β − 1 k 1 − 1 ! k 2 − β − 1 ! ⋅ ∫ a x   ( t − a ) k 1 − 1 ( b − t ) k 2 − β − 1 d t , β = 0 , 1 , … , k 2 − 1 {:[h_(beta,1)(x)=((m-beta-1)!(x-a)^(beta))/(beta!(b-a)^(m-beta-1)(k_(1)-beta-1)!(k_(2)-1)!)],[*int_(x)^(b)(t-a)^(k_(1))beta-1(b-t)^(k_(2)-1)dt","quad beta=0","1","dots","k_(1)-1],[h_(beta,2)(x)=((-1)^(beta)(m-beta-1)!(b-x)^(beta))/(beta!(b-a)^(m-beta-1)(k_(1)-1)!(k_(2)-beta-1)!)],[*int_(a)^(x)(t-a)^(k_(1)-1)(b-t)^(k_(2)-beta-1)dt","quad beta=0","1","dots","k_(2)-1]:}\begin{aligned} & h_{\beta, 1}(x)=\frac{(m-\beta-1)!(x-a)^{\beta}}{\beta!(b-a)^{m-\beta-1}\left(k_{1}-\beta-1\right)!\left(k_{2}-1\right)!} \\ & \cdot \int_{x}^{b}(t-a)^{k_{1}} \beta-1(b-t)^{k_{2}-1} d t, \quad \beta=0,1, \ldots, k_{1}-1 \\ & h_{\beta, 2}(x)=\frac{(-1)^{\beta}(m-\beta-1)!(b-x)^{\beta}}{\beta!(b-a)^{m-\beta-1}\left(k_{1}-1\right)!\left(k_{2}-\beta-1\right)!} \\ & \cdot \int_{a}^{x}(t-a)^{k_{1}-1}(b-t)^{k_{2}-\beta-1} d t, \quad \beta=0,1, \ldots, k_{2}-1 \end{aligned}hb,1(x)=(m−b−1)!(x−a)bb!(b−a)m−b−1(k1−b−1)!(k2−1)!⋅∫xb(t−a)k1b−1(b−t)k2−1dt,b=0,1,…,k1−1hb,2(x)=(−1)b(m−b−1)!(b−x)bb!(b−a)m−b−1(k1−1)!(k2−b−1)!⋅∫ax(t−a)k1−1(b−t)k2−b−1dt,b=0,1,…,k2−1
Consider again the generalized interpolation polynomials (5) where we have (3) and k = max ( k 1 , k 2 ) k = max k 1 , k 2 k=max(k_(1),k_(2))k=\max \left(k_{1}, k_{2}\right)k=max(k1,k2).
We can immediately see that the operator F 0 [ f ∣ x ] F 0 [ f ∣ x ] F_(0)[f∣x]F_{0}[f \mid x]F0[f∣x]keeps the sign and the formula
d F 0 [ f ∣ x ] d x = ( m − 1 ) ! ( x − a ) k 1 − 1 ( b − x ) k 2 − 1 ( k 1 − 1 ) ! ( k 2 − 1 ) ! ( b − a ) m − 2 [ a , b ; f ] d F 0 [ f ∣ x ] d x = ( m − 1 ) ! ( x − a ) k 1 − 1 ( b − x ) k 2 − 1 k 1 − 1 ! k 2 − 1 ! ( b − a ) m − 2 [ a , b ; f ] (dF_(0)[f∣x])/(dx)=((m-1)!(x-a)^(k_(1)-1)(b-x)^(k_(2)-1))/((k_(1)-1)!(k_(2)-1)!(b-a)^(m-2))[a,b;f]\frac{d F_{0}[f \mid x]}{d x}=\frac{(m-1)!(x-a)^{k_{1}-1}(b-x)^{k_{2}-1}}{\left(k_{1}-1\right)!\left(k_{2}-1\right)!(b-a)^{m-2}}[a, b ; f]dF0[f∣x]dx=(m−1)!(x−a)k1−1(b−x)k2−1(k1−1)!(k2−1)!(b−a)m−2[a,b;f]
shows us that it also preserves the monotonicity of the function on the interval [ a , b ] [ a , b ] [a,b][a, b][a,b].
If we do the calculations we also find
d 2 F 1 [ f ∣ x ] d x 2 = ( m − 2 ) ! ( x − a ) k 1 − 2 ( b − x ) k 2 − 2 ( k 1 − 1 ) ! ( k 2 − 1 ) ! ( b − a ) m − 2 . ⋅ { ( k 1 − 1 ) [ ( k 2 − 1 ) a + k 1 b − ( m − 1 ) x ] [ a , a , b ; f ] + + ( k 2 − 1 ) [ ( m − 1 ) x − k 2 a − ( k 1 − 1 ) b ] [ a , b , b ; f ] } d 2 F 1 [ f ∣ x ] d x 2 = ( m − 2 ) ! ( x − a ) k 1 − 2 ( b − x ) k 2 − 2 k 1 − 1 ! k 2 − 1 ! ( b − a ) m − 2 . ⋅ k 1 − 1 k 2 − 1 a + k 1 b − ( m − 1 ) x [ a , a , b ; f ] + + k 2 − 1 ( m − 1 ) x − k 2 a − k 1 − 1 b [ a , b , b ; f ] {:[(d^(2)F_(1)[f∣x])/(dx^(2))=((m-2)!(x-a)^(k_(1)-2)(b-x)^(k_(2)-2))/((k_(1)-1)!(k_(2)-1)!(b-a)^(m-2)).],[*{(k_(1)-1)[(k_(2)-1)a+k_(1)b-(m-1)x][a,a,b;f]+:}],[{:+(k_(2)-1)[(m-1)x-k_(2)a-(k_(1)-1)b][a,b,b;f]}]:}\begin{gathered} \frac{d^{2} F_{1}[f \mid x]}{d x^{2}}=\frac{(m-2)!(x-a)^{k_{1}-2}(b-x)^{k_{2}-2}}{\left(k_{1}-1\right)!\left(k_{2}-1\right)!(b-a)^{m-2}} . \\ \cdot\left\{\left(k_{1}-1\right)\left[\left(k_{2}-1\right) a+k_{1} b-(m-1) x\right][a, a, b ; f]+\right. \\ \left.+\left(k_{2}-1\right)\left[(m-1) x-k_{2} a-\left(k_{1}-1\right) b\right][a, b, b ; f]\right\} \end{gathered}d2F1[f∣x]dx2=(m−2)!(x−a)k1−2(b−x)k2−2(k1−1)!(k2−1)!(b−a)m−2.⋅{(k1−1)[(k2−1)a+k1b−(m−1)x][a,a,b;f]++(k2−1)[(m−1)x−k2a−(k1−1)b][a,b,b;f]}
and this formula shows us that the operator F 1 [ f ∣ x ] F 1 [ f ∣ x ] F_(1)[f∣x]F_{1}[f \mid x]F1[f∣x]preserves the usual nonconcavity (of order 1) on the interval
[ k 2 a + ( k 1 − 1 ) b m − 1 , ( k 2 − 1 ) a + k 1 b m − 1 ] . k 2 a + k 1 − 1 b m − 1 , k 2 − 1 a + k 1 b m − 1 . [(k_(2)a+(k_(1)-1)b)/(m-1),((k_(2)-1)a+k_(1)b)/(m-1)].\left[\frac{k_{2} a+\left(k_{1}-1\right) b}{m-1}, \frac{\left(k_{2}-1\right) a+k_{1} b}{m-1}\right] .[k2a+(k1−1)bm−1,(k2−1)a+k1bm−1].
This property is similar to that which expresses that the operator F k − 1 [ f ∣ x ] = L ( a , a , … , a ⏟ , b , b , … , b ⏟ ; f ∣ x ) F k − 1 [ f ∣ x ] = L ( a , a , … , a ⏟ , b , b , … , b ⏟ ; f ∣ x ) F_(k-1)[f∣x]=L(ubrace(a,a,dots,aubrace),ubrace(b,b,dots,bubrace);f∣x)F_{k-1}[f \mid x]=L(\underbrace{a, a, \ldots, a}, \underbrace{b, b, \ldots, b} ; f \mid x)Fk−1[f∣x]=L(a,a,…,a⏟,b,b,…,b⏟;f∣x)preserves the non-concavity of order m − 3 m − 3 m-3m-3m−3on the interval
[ k 1 a + ( k 2 − 1 ) b m − 1 , ( k 1 − 1 ) a + k 2 b m − 1 ] k 1 a + k 2 − 1 b m − 1 , k 1 − 1 a + k 2 b m − 1 [(k_(1)a+(k_(2)-1)b)/(m-1),((k_(1)-1)a+k_(2)b)/(m-1)]\left[\frac{k_{1} a+\left(k_{2}-1\right) b}{m-1}, \frac{\left(k_{1}-1\right) a+k_{2} b}{m-1}\right][k1a+(k2−1)bm−1,(k1−1)a+k2bm−1]
and which results as a limiting case from a property already established for the Lagrange polynomial (on distinct nodes) [4].
10. The properties of non-conservation and conservation of nonconcavity of order n n nnnoperators (5) are to be compared with the very remarkable property of the SN Bernstein polynomial ∑ α = 0 m ( m α ) f ( α m ) x α ( 1 − x ) m − α ∑ α = 0 m   m α f α m x α ( 1 − x ) m − α sum_(alpha=0)^(m)((m)/( alpha))f((alpha )/(m))x^(alpha)(1-x)^(m-alpha)\sum_{\alpha=0}^{m}\left(\frac{m}{\alpha}\right) f\left(\frac{\alpha}{m}\right) x^{\alpha}(1-x)^{m-\alpha}∑a=0m(ma)f(am)xa(1−x)m−ato keep on the interval [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1][0,1], any convexity property of the function f f fffdefined on this interval [2].
General considerations on the preservation of the convexity appearance by generalized interpolation polynomials of the form ∑ α = 0 m f ( x α ) P α ( x ) ∑ α = 0 m   f x α P α ( x ) sum_(alpha=0)^(m)f(x_(alpha))P_(alpha)(x)\sum_{\alpha=0}^{m} f\left(x_{\alpha}\right) P_{\alpha}(x)∑a=0mf(xa)Pa(x)were made in one of my previous works [4].

BIBLIOGRAPHY

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ON THE CONSERVATION OF THE CONVEXITY CHARACTER OF FUNCTIONS THROUGH INTERPOLATION

Summary

Continuing research on the conservation of convexity allure [ 3 , 4 , 5 ] [ 3 , 4 , 5 ] [3,4,5][3,4,5][3,4,5], the Lagrange-Hermite polynomial is considered L ( x 1 , x 2 , … , x m ; f ∣ x ) L x 1 , x 2 , … , x m ; f ∣ x L(x_(1),x_(2),dots,x_(m);f∣x)L\left(x_{1}, x_{2}, \ldots, x_{m} ; f \mid x\right)L(x1,x2,…,xm;f∣x), where the nodes x α , α = 1 , 2 , … , m x α , α = 1 , 2 , … , m x_(alpha),alpha=1,2,dots,mx_{\alpha}, \alpha=1,2, \ldots, mxa,a=1,2,…,mare not necessarily distinct. They are denoted by
F γ [ f ∣ x ] F γ [ f ∣ x ] F_(gamma)[f∣x]F_{\gamma}[f \mid x]Fc[f∣x]the sum of the terms in this polynomial that contain only the values ​​of the derivatives at nodes up to order γ γ gamma\gammacinclusive. Theorems 1,2 express cases when the operator F γ [ f ∣ x ] F γ [ f ∣ x ] F_(gamma)[f∣x]F_{\gamma}[f \mid x]Fc[f∣x]does not preserve non-concavity of the order n n nnnon no interval. Theorem 3 refers to the case of all confused nodes. Then F γ [ f ∣ x ] F γ [ f ∣ x ] F_(gamma)[f∣x]F_{\gamma}[f \mid x]Fc[f∣x]return to Taylor polynomials relative to the function f f fff. Theorem 4 states some properties of the conservation of convexity in the case when we have only two distinct nodes, one of which is simple. The paper also contains some results concerning the conservation of convexity in the general case of two distinct nodes.

[MR0234174, Zbl 0172.08102]

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1968

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