Newton Interpolating Series at 𝑚 Distinct Points with Coefficients in a Real Banach Algebra

Abstract

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

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,…,𝑥_𝑚−1, where 𝑚≥2, of multiplicity 𝑛_𝑖, 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 Section 2 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 at 𝑚 Distinct Points

Let 𝑥₀,𝑥₁,…,𝑥_𝑚−1 be a set of real numbers which satisfy 𝑥₀<𝑥₁<⋯<𝑥_𝑚−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 polynomials𝑢_𝑖(𝑥)=^𝑚−1_𝑘=0𝑥−𝑥_𝑘^{𝑟(𝑖)𝑞(𝑖)−1}_𝑘=0𝑥−𝑥_𝑘,𝑖=0,1,2,…, (2.1) where 𝑥 is a real variable. Hence𝑢_𝑖(𝑥)=𝑢_𝑚^𝑞(𝑖)(𝑥)𝑢_𝑟(𝑖)(𝑥). (2.2) If (𝒜,‖‖) is a real Banach algebra, we call an infinite series of the form^∞_𝑖=0𝐴_𝑖𝑢_𝑖(𝑥), (2.3) where 𝐴_𝑖∈𝒜 and 𝑥 commutes with the elements of 𝒜, a Newton interpolating series with coefficients in 𝒜 at 𝑥₀,𝑥₁,…,𝑥_𝑚−1. We choose a real number 𝑥 and consider the partial sum 𝑆_𝑛,𝑥=∑^𝑛_𝑖=0𝐴_𝑖𝑢_𝑖(𝑥). The series (2.3) is called convergent at 𝑥 if there exists 𝑆_𝑥∈𝒜 such that the sequence {𝑆_𝑛,𝑥}_𝑛∈ℕ converges to 𝑆_𝑥 in 𝒜 that is lim_𝑛→∞‖𝑆_𝑥−𝑆_𝑛,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)=𝑆_𝑥, 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 𝑥 and |𝑢_𝑚(𝑥)|≠0, then there exists an interval 𝐼_𝑥 which contains 𝑥 and at least one of 𝑥_𝑖 such that the series converges absolutely on 𝐼_𝑥⧵{𝑥} and converges uniformly on every closed interval [𝑎,𝑏]⊂𝐼_𝑥.

Proof. Since 𝑢_𝑚(𝑥) is a polynomial of degree 𝑚 having 𝑚 distinct real roots, it follows that there exists an interval 𝐼_𝑥 which contains 𝑥 and at least one of 𝑥_𝑖 such that 𝑢_𝑚(𝑥)<𝑢_𝑚(𝑥), (2.4) for every 𝑥∈𝐼_𝑥⧵{𝑥}. Because the series converges at 𝑥, there exists 𝑀>0 such that, for every 𝑖, 𝐴_𝑖𝑢_𝑖(𝑥)<𝑀. (2.5) If 𝑥∈𝐼_𝑥, then according to (2.2) and (2.5) 𝐴_𝑖𝑢_𝑖(𝑥)<𝑀𝑣^{𝑖−𝑟(𝑖)}(𝑥𝑢)^𝑟(𝑖)(𝑥)𝑢_𝑟(𝑖)(𝑥), (2.6) where 𝑣(𝑥)=𝑢_𝑚(𝑥)𝑢_𝑚(𝑥)^1/𝑚. (2.7) Let 𝑀^′=max_{0≤𝑟≤𝑚−1}𝑢_𝑟(𝑥)𝑢_𝑟(𝑥). (2.8) Now, from (2.4), (2.6), and (2.8), we obtain ‖𝐴_𝑖𝑢_𝑖(𝑥)‖<𝑀𝑀′𝑣^𝑖−𝑚(𝑥), 𝑖=0,1,…, where 𝑣(𝑥)<1. Hence, the series converges absolutely at 𝑥.
If [𝑎,𝑏]⊂𝐼_𝑥, then we consider 𝑥∈[𝑎,𝑏]. Let𝑀^′′=max_{0≤𝑟≤𝑚−1}max_{𝑥∈[𝑎,𝑏]}𝑢_𝑟(𝑥)𝑢_𝑟(𝑥). (2.9) Then there exists 𝑐∈[𝑎,𝑏] such that, for every 𝑥∈[𝑎,𝑏], 𝑣(𝑥)≤𝑣(𝑐)<1 (2.10) and ‖𝐴_𝑖𝑢_𝑖(𝑥)‖≤𝑀𝑀^′′𝑣^𝑖−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 𝑃_1,𝑚(𝑥)𝑃=1,_𝑖,𝑚(𝑥)=𝑢^′_𝑖(𝑥),if𝑞(𝑖)𝑢𝑖≤𝑚,+1^′_𝑟(𝑖)(𝑥)𝑣_𝑟(𝑖)(𝑥)+𝑞(𝑖)𝑢_𝑟(𝑖)(𝑥)𝑣^′_𝑟(𝑖)(𝑥),if𝑖>𝑚, (2.11) where 𝑣_𝑟(𝑖)(𝑥)=^𝑚−1_{(𝑖)𝑘=𝑟}𝑥−𝑥_𝑘. (2.12) If 𝑡≤𝑚, then we denote by 𝑃_{𝑖,𝑚;𝑡}(𝑧₁;𝑧₂;…;𝑧_𝑡) the divided difference of 𝑃_𝑖,𝑚(𝑥) with respect to 𝑡 distinct points 𝑧₁,𝑧₂,…,𝑧_𝑡. Thus, 𝑃_𝑖,𝑚;1(𝑧₁)=𝑃_𝑖,𝑚(𝑧₁) and for 𝑗≥2𝑃_{𝑖,𝑚;𝑗}𝑧₁;𝑧₂;…;𝑧_𝑗=𝑃_{𝑖,𝑚;𝑗−1}𝑧₂;…;𝑧_𝑗−𝑃_{𝑖,𝑚;𝑗−1}𝑧₁;…;𝑧_𝑗−1𝑧_𝑗−𝑧₁. (2.13) Since, for 𝑖≥𝑚+1, 𝑢^′_𝑖(𝑥)=𝑢_𝑖−𝑚(𝑥)𝑃_𝑖,𝑚(𝑥), then by (2.1), (2.2), (2.11) and by Newton's interpolation formula, it follows that𝑢^′_𝑖(𝑥)=^𝑖𝑘=1𝑃_{𝑖,𝑚;𝑘}𝑥₀;𝑥₁;…;𝑥_𝑘−1𝑢_𝑘−1(𝑥),if𝑖≤𝑚,_𝑚𝑘=1𝑃_{𝑖,𝑚;𝑚−𝑘+1}𝑥_𝑟(𝑖);…;𝑥_{𝑟(𝑖)+𝑚−𝑘}𝑢_𝑖−𝑘(𝑥),if𝑖≥𝑚+1, (2.14) where, for 𝑠≥𝑚, 𝑥_𝑠=𝑥_𝑟(𝑠).

The series^∞_𝑖=0𝐴_𝑖⁽¹⁾𝑢_𝑖(𝑥), (2.15) where, if 𝑖≤𝑚−1,𝐴_𝑖⁽¹⁾=^𝑚−𝑖_𝑗=1𝑃_{𝑖+𝑗,𝑚;𝑖+1}𝑥₀;𝑥₁;…;𝑥_𝑖𝐴_𝑖+𝑗+^𝑚_{𝑗=𝑚−𝑖+1}𝑃_{𝑖+𝑗,𝑚;𝑚−𝑗+1}𝑥_{𝑟(𝑖+𝑗)};…;𝑥_{𝑟(𝑖+𝑗)+𝑚−𝑗}𝐴_𝑖+𝑗 (2.16) and, if 𝑖≥𝑚,𝐴_𝑖⁽¹⁾=^𝑚_𝑗=1𝑃_{𝑖+𝑗,𝑚;𝑚−𝑗+1}𝑥_{𝑟(𝑖+𝑗)};𝑥_{𝑟(𝑖+𝑗)+1};…;𝑥_{𝑟(𝑖+𝑗)+𝑚−𝑗}𝐴_𝑖+𝑗, (2.17) is called the derived series of (2.3).

Now, let us denote by ℒ_𝑛(ℝ,𝒜) the complete real algebra of bounded multilinear forms 𝑓∶ℝ^𝑛→𝒜. Then we can consider 𝒜=ℒ(ℝ,𝒜), and hence 𝒜=ℒ_𝑛(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 𝑢_𝑚(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 𝑆^′(𝑥)=^∞_𝑖=0𝐴_𝑖⁽¹⁾𝑢_𝑖(𝑥), (2.18) where 𝐴_𝑖⁽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. If 𝑥∈𝐼, since 𝑢_𝑚(𝑥) is a polynomial and 𝐼 does not contain a local extremum of 𝑢_𝑚(𝑥), then we can choose 𝑥∈𝐼 such that 𝑢_𝑚(𝑥)<𝑢_𝑚(𝑥). (2.19) Since the series (2.3) is convergent at 𝑥, there exists 𝑀>0 such that 𝐴_𝑖𝑢_𝑖(𝑥)<𝑀,𝑖=0,1,…. (2.20) Hence, for 𝑘∈{0,1,…𝑚−1}, 𝐴_𝑖+𝑘𝑢_𝑖(𝑥)=𝐴_𝑖+𝑘𝑢_𝑖+𝑘(𝑥𝑢)^𝑖(𝑥)𝑢_𝑖+𝑘(𝑥)𝑢_𝑖(𝑥)𝑢_𝑖(𝑥)𝑢≤𝑀′^𝑖(𝑥)𝑢_𝑖(𝑥), (2.21) where 𝑀′=𝑀max_{0≤𝑛≤𝑚−1}𝑢_𝑖(𝑥)𝑢_𝑖+𝑛(𝑥). (2.22) Here we remark that, from (2.2), for a fixed 𝑥, 𝑢_𝑖(𝑥)/𝑢_𝑖+𝑛(𝑥) takes only a finite number of values, for 𝑖∈ℕ.
Since (see, e.g., [5], page 121)𝑃_{𝑖,𝑚;𝑡}𝑧₀;𝑧₁;…;𝑧_𝑡−1=^𝑡−1_𝑘=0𝑃_𝑖,𝑚𝑧_𝑘𝑢^′_𝑡𝑧_𝑘,𝑡≤𝑚, (2.23) where we consider 𝑧₀<𝑧₁<⋯<𝑧_𝑚−1 and 𝑢_𝑖(𝑧)=^𝑚−1_𝑘=0𝑧−𝑧_𝑘^{𝑟(𝑖)𝑞(𝑖)−1}_𝑘=0𝑧−𝑧_𝑘, (2.24) by (2.11) and (2.23), for 𝑗=1,2,…,𝑚, we obtain 𝑃_{𝑖+𝑗,𝑚;𝑚−𝑗+1}𝑥_{𝑟(𝑖+𝑗)};𝑥_{𝑟(𝑖+𝑗)+1};…;𝑥_{𝑟(𝑖+𝑗)+𝑚−𝑗}=𝛼_𝑖+𝑗𝑞(𝑖)+𝛽_𝑖+𝑗, (2.25) where 𝛼_𝑖+𝑗=𝛼_𝑖+𝑗(𝑥₀,𝑥₁,…𝑥_𝑚−1),𝛽_𝑖+𝑗=𝛽_𝑖+𝑗(𝑥₀,𝑥₁,…𝑥_𝑚−1) take only a finite number of values. Then there exists 𝐶₁>0 such that, for all 𝑖 and 𝑗, 𝑃_{𝑖+𝑗,𝑚;𝑚−𝑗+1}𝑥_{𝑟(𝑖+𝑗)};𝑥_{𝑟(𝑖+𝑗)+1};…;𝑥_{𝑟(𝑖+𝑗)+𝑚−𝑗}<𝐶₁𝑞(𝑖). (2.26) By (2.2) there exists 𝐶₂=𝐶₂(𝑥,𝑥)>0, independent of 𝑖, such that 𝑢_𝑖(𝑥)𝑢_𝑖(𝑥)<𝐶₂𝑢_𝑚(𝑥)𝑢_𝑚(𝑥)^𝑞(𝑖). (2.27) 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 𝐴_𝑖⁽¹⁾𝑢_𝑖(𝑥)≤𝐶₃𝑞(𝑖𝑢)^𝑚(𝑥)𝑢_𝑚(𝑥)^𝑞(𝑖). (2.28) 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, if 𝑥∈𝐼, because for 𝑥=𝑥_𝑖, 𝑆(𝑥) is a polynomial function and Theorem 2.2 follows easily, then we may suppose that 𝑢_𝑚(𝑥)≠0. Then, by Theorem 2.1, we can choose a compact interval 𝐼_𝑥⊂𝐼 such that 𝑥 belongs to its interior, 𝑥_𝑖∉𝐼_𝑥, for every 𝑖, and the series (2.3) converges absolutely on 𝐼_𝑥. We consider 𝛿>0 small enough such that for every ℎ∈ℝ,|ℎ|≤𝛿, 𝑥+ℎ∈𝐼_𝑥. Then𝑆(𝑥+ℎ)−𝑆(𝑥)ℎ=^∞_𝑖=1𝛼_𝑖(ℎ), (2.29) where 𝛼_𝑖(ℎ)=𝐴_𝑖𝑢_𝑖(𝑥+ℎ)−𝑢_𝑖(𝑥)ℎ=𝐴_𝑖𝑢^′_𝑖𝜉_𝑖, (2.30) where 𝜉_𝑖 is a point lying between 𝑥 and 𝑥+ℎ.
If 𝑦∈𝐼_𝑥, then it is different from 𝑥_𝑖, for every 𝑖, and, by (2.2), 𝑢^′_𝑖𝑦=𝑞(𝑖)𝑢_𝑚^{𝑞(𝑖)−1}𝑦𝑢^′_𝑚𝑦𝑢_𝑟(𝑖)𝑦+𝑢_𝑚^𝑞(𝑖)𝑦𝑢^′_𝑟(𝑖)𝑦=𝑢^′_𝑚𝑦𝑢_𝑚𝑦𝑞(𝑖)𝑢_𝑖𝑦+𝑢^′_𝑟(𝑖)𝑦𝑢_𝑟(𝑖)𝑦𝑢_𝑖𝑦. (2.31) We put 𝑀₁=max_{0≤𝑗≤𝑚}max_{𝑦∈𝐼𝑥}𝑢^′_𝑗𝑦𝑢_𝑗𝑦,𝑀₂=max_{0≤𝑗≤𝑚}max_{𝑦∈𝐼𝑥}𝑢_𝑗𝑦,𝑚₁=min_{0≤𝑗≤𝑚}min_{𝑦∈𝐼𝑥}𝑢^′_𝑗𝑦𝑢_𝑗𝑦,𝑚₂=min_{0≤𝑗≤𝑚}min_{𝑦∈𝐼𝑥}𝑢_𝑗𝑦. (2.32)(2.33) Since, for 𝑦∈𝐼_𝑥, the series ∑^∞_𝑖=0𝐴_𝑖𝑢_𝑖(𝑦) converges absolutely, it follows that the series ∑^∞_𝑖=0‖𝐴_𝑖‖|𝑢_𝑖(𝑦)| converges. Hence, because 𝑚₂>0, we obtain that the series ∑^∞_𝑖=0𝐴_𝑖𝑢_𝑚^𝑞(𝑖)(𝑦) converges absolutely. Similarly, since the series ∑^∞_𝑖=0𝐴_𝑖𝑢^′_𝑖(𝑦) and ∑^∞_𝑖=0𝐴_𝑖(𝑢^′_𝑟(𝑖)(𝑦)/𝑢_𝑟(𝑖)(𝑦))𝑢_𝑖(𝑦) converge absolutely, from (2.31), it follows that the series ∑^∞_𝑖=0𝐴_𝑖(𝑢^′_𝑚(𝑦)/𝑢_𝑚(𝑦))𝑞(𝑖)𝑢_𝑖(𝑦) converges absolutely. Then, by (2.2), the series ∑^∞_𝑖=0‖𝐴_𝑖‖𝑞(𝑖)|𝑢_𝑚^𝑞(𝑖)(𝑦)||𝑢_𝑟(𝑖)(𝑦)| converges, and thus ∑^∞_𝑖=0𝐴_𝑖𝑞(𝑖)𝑢_𝑚^𝑞(𝑖)(𝑦) converges absolutely.
We consider 𝑥₀∈𝐼_𝑥 such that𝑢_𝑚𝑥₀=max_{𝑦∈𝐼𝑥}𝑢_𝑚𝑦, (2.34) and we put 𝑀₃=max_{0≤𝑗≤𝑚}max_{𝑦∈𝐼𝑥}𝑢^′_𝑗𝑦. (2.35) Then, from (2.31)–(2.35), for every 𝑖, 𝑢^′𝜉_𝑖≤𝑞(𝑖)𝑀₂𝑀₃𝑢_𝑚^{𝑞(𝑖)−1}𝑥₀+𝑀₃𝑢_𝑚^𝑞(𝑖)𝑥₀≤𝑀₂𝑀₃𝑚₂𝑞(𝑖)𝑢_𝑚^𝑞(𝑖)𝑥₀+𝑀₃𝑢_𝑚^𝑞(𝑖)𝑥₀. (2.36) Hence, the series ∑^∞_𝑖=0𝐴_𝑖𝑢′(𝜉_𝑖)=∑^∞_𝑖=1𝛼_𝑖(ℎ) converges absolutely and uniformly on [−𝛿,𝛿]. If we put 𝛼_𝑖(0)=𝐴_𝑖𝑢^′_𝑖(𝑥), then it follows that the functions 𝛼_𝑖(ℎ) are continuous on [−𝛿,𝛿]. Thus, the sum of the series (2.29) is continuous and lim_ℎ→0𝑆(𝑥+ℎ)−𝑆(𝑥)ℎ=^∞_𝑖=1𝛼_𝑖(0)=^∞_𝑖=0𝐴_𝑖𝑢^′_𝑖(𝑥). (2.37) Hence, 𝑆(𝑥+ℎ)−𝑆(𝑥)=^∞_𝑖=0𝐴_𝑖𝑢^′_𝑖(𝑥)ℎ+𝑜(ℎ), (2.38) and this completes the proof of the theorem.

3. Representable Functions into Newton Interpolating Series at 𝑚 Distinct Points

If 𝐼⊂ℝ is an interval which contains 𝑥₀,𝑥₁,…,𝑥_𝑚−1 such that 𝑥₀<𝑥₁<⋯<𝑥_𝑚−1, 𝑚≥2, then we consider a function 𝐹∶𝐼→𝒜. We say that the function 𝐹 is representable into a Newton interpolating series at 𝑥₀,𝑥₁,…,𝑥_𝑚−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.

When 𝑥_𝑗=𝑗+1, 𝐼=𝒜=ℝ, and 𝐹(𝑥)=𝑒^𝛼𝑥, it is known that the coefficients 𝐴_𝑖 and 𝐹 are related through the complex integral formula (see [1], Chapter 2, Section 3) 𝐴_𝑖=12𝜋√−1^|𝑧|=𝑖𝐹(𝑧)𝑢_𝑖+1(𝑧)𝑑𝑧. (3.1)

If 𝐹 is representable into a Newton interpolating series at 𝑥₀,𝑥₁,…,𝑥_𝑚−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 𝐴_𝑖.

Theorem 3.1. Suppose that 𝐹∶𝐼→𝒜 is a function which is representable into a Newton interpolating series Σ of the form (2.3) at 𝑥₀,𝑥₁,…,𝑥_𝑚−1∈𝐼. Then, for every 𝑛, there exists 𝐽_𝑛=⋃^𝑚−1_𝑖=0𝑉_𝑖,𝑛⊂𝐼, where, for every 𝑖, 𝑉_𝑖,𝑛 is an open interval which contains 𝑥_𝑖, 𝐹 has Fréchet derivative of order 𝑛 on 𝑉_𝑖,𝑛, the series Σ is uniquely defined, and for each 𝑖𝐴_𝑖=1𝑞(𝑖)!^𝑟(𝑖)_𝑗=0𝐹(𝑥)∏^𝑟(𝑖)_{𝑘=0,𝑘≠𝑗}𝑥−𝑥_𝑘∏^𝑚−1_{𝑘=0,𝑘≠𝑗}𝑥−𝑥_𝑘^{𝑞(𝑖)(𝑞(𝑖))}_{𝑥=𝑥𝑗}+1𝑞(𝑖)−1!^𝑚−1_{(𝑖)𝑗=𝑟+1}𝐹(𝑥)∏^𝑟(𝑖)_𝑘=0𝑥−𝑥_𝑘∏^𝑚−1_{𝑘=0,𝑘≠𝑗}𝑥−𝑥_𝑘^{𝑞(𝑖)(𝑞(𝑖)−1)}_{𝑥=𝑥𝑗}, (3.2) where (𝐹(𝑥))⁽⁻¹⁾ and 1/(−1)! mean 0.

If 𝐹∶𝐼→𝒜 is representable into a Newton interpolating series at 𝑥₀,𝑥₁,…,𝑥_𝑚−1∈𝐼, then the series of the form (2.3), where 𝐴_𝑖 are given in (3.2), is called the Newton interpolating series at 𝑥₀,𝑥₁,…,𝑥_𝑚−1 of the function F.

Given a Newton interpolating series of the form (2.3), we put 𝑁_𝑛(𝑥)=^𝑛_𝑖=0𝐴_𝑖𝑢_𝑖(𝑥). (3.3)

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 algebra 𝒜 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 𝐹(𝑏)−𝐹(𝑎)=𝐹′(𝑐)(𝑏−𝑎). (3.4)

Proof. If 𝑥∈[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 𝑔_ℎ1∶[𝑎,𝑏]→ℝ, defined by 𝑔_ℎ1(𝑥)=𝜑(𝑥,ℎ₁). Then, by Lagrange's theorem, there exists 𝑐∈(𝑎,𝑏) such that 𝑔_ℎ1(𝑏)−𝑔_ℎ1(𝑎)=𝑔^′_ℎ1(𝑐)(𝑏−𝑎). Hence, the lemma is proved.

Corollary 3.3. Suppose that 𝐹∶[𝑥₀,𝑥_𝑚−1]→𝒜 has Fréchet derivatives of all orders on [𝑥₀,𝑥_𝑚−1] and it is representable into a Newton interpolating series at 𝑥₀,𝑥₁,…,𝑥_𝑚−1. If its Newton interpolating series (2.3) has the partial sums given in (3.3), then, for a fixed 𝑛, the polynomial 𝑁_𝑛(𝑥) satisfies 𝑁^(𝛼𝑗)_𝑛𝑥_𝑗=𝐹^(𝛼𝑗)𝑥_𝑗, (3.5) where 0≤𝑗≤𝑚−1 and 𝛼_𝑗∈0,1,…,𝑞(𝑛),if(𝑛),(𝑛),𝑗≤𝑟0,1,…,𝑞−1if𝑗≥𝑟(𝑛)+1. (3.6) Furthermore, if 𝒜 is the dual algebra of a real Banach algebra 𝒜₁, then, for every 𝑥∈[𝑥₀,𝑥_𝑚−1], there exists 𝜉=𝜉(𝑥,𝑛)∈(𝑥₀,𝑥_𝑚−1) such that 𝐹(𝑥)=𝑁_𝑛(𝑥)+𝐹^(𝑛+1)(𝜉)()!𝑢𝑛+1^𝑛+1(𝑥). (3.7)

Proof. It follows immediately that every polynomial function is representable into a finite Newton interpolating series at 𝑥₀,𝑥₁,…,𝑥_𝑚−1. Then the coefficients 𝐴_𝑖,𝑖≤𝑛, are given in (3.2), where 𝐹 is replaced by 𝑁_𝑛. Since 𝑁_𝑛(𝑥) are the partial sums of the Newton interpolating series at 𝑥₀,𝑥₁,…,𝑥_𝑚−1 of the function 𝐹, it follows that 𝐴_𝑖=1𝑞(𝑖)!^𝑟(𝑖)_𝑗=0𝐹(𝑥)∏^𝑟(𝑖)_{𝑘=0,𝑘≠𝑗}𝑥−𝑥_𝑘∏^𝑚−1_{𝑘=0,𝑘≠𝑗}𝑥−𝑥_𝑘^{𝑞(𝑖)(𝑞(𝑖))}_{𝑥=𝑥𝑗}+1𝑞(𝑖)−1!^𝑚−1_{(𝑖)𝑗=𝑟+1}𝐹(𝑥)∏^𝑟(𝑖)_𝑘=0𝑥−𝑥_𝑘∏^𝑚−1_{𝑘=0,𝑘≠𝑗}𝑥−𝑥_𝑘^{𝑞(𝑖)(𝑞(𝑖)−1)}_{𝑥=𝑥𝑗}=1𝑞(𝑖)!^𝑟(𝑖)_𝑗=0𝑁_𝑛(𝑥)∏^𝑟(𝑖)_{𝑘=0,𝑘≠𝑗}𝑥−𝑥_𝑘∏^𝑚−1_{𝑘=0,𝑘≠𝑗}𝑥−𝑥_𝑘^{𝑞(𝑖)(𝑞(𝑖))}_{𝑥=𝑥𝑗}+1𝑞(𝑖)−1!^𝑚−1_{(𝑖)𝑗=𝑟+1}𝑁_𝑛(𝑥)∏^𝑟(𝑖)_𝑘=0𝑥−𝑥_𝑘∏^𝑚−1_{𝑘=0,𝑘≠𝑗}𝑥−𝑥_𝑘^{𝑞(𝑖)(𝑞(𝑖)−1)}_{𝑥=𝑥𝑗}. (3.8) 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)−𝑁_𝑛(𝑥)+𝐾𝑢_𝑛+1(𝑥), where 𝐾 is a constant from 𝒜.

Theorem 3.4. One considers 𝒜 and 𝒜₁ two Banach real algebras such that 𝒜=𝒜₁^∗. One supposes that 𝐹∶[𝑥₀,𝑥_𝑚−1]→𝒜 has Fréchet derivatives of all orders on [𝑥₀,𝑥_𝑚−1]. If 𝑀_𝑠=sup_{𝑥∈[𝑥0,𝑥𝑚−1]}‖𝐹^(𝑠)(𝑥)‖ and the sequence {𝑧_𝑠} given by 𝑧_𝑠=𝑥_𝑚−1−𝑥₀^𝑠𝑀_𝑠𝑠!,𝑠=0,1,…, (3.9) converges to zero, then 𝐹 is the sum of its Newton interpolating series at 𝑥₀,𝑥₁,…,𝑥_𝑚−1 on [𝑥₀,𝑥_𝑚−1].

Proof. From (3.7) it follows that 𝐹(𝑥)−𝑁_𝑛(𝑥≤𝑀)^𝑛+1()!𝑢𝑛+1^𝑛+1(𝑥), (3.10) and by (2.2) we obtain that max_{𝑥𝑥∈0,𝑥𝑚−1}𝑢_𝑛+1(𝑥)≤𝑥_𝑚−1−𝑥₀^𝑛+1. (3.11) Now (3.10) and (3.11) imply 𝐹(𝑥)−𝑁_𝑛(𝑥≤𝑥)^𝑚−1−𝑥₀^𝑛+1𝑀_𝑛+1()!.𝑛+1 (3.12) Hence, 𝑁_𝑛(𝑥) converges uniformly to 𝐹 on [𝑥₀,𝑥_𝑚−1].

In the last theorem of this section we study the algebra of functions which are the sums of their Newton interpolating series at 𝑥₀,𝑥₁,…,𝑥_𝑚−1.

If 𝒜 is a Banach real algebra, then we denote by 𝒩(𝑥₀,𝑥₁,…,𝑥_𝑚−1) the set of all functions 𝐹∶[𝑥₀,𝑥_𝑚−1]→𝒜 which are representable into a Newton interpolating series at 𝑥₀,𝑥₁,…,𝑥_𝑚−1. Thus, by Theorem 3.1, every 𝐹∈𝒩(𝑥₀,𝑥₁,…,𝑥_𝑚−1) is the sum of its Newton interpolating series at 𝑥₀,𝑥₁,…,𝑥_𝑚−1. We take 𝐹(𝑥)=^∞_𝑖=0𝐴_𝑖𝑢_𝑖(𝑥),𝐺(𝑥)=^∞_𝑖=0𝐵_𝑖𝑢_𝑖(𝑥) (3.13) two elements of 𝒩(𝑥₀,𝑥₁,…,𝑥_𝑚−1). Our objective is to define the addition and the multiplication of 𝐹 and 𝐺. We begin with a lemma.

Lemma 3.5. If 𝑢_𝑖 are given in (2.1), then, for every 𝑖,𝑗∈{1,2,…,𝑚}, there exist nonnegative real numbers 𝐶_𝑘(𝑖,𝑗), 𝑘∈{max{𝑖,𝑗},max{𝑖,𝑗}+1,…,𝑖+𝑗}, uniquely defined, such that 𝑢_𝑖𝑢_𝑗=^𝑖+𝑗_{𝑘=max{𝑖,𝑗}}𝐶_𝑘𝑖,𝑗𝑢_𝑘. (3.14)

Proof. If 𝑗≤𝑖<𝑚, then it follows that 𝑢_𝑖𝑢_𝑗=𝑢_𝑖𝑢_𝑗−1𝑥−𝑥_𝑗−1=𝑢_𝑖+1𝑢_𝑗−1+𝑥_𝑖−𝑥_𝑗−1𝑢_𝑖𝑢_𝑗−1, (3.15) and if 𝑖=𝑚, then 𝑢_𝑖𝑢_𝑗=𝑢_𝑖+𝑗. Now, for every 𝑖, the induction on 𝑗 completes the proof of the lemma.

We define addition and multiplication of 𝐹 and 𝐺 as follows: (𝐹+𝐺)(𝑥)=^∞_𝑖=0𝐴_𝑖+𝐵_𝑖𝑢_𝑖(𝑥),(𝐹𝐺)(𝑥)=^∞_𝑖=0𝑃_𝑖𝑢_𝑖(𝑥), (3.16) where 𝑃_𝑖=_{(𝑖)(𝛼,𝛽)∈𝐷}𝐶_{𝑖−(𝑞(𝛼)+𝑞(𝛽))𝑚}𝑟(𝛼),𝑟𝛽𝑎_𝛼𝑏_𝛽 (3.17)𝐷(𝑖)={(𝛼,𝛽)∈ℕ²∶max{𝛼+𝑞(𝛽)𝑚,𝛽+𝑞(𝛼)𝑚}≤𝑖≤𝛼+𝛽} and 𝐶_𝑘(𝑠,𝑡) are given in (3.14).

Let 𝑥∈[𝑥₀,𝑥_𝑚−1] be such that𝑢_𝑚(𝑥)=sup_{𝑥𝑥∈0,𝑥𝑚−1}𝑢_𝑚(𝑥). (3.18) Since 𝑢_𝑖(𝑥)>0, for every 𝑥>𝑥_𝑚−1, and the roots of 𝑢^′_𝑖(𝑥) belong to [𝑥₀,𝑥_𝑚−1], there exists 𝑧>𝑥_𝑚−1, uniquely determined such that𝑢_𝑚(𝑧)=𝑢_𝑚(𝑥). (3.19) We put 𝐾₁=min_{0≤𝑖≤𝑚−1}𝑢_𝑖(𝑥,𝐾₂)=max_{0≤𝑖≤𝑚−1}𝑢_𝑖(𝑧). (3.20)

Then, by (2.2),𝑢_𝑖(𝑧)=𝑢_𝑚^𝑞(𝑖)(𝑧)𝑢_𝑟(𝑖)(𝑧)=𝑢_𝑚^𝑞(𝑖)(𝑥)𝑢_𝑟(𝑖)(𝑥𝑢)^𝑟(𝑖)(𝑧)𝑢_𝑟(𝑖)(𝑥)≤𝐾₂𝐾₁𝑢_𝑖(𝑥). (3.21) If 𝐹(𝑥)=∑^∞_𝑖=0𝐴_𝑖𝑢_𝑖(𝑥)∈𝒩(𝑥₀,…,𝑥_𝑚−1), then the series ∑^∞_𝑖=0𝐴_𝑖𝑢_𝑖(𝑥) converges absolutely and, by (3.21), the series ∑^∞_𝑖=0‖𝐴_𝑖‖𝑢_𝑖(𝑧) converges. We define‖𝐹‖_𝒩=^∞_𝑖=0𝐴_𝑖𝑢_𝑖(𝑧). (3.22)

Theorem 3.6. If the addition and the multiplication are defined by (3.16), then 𝒩(𝑥₀,…,𝑥_𝑚−1) becomes a real algebra. Furthermore, ‖‖_𝒩 defined by (3.22) is a norm, and 𝒩(𝑥₀,…,𝑥_𝑚−1) becomes a real Banach algebra with respect to ‖‖_𝒩.

Proof. If 𝐹,𝐺∈𝒩(𝑥₀,…,𝑥_𝑚−1), then their Newton interpolating series at 𝑥₀,𝑥₁,…,𝑥_𝑚−1 converge absolutely on [𝑥₀,𝑥_𝑚−1] and converge uniformly to 𝐹 and 𝐺, respectively. By (3.16) it follows easily that the Newton interpolating series at 𝑥₀,𝑥₁,…,𝑥_𝑚−1 of 𝐹+𝐺 and 𝐹𝐺 have the same properties. Thus, 𝒩(𝑥₀,…,𝑥_𝑚−1) is a real algebra.Since 𝑢_𝑖𝑢_𝑗𝒩=^𝑖+𝑗_𝑘=𝑖𝐶_𝑘𝑖,𝑗𝑢_𝑘𝒩=^𝑖+𝑗_𝑘=𝑖𝐶_𝑘𝑖,𝑗𝑢_𝑘(𝑧)=𝑢_𝑖𝒩𝑢_𝑗𝒩, (3.23) for every 𝑖,𝑗∈{1,2,…,𝑚}, 𝑖≥𝑗, it follows that ‖𝐹𝐺‖_𝒩≤‖𝐹‖_𝒩‖𝐺‖_𝒩 and 𝒩(𝑥₀,…,𝑥_𝑚−1) is a normed algebra.
Now, we will prove that 𝒩(𝑥₀,…,𝑥_𝑚−1) is a real Banach algebra. Consider {𝐹_𝑛=∑^∞_𝑗=0𝐴_𝑗,𝑛𝑢_𝑗}_𝑛∈ℕ a Cauchy sequence of 𝒩(𝑥₀,…,𝑥_𝑚−1). Let 𝜖>0 be arbitrary. Then there exists 𝑁(𝜖)∈ℕ such that, for all 𝑛≥𝑁(𝜖) and 𝑝=1,2,…, ‖𝐹_𝑛+𝑝−𝐹_𝑛‖_𝒩<𝜖. From (3.22), we obtain^∞_𝑗=0𝐴_{𝑗,𝑛+𝑝}−𝐴_𝑗,𝑛𝑢_𝑗(𝑧)<𝜖. (3.24) Thus, for each fixed 𝑗, 𝑛≥𝑁(𝜖) and 𝑝=1,2,…, ‖𝐴_{𝑗,𝑛+𝑝}−𝐴_𝑗,𝑛‖𝑢_𝑗(𝑧)<𝜖. Hence, {𝐴_𝑗,𝑛}_𝑛∈ℕ is a Cauchy sequence in 𝒜 and, since 𝒜 is a complete algebra, this sequence converges. We set 𝐴_𝑗=lim_𝑛→∞𝐴_𝑗,𝑛, and we consider the formal series 𝐹=^∞_𝑗=0𝐴_𝑗𝑢_𝑗. (3.25) We have to show that 𝐹∈𝒩(𝑥₀,𝑥₁,…,𝑥_𝑚−1) and lim_𝑛→∞𝐹_𝑛=𝐹. From (3.24), ^𝑠_𝑗=0𝐴_{𝑗,𝑛+𝑝}−𝐴_𝑗,𝑛𝑢_𝑗(𝑧)<𝜖. (3.26) Fix now some 𝑛≥𝑁(𝜖) and let 𝑝→∞ and then 𝑠→∞. Hence, for all 𝑛≥𝑁(𝜖), we obtain ^∞_𝑗=0𝐴_𝑗−𝐴_𝑗,𝑛𝑢_𝑗(𝑧)≤𝜖. (3.27) Since, for 𝑥∈[𝑥₀,𝑥_𝑚−1], 𝑛₀≥𝑁(𝜖), 𝑝′=0,1,…, ‖∑^𝑖+𝑝′_𝑗=𝑖𝐴_𝑗𝑢_𝑗(𝑥)‖≤∑^∞_𝑗=𝑖‖𝐴_𝑗−𝐴_𝑗,𝑛0‖𝑢_𝑗(𝑧)+∑^∞_𝑗=𝑖‖𝐴_𝑗,𝑛0‖𝑢_𝑗(𝑧) and 𝐹_𝑛0∈𝒩(𝑥₀,…,𝑥_𝑚−1), from (3.27), it follows that 𝐹 is a function defined on [𝑥₀,𝑥_𝑚−1] and it is representable into the Newton interpolation series (3.25). Thus, 𝐹∈𝒩(𝑥₀,…, 𝑥_𝑚−1). Finally, from (3.27), ‖𝐹−𝐹_𝑛‖_𝒩≤𝜖 and lim_𝑛→∞𝐹_𝑛=𝐹.

Remark 3.7. Let 𝒜[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,…,𝑥_𝑚−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,…,𝑥_𝑚−1). Hence, it follows that if 𝐹∶𝐼→ℝ, where 𝐼 is an open interval which contains 𝑥_𝑖, for every 𝑖, and its Taylor series at 𝑥₀ converges on 𝐼, then the restriction of 𝐹 to [𝑥₀,𝑥_𝑚−1] belongs to 𝒩(𝑥₀,…,𝑥_𝑚−1). However, 𝒩(𝑥₀,…,𝑥_𝑚−1) contains also functions which are not Fréchet differentiable on [𝑥₀,𝑥_𝑚−1]. For example, we take 𝒜=ℝ, 𝑚=2, 𝑥₀=−1, 𝑥₁=1 and 𝐹(𝑥)=|𝑥|. Then, 𝐹(𝑥)=1−1−𝑥²=1+^∞_𝑘=1(−1)^𝑘+1∏^𝑘−2_𝑖=1(𝑘+𝑖)(𝑘−1)!2^2𝑘−1𝑥²−1^𝑘, (3.28) for 𝑥∈[−1,1], which implies that 𝐹∈𝒩(−1,1).