# Newton Interpolating Series atm Distinct Points with Coefficients in a Real Banach Algebra

Authors: Ghiocel Groza; Nicolae Pop
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

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## Abstract

A real Banach algebra of Newton interpolating series, used to approximate
the solutions of multipoint boundary value problems for ODE's, is studied.


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## Body

## 1. Introduction

Let(𝒜,∥∥) be a real Banach algebra. In this paper we study the so-called Newton interpolating series having the coefficients in (𝒜,∥∥), defined in Section 2, by means of Newton interpolating polynomials at m distinct nodes, x0,x1,…,xm-1, where m≥2, of multiplicity ni, i=0,1,…,m-1. These series are useful generalizations of power series which, in particular forms, were used in number theory to prove the transcendence of some values of exponential series [1, Chapter 2]. Knowing the importance of power series in the theory of initial value problems for differential equations, it seems to be useful to study Newton interpolating series in order to find the solution of the multipoint boundary value problems for differential equations. In [2] there are some numerical results in the particular case 𝒜=ℝ. Other methods used to solve similar interpolation problems can be found in [3].In Section2 we obtain results concerning Newton interpolating series and their derivatives (Theorems 2.1 and 2.2). Section 3 deals with functions which are representable into Newton interpolating series. Theorem 3.4 gives a criterion for a function to be representable into a Newton interpolating series (see Remark 3.7). The coefficients of this series are computed in Theorem 3.1. We prove that, with respect to a suitable norm, the algebra of these functions becomes a real Banach algebra (Theorem 3.6). This adds a new completion of the ring of polynomials 𝒜[x], where x commutes with the elements of 𝒜, to the usual real Banach spaces, where 𝒜[x] is a dense subset. Other examples can be found in [4].


## 2. Newton Interpolating Series atm Distinct Points

Letx0,x1,…,xm-1 be a set of real numbers which satisfy x0<x1<⋯<xm-1, where m≥2. If i is a nonnegative integer, then we consider the quotient q(i) and the remainder r(i) obtained when i is divided by m. We construct the polynomials(2.1)ui(x)=∏k=0m-1(x-xk)q(i)∏k=0r(i)-1(x-xk),i=0,1,2,…,
where x is a real variable. Hence(2.2)ui(x)=umq(i)(x)ur(i)(x).
If (𝒜,∥∥) is a real Banach algebra, we call an infinite series of the form(2.3)∑i=0∞Aiui(x),
where Ai∈𝒜 and x commutes with the elements of 𝒜, a Newton interpolating series with coefficients in 𝒜 at x0,x1,…,xm-1. We choose a real number x and consider the partial sum Sn,x=∑i=0nAiui(x). The series (2.3) is called convergent at x if there exists Sx∈𝒜 such that the sequence {Sn,x}n∈ℕ converges to Sx in 𝒜 that is limn→∞∥Sx-Sn,x∥=0. If I is a set and the series is convergent at x, for every x∈I, then we say that the series converges on I. In this case the function S:I→𝒜, given by S(x)=Sx, is called the sum of the series (2.3) on I.Theorem 2.1.
Let(𝒜,∥∥) be a real Banach algebra. If a Newton interpolating series of the form  (2.3) converges at a real number x̅ and |um(x̅)|≠0, then there exists an interval Ix̅ which contains x̅ and at least one of xi such that the series converges absolutely on Ix̅∖{x̅} and converges uniformly on every closed interval [a,b]⊂Ix̅.Proof.
Sinceum(x) is a polynomial of degree m having m distinct real roots, it follows that there exists an interval Ix¯ which contains x¯ and at least one of xi such that
(2.4)|um(x)|<|um(x¯)|,
for every x∈Ix¯∖{x¯}. Because the series converges at x¯, there exists M>0 such that, for every i,
(2.5)‖Aiui(x¯)‖<M.
If x∈Ix¯, then according to (2.2) and (2.5)
(2.6)‖Aiui(x)‖<Mvi-r(i)(x)|ur(i)(x)ur(i)(x¯)|,
where
(2.7)v(x)=|um(x)um(x¯)|1/m.
Let
(2.8)M′=max⁡0≤r≤m-1|ur(x)ur(x¯)|.
Now, from (2.4), (2.6), and (2.8), we obtain ∥Aiui(x)∥<MM′vi-m(x), i=0,1,…, where v(x)<1. Hence, the series converges absolutely at x.
If[a,b]⊂Ix¯, then we consider x∈[a,b]. Let(2.9)M′′=max⁡0≤r≤m-1max⁡x∈[a,b]|ur(x)ur(x¯)|.
Then there exists c∈[a,b] such that, for every x∈[a,b],
(2.10)v(x)≤v(c)<1
and ∥Aiui(x)∥≤MM′′vi-m(c). Thus, from (2.9), (2.10), and Weierstrass test, the series is uniformly convergent on [a,b].Given a Newton interpolating series of the form (2.3), we consider the polynomials (2.11)P1,m(x)=1,Pi,m(x)={ui′(x),ifi≤m,(q(i)+1)ur(i)′(x)vr(i)(x)+q(i)ur(i)(x)vr(i)′(x),ifi>m,
where (2.12)vr(i)(x)=∏k=r(i)m-1(x-xk).
If t≤m, then we denote by Pi,m;t(z1;z2;…;zt) the divided difference of Pi,m(x) with respect to t distinct points z1,z2,…,zt. Thus, Pi,m;1(z1)=Pi,m(z1) and for j≥2(2.13)Pi,m;j(z1;z2;…;zj)=Pi,m;j-1(z2;…;zj)-Pi,m;j-1(z1;…;zj-1)zj-z1.
Since, for i≥m+1, ui′(x)=ui-m(x)Pi,m(x), then by (2.1), (2.2), (2.11) and by Newton's interpolation formula, it follows that(2.14)ui′(x)={∑k=1iPi,m;k(x0;x1;…;xk-1)uk-1(x),ifi≤m,∑k=1mPi,m;m-k+1(xr(i);…;xr(i)+m-k)ui-k(x),ifi≥m+1,
where, for s≥m, xs=xr(s).The series(2.15)∑i=0∞Ai(1)ui(x),
where, if i≤m-1,(2.16)Ai(1)=∑j=1m-iPi+j,m;i+1(x0;x1;…;xi)Ai+j+∑j=m-i+1mPi+j,m;m-j+1(xr(i+j);…;xr(i+j)+m-j)Ai+j
and, if i≥m,(2.17)Ai(1)=∑j=1mPi+j,m;m-j+1(xr(i+j);xr(i+j)+1;…;xr(i+j)+m-j)Ai+j,
is called the derived series of (2.3).Now, let us denote byℒn(ℝ,𝒜) the complete real algebra of bounded multilinear forms f:ℝn→𝒜. Then we can consider 𝒜=ℒ(ℝ,𝒜), and hence 𝒜=ℒn(ℝ,𝒜). Thus, if I⊂ℝ is an interval and F:I→𝒜, then the Fréchet derivatives of this function at a point of I can be considered an element of 𝒜.Theorem 2.2.
Let(𝒜,∥∥) be a real Banach algebra. If the series  (2.3)  converges, on an open interval  I which does not contain a local extremum of  um(x), to  S, then the derived series of   (2.3)  converges absolutely on  I, there exists the Fréchet derivative  S′(x) of  S(x) at  x∈I, and
(2.18)S′(x)=∑i=0∞Ai(1)ui(x),
where Ai(1) are given in  (2.16), (2.17). Moreover, the series  (2.15) can be obtained by termwise differentiation from the series  (2.3).Proof.
Ifx∈I, since um(x) is a polynomial and I does not contain a local extremum of um(x), then we can choose x̃∈I such that
(2.19)|um(x)|<|um(x̃)|.
Since the series (2.3) is convergent at x̃, there exists M>0 such that
(2.20)‖Aiui(x̃)‖<M,i=0,1,….
Hence, for k∈{0,1,…m-1},
(2.21)‖Ai+kui(x)‖=‖Ai+kui+k(x̃)‖|ui(x̃)ui+k(x̃)||ui(x)ui(x̃)|≤M′|ui(x)ui(x̃)|,
where
(2.22)M′=Mmax⁡0≤n≤m-1|ui(x̃)ui+n(x̃)|.
Here we remark that, from (2.2), for a fixed x̃, ui(x̃)/ui+n(x̃) takes only a finite number of values, for i∈ℕ.
Since (see, e.g., [5], page 121)(2.23)Pi,m;t(z0;z1;…;zt-1)=∑k=0t-1Pi,m(zk)ut′(zk),t≤m,
where we consider z0<z1<⋯<zm-1 and
(2.24)ui(z)=∏k=0m-1(z-zk)q(i)∏k=0r(i)-1(z-zk),
by (2.11) and (2.23), for j=1,2,…,m, we obtain
(2.25)Pi+j,m;m-j+1(xr(i+j);xr(i+j)+1;…;xr(i+j)+m-j)=αi+jq(i)+βi+j,
where αi+j=αi+j(x0,x1,…xm-1),βi+j=βi+j(x0,x1,…xm-1) take only a finite number of values. Then there exists C1>0 such that, for all i and j,
(2.26)|Pi+j,m;m-j+1(xr(i+j);xr(i+j)+1;…;xr(i+j)+m-j)|<C1q(i).
By (2.2) there exists C2=C2(x,x̃)>0, independent of i, such that
(2.27)|ui(x)ui(x̃)|<C2|um(x)um(x̃)|q(i).
Hence, by using (2.16), (2.17), (2.21), (2.26), and (2.27), it follows that there exists C3>0, independent of i, such that
(2.28)|Ai(1)ui(x)|≤C3q(i)|um(x)um(x̃)|q(i).
Then, by (2.19), the series (2.15) converges absolutely at x and also on I. By (2.14) it follows that, for x∈I, the series (2.15) can be obtained by termwise differentiation from the series (2.3).
Now, ifx∈I, because for x=xi, S(x) is a polynomial function and Theorem 2.2 follows easily, then we may suppose that  um(x)≠0. Then, by Theorem 2.1, we can choose a compact interval Ix⊂I such that x belongs to its interior, xi∉Ix, for every i, and the series (2.3) converges absolutely on Ix. We consider δ>0  small enough such that for every h∈ℝ,|h|≤δ, x+h∈Ix. Then(2.29)S(x+h)-S(x)h=∑i=1∞αi(h),
where
(2.30)αi(h)=Aiui(x+h)-ui(x)h=Aiu′i(ξi),
where ξi is a point lying between x and x+h.
Ify∈Ix, then it is different from xi, for every i, and, by (2.2),
(2.31)ui′(y)=q(i)umq(i)-1(y)um′(y)ur(i)(y)+umq(i)(y)ur(i)′(y)=um′(y)um(y)q(i)ui(y)+ur(i)′(y)ur(i)(y)ui(y).
We put
(2.32)M1=max⁡0≤j≤mmax⁡y∈Ix|uj′(y)uj(y)|,M2=max⁡0≤j≤mmax⁡y∈Ix|uj(y)|,(2.33)m1=min⁡0≤j≤mmin⁡y∈Ix|uj′(y)uj(y)|,m2=min⁡0≤j≤mmin⁡y∈Ix|uj(y)|.
Since, for y∈Ix, the series ∑i=0∞Aiui(y) converges absolutely, it follows that the series ∑i=0∞∥Ai∥|ui(y)| converges. Hence, because m2>0, we obtain that the series ∑i=0∞Aiumq(i)(y) converges absolutely. Similarly, since the series ∑i=0∞Aiui′(y) and ∑i=0∞Ai(ur(i)′(y)/ur(i)(y))ui(y) converge absolutely, from (2.31), it follows that the series ∑i=0∞Ai(um′(y)/um(y))q(i)ui(y) converges absolutely. Then, by (2.2), the series ∑i=0∞∥Ai∥q(i)|umq(i)(y)||ur(i)(y)| converges, and thus ∑i=0∞Aiq(i)umq(i)(y) converges absolutely.
We considerx0∈Ix such that(2.34)|um(x0)|=max⁡y∈Ix|um(y)|,
and we put
(2.35)M3=max⁡0≤j≤mmax⁡y∈Ix|uj′(y)|.
Then, from (2.31)–(2.35), for every i,
(2.36)|u′(ξi)|≤q(i)M2M3|umq(i)-1(x0)|+M3|umq(i)(x0)|≤M2M3m2q(i)|umq(i)(x0)|+M3|umq(i)(x0)|.
Hence, the series ∑i=0∞Aiu′(ξi)=∑i=1∞αi(h) converges absolutely and uniformly on [-δ,δ]. If we put αi(0)=Aiui′(x), then it follows that the functions αi(h) are continuous on [-δ,δ]. Thus, the sum of the series (2.29) is continuous and
(2.37)lim⁡h→0S(x+h)-S(x)h=∑i=1∞αi(0)=∑i=0∞Aiui′(x).
Hence,
(2.38)S(x+h)-S(x)=∑i=0∞Aiui′(x)h+o(h),
and this completes the proof of the theorem.


## 3. Representable Functions into Newton Interpolating Series  atm Distinct Points

IfI⊂ℝ is an interval which contains x0,x1,…,xm-1 such that x0<x1<⋯<xm-1, m≥2, then we consider a function F:I→𝒜. We say that the function F is representable into a Newton interpolating series atx0,x1,…,xm-1 if there exists a series of the form (2.3), which converges absolutely on I and converges uniformly to F on I. Hence, it follows that F is a continuous function.Whenxj=j+1, I=𝒜=ℝ, and F(x)=eαx, it is known that the coefficients Ai and F are related through the complex integral formula (see [1], Chapter 2, Section 3) (3.1)Ai=12π-1∫|z|=iF(z)ui+1(z)dz.IfF is representable into a Newton interpolating series at x0,x1,…,xm-1, by using Theorems 2.1 and 2.2, similarly as in [2, Theorem 3], then it follows the following result which gives a formula for Ai.Theorem 3.1.
Suppose thatF:I→𝒜 is a function which is representable into a Newton interpolating series Σ of the form  (2.3) at x0,x1,…,xm-1∈I. Then, for every n, there exists Jn=⋃i=0m-1Vi,n⊂I, where, for every i, Vi,n is an open interval which contains xi, F has Fréchet derivative of order n on Vi,n, the series Σ is uniquely defined, and for each i(3.2)Ai=1q(i)!∑j=0r(i)(F(x)∏k=0,k≠jr(i)(x-xk)∏k=0,k≠jm-1(x-xk)q(i))x=xj(q(i))+1(q(i)-1)!∑j=r(i)+1m-1(F(x)∏k=0r(i)(x-xk)∏k=0,k≠jm-1(x-xk)q(i))x=xj(q(i)-1),
where  (F(x))(-1) and  1/(-1)! mean   0.IfF:I→𝒜 is representable into a Newton interpolating series at x0,x1,…,xm-1∈I, then the series of the form (2.3), where Ai are given in (3.2), is called the Newton interpolating series atx0,x1,…,xm-1 of the function F.Given a Newton interpolating series of the form (2.3), we put
(3.3)Nn(x)=∑i=0nAiui(x).Now, in order to prove a corollary of the theorem, we need the following form of Lagrange's theorem.Lemma 3.2.
One considers a real Banach algebra𝒜 and 𝒜* the dual of 𝒜. If F:[a,b]→𝒜*, where a<b is Fréchet differentiable on [a,b], then there exists c∈(a,b) such that
(3.4)F(b)-F(a)=F′(c)(b-a).Proof.
Ifx∈[a,b] and h1∈𝒜, then we consider the real function φ given by φ(x,h1)=F(x)(h1). Then, for every h1∈𝒜, h2∈ℝ such that x+h2∈[a,b],  φ(x+h2,h1)-φ(x,h1)=F(x+h2)(h1)-F(x)(h1)=F′(x)(h2)(h1)+o(h2) and φ has Fréchet derivative with respect to x. Now, for h1∈𝒜, we consider the real function gh1:[a,b]→ℝ, defined by gh1(x)=φ(x,h1). Then, by Lagrange's theorem, there exists c∈(a,b) such that gh1(b)-gh1(a)=gh1′(c)(b-a). Hence, the lemma is proved.Corollary 3.3.
Suppose thatF:[x0,xm-1]→𝒜 has Fréchet derivatives of all orders on [x0,xm-1] and it is representable into a Newton interpolating series at x0,x1,…,xm-1. If its Newton interpolating series (2.3) has the partial sums given in  (3.3), then, for a fixed n, the polynomial Nn(x) satisfies
(3.5)Nn(αj)(xj)=F(αj)(xj),
where 0≤j≤m-1 and
(3.6)αj∈{{0,1,…,q(n)},ifj≤r(n),{0,1,…,q(n)-1},ifj≥r(n)+1.
Furthermore, if 𝒜 is the dual algebra of a real Banach algebra 𝒜1, then, for every x∈[x0,xm-1], there exists ξ=ξ(x,n)∈(x0,xm-1) such that
(3.7)F(x)=Nn(x)+F(n+1)(ξ)(n+1)!un+1(x).Proof.
It follows immediately that every polynomial function is representable into a finite Newton interpolating series atx0,x1,…,xm-1. Then the coefficients Ai,i≤n, are given in (3.2), where F is replaced by Nn. Since Nn(x) are the partial sums of the Newton interpolating series at x0,x1,…,xm-1 of the function F, it follows that
(3.8)Ai=1q(i)!∑j=0r(i)(F(x)∏k=0,k≠jr(i)(x-xk)∏k=0,k≠jm-1(x-xk)q(i))x=xj(q(i))+1(q(i)-1)!∑j=r(i)+1m-1(F(x)∏k=0r(i)(x-xk)∏k=0,k≠jm-1(x-xk)q(i))x=xj(q(i)-1)=1q(i)!∑j=0r(i)(Nn(x)∏k=0,k≠jr(i)(x-xk)∏k=0,k≠jm-1(x-xk)q(i))x=xj(q(i))+1(q(i)-1)!∑j=r(i)+1m-1(Nn(x)∏k=0r(i)(x-xk)∏k=0,k≠jm-1(x-xk)q(i))x=xj(q(i)-1).
Then (3.5) follows, by mathematical induction on n, by using (3.8), with i=0,1,…. We observe, finally, that (3.7) gives the error of the interpolating formula and it is obtained by applying the generalization of Rolle's theorem, which is a consequence of Lemma 3.2, for the function H(x)=F(x)-Nn(x)+Kun+1(x), where K is a constant from 𝒜.Theorem 3.4.
One considers𝒜 and 𝒜1 two Banach real algebras such that 𝒜=𝒜1*. One supposes that F:[x0,xm-1]→𝒜 has Fréchet derivatives of all orders on [x0,xm-1]. If Ms=supx∈[x0,xm-1]∥F(s)(x)∥ and the sequence {zs} given by
(3.9)zs=(xm-1-x0)sMss!,s=0,1,…,
converges to zero, then F is the sum of its Newton interpolating series at x0,x1,…,xm-1 on [x0,xm-1].Proof.
From (3.7) it follows that
(3.10)‖F(x)-Nn(x)‖≤Mn+1(n+1)!|un+1(x)|,
and by (2.2) we obtain that
(3.11)max⁡x∈[x0,xm-1]|un+1(x)|≤(xm-1-x0)n+1.
Now (3.10) and (3.11) imply
(3.12)‖F(x)-Nn(x)‖≤(xm-1-x0)n+1Mn+1(n+1)!.
Hence, Nn(x) converges uniformly to F on [x0,xm-1].In the last theorem of this section we study the algebra of functions which are the sums of their Newton interpolating series atx0,x1,…,xm-1.If𝒜 is a Banach real algebra, then we denote by 𝒩(x0,x1,…,xm-1) the set of all functions F:[x0,xm-1]→𝒜 which are representable into a Newton interpolating series at x0,x1,…,xm-1. Thus, by Theorem 3.1, every F∈𝒩(x0,x1,…,xm-1) is the sum of its Newton interpolating series at x0,x1,…,xm-1. We take (3.13)F(x)=∑i=0∞Aiui(x),G(x)=∑i=0∞Biui(x)
two elements of 𝒩(x0,x1,…,xm-1). Our objective is to define the addition and the multiplication of F and G. We begin with a lemma.Lemma 3.5.
Ifui are given in  (2.1), then, for every i,j∈{1,2,…,m}, there exist nonnegative real numbers Ck(i,j), k∈{max{i,j},max{i,j}+1,…,i+j}, uniquely defined, such that
(3.14)uiuj=∑k=max⁡⁡{i,j}i+jCk(i,j)uk.Proof.
Ifj≤i<m, then it follows that
(3.15)uiuj=uiuj-1(x-xj-1)=ui+1uj-1+(xi-xj-1)uiuj-1,
and if i=m, then uiuj=ui+j. Now, for every i, the induction on j completes the proof of the lemma.We define addition and multiplication ofF and G as follows: (3.16)(F+G)(x)=∑i=0∞(Ai+Bi)ui(x),(FG)(x)=∑i=0∞Piui(x),
where
(3.17)Pi=∑(α,β)∈D(i)Ci-(q(α)+q(β))m(r(α),r(β))aαbβD(i)={(α,β)∈ℕ2:max{α+q(β)m,β+q(α)m}≤i≤α+β} and Ck(s,t) are given in (3.14).Letx¯∈[x0,xm-1] be such that(3.18)|um(x¯)|=sup⁡x∈[x0,xm-1]|um(x)|.
Since ui(x)>0, for every x>xm-1, and the roots of ui′(x) belong to [x0,xm-1], there exists z>xm-1, uniquely determined such that(3.19)um(z)=|um(x¯)|.
We put
(3.20)K1=min⁡0≤i≤m-1|ui(x¯)|,K2=max⁡0≤i≤m-1|ui(z)|.Then, by (2.2),(3.21)ui(z)=umq(i)(z)ur(i)(z)=|umq(i)(x¯)||ur(i)(x¯)||ur(i)(z)ur(i)(x¯)|≤K2K1|ui(x¯)|.
If F(x)=∑i=0∞Aiui(x)∈𝒩(x0,…,xm-1), then the series ∑i=0∞Aiui(x¯) converges absolutely and, by (3.21), the series ∑i=0∞∥Ai∥ui(z) converges. We define(3.22)‖F‖N=∑i=0∞‖Ai‖ui(z).Theorem 3.6.
If the addition and the multiplication are defined by  (3.16), then 𝒩(x0,…,xm-1) becomes a real algebra. Furthermore, ∥∥𝒩 defined by  (3.22) is a norm, and 𝒩(x0,…,xm-1) becomes a real Banach algebra with respect to ∥∥𝒩.Proof.
IfF,G∈𝒩(x0,…,xm-1), then their Newton interpolating series at x0,x1,…,xm-1 converge absolutely on [x0,xm-1] and converge uniformly to F and G, respectively. By (3.16) it follows easily that the Newton interpolating series at x0,x1,…,xm-1 of F+G and FG have the same properties. Thus, 𝒩(x0,…,xm-1) is a real algebra.Since
(3.23)‖uiuj‖N=‖∑k=ii+jCk(i,j)uk‖N=∑k=ii+jCk(i,j)uk(z)=‖ui‖N‖uj‖N,
for every i,j∈{1,2,…,m}, i≥j, it follows that ∥FG∥𝒩≤∥F∥𝒩∥G∥𝒩 and 𝒩(x0,…,xm-1) is a normed algebra.
Now, we will prove that𝒩(x0,…,xm-1) is a real Banach algebra. Consider {Fn=∑j=0∞Aj,nuj}n∈ℕ a Cauchy sequence of 𝒩(x0,…,xm-1). Let ϵ>0 be arbitrary. Then there exists N(ϵ)∈ℕ such that, for all n≥N(ϵ) and p=1,2,…, ∥Fn+p-Fn∥𝒩<ϵ. From (3.22), we obtain(3.24)∑j=0∞‖Aj,n+p-Aj,n‖uj(z)<ϵ.
Thus, for each fixed j, n≥N(ϵ) and p=1,2,…, ∥Aj,n+p-Aj,n∥uj(z)<ϵ. Hence, {Aj,n}n∈ℕ is a Cauchy sequence in 𝒜 and, since 𝒜 is a complete algebra, this sequence converges. We set Aj=limn→∞Aj,n, and we consider the formal series
(3.25)F=∑j=0∞Ajuj.
We have to show that F∈𝒩(x0,x1,…,xm-1) and limn→∞Fn=F. From (3.24),
(3.26)∑j=0s‖Aj,n+p-Aj,n‖uj(z)<ϵ.
Fix now some n≥N(ϵ) and let p→∞ and then s→∞. Hence, for all n≥N(ϵ), we obtain
(3.27)∑j=0∞‖Aj-Aj,n‖uj(z)≤ϵ.
Since, for x∈[x0,xm-1],  n0≥N(ϵ),  p′=0,1,…, ∥∑j=ii+p′Ajuj(x)∥≤∑j=i∞∥Aj-Aj,n0∥uj(z)+∑j=i∞∥Aj,n0∥uj(z) and Fn0∈𝒩(x0,…,xm-1), from (3.27), it follows that F is a function defined on [x0,xm-1] and it is representable into the Newton interpolation series (3.25). Thus, F∈𝒩(x0,…,  xm-1). Finally, from (3.27), ∥F-Fn∥𝒩≤ϵ and limn→∞Fn=F.Remark 3.7.
Let𝒜[x] be the ring of polynomials with coefficients in 𝒜, where x commutes with the elements of 𝒜. With respect to the topology defined by the norm given in (3.22), 𝒜[x] is a dense subset of 𝒩(x0,…,xm-1). If F satisfies the hypotheses of Theorem 3.4 and its Newton interpolating series converges absolutely at x¯ defined in (3.18), then F∈𝒩(x0,…,xm-1). Hence, it follows that if F:I→ℝ, where I is an open interval which contains xi, for every i, and its Taylor series at x0 converges on I, then the restriction of F to [x0,xm-1] belongs to 𝒩(x0,…,xm-1). However, 𝒩(x0,…,xm-1) contains also functions which are not Fréchet differentiable on [x0,xm-1]. For example, we take 𝒜=ℝ, m=2, x0=-1, x1=1 and F(x)=|x|. Then,
(3.28)F(x)=1-(1-x2)=1+∑k=1∞(-1)k+1∏i=1k-2(k+i)(k-1)!22k-1(x2-1)k,
for x∈[-1,1], which implies that F∈𝒩(-1,1).



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*Source: 365491-2011-04-27.xml*