Error Inequalities in Polynomial Interpolation and Their Applications

In the year 1929 Lidstone [15] introduced a generalization of Taylor’s series, it approximates a given function in the neighborhood of two points instead of one. From the practical point of view such a development is very useful; and in terms of completely continuous functions it has been characterized in the work of Boas [9], Poritsky [19], Schoenberg [20], Whittaker [28, 29], Widder [30, 31], and others. In the field of approximation theory [12,27] the Lidstone interpolating polynomial P _(1.1.1)(t) of degree (2m — 1) satisfies the Lidstone conditions

Let —∞ < a< b< ∞, and a ≤ a₁ < a₂ < … < a_r ≤ b (r ≥ 2) be the given points. It is well known that the Hermite interpolating polynomial P _(2.1.1)(t) of degree (n − 1) (n ≥r) satisfying the Hermite conditions exists uniquely [10,22].

Let —∞ < a < b < ∞, and a ≤ a ₁ ≤ a ₂ ≤ … ≤ a _n ≤ b be the given points. It is easily seen that the Abel 7#x2014; Gontscharoff interpolating polynomial P _(3.1.1)(t) of degree (n − 1) satisfying the Abel — Gontscharoff conditions exists uniquely [10,15].

Results of Chapters 1 – 3 are used here to obtain the best possible / sharp error bounds for the derivatives of several other interpolating polynomials. Some of these interpolating polynomials satisfy : (i) (n,p) and (p,n) conditions, which arise in determining the intervals of nonoscillation for the linear differential equations, (ii) particular cases of two — point Birkhoff’s conditions, (iii) two — point Abel — Gontscharoff — Hermite conditions, and (iv) two — point Abel — Gontscharoff — Lidstone conditions.

Although polynomials have several attractive features, polynomial interpolation of a given function often has the drawback of producing approximations that may be wildly oscillatory. To overcome this difficulty we divide the interval of interest into small subintervals and in each subinterval consider polynomials of relatively low degree and finally ‘piece together’ these polynomials. This subject has steadily developed over the past fifty years, and at present there are thousands of research papers on piecewise — polynomial interpolation and their applications. The plan of this chapter is as follows: In Section5.2 we collect some results from analysis which will be used repeatedly throughout the remaining monograph. In Section5.3 we shall follow the general treatment of Birkhoff, Ciarlet, Schultz and Varga [8,10,11] to provide an explicit representation of piecewise - Hermite interpolates. This representation is then used to obtain error bounds for the derivatives of piecewise — Hermite interpolates in L _∞ and L ₂ — norms. These bounds improve on those given by Schultz [21], and extend to cases not considered by him. Section5.4 contains an explicit representation of piecewise — Lidstone interpolates and the error bounds for its derivatives in L _∞, and L ₂ — norms. Results of Sections5.3 and 5.4 are extended to two variables in Sections5.5 and 5.6, respectively.

Spline interpolation is an improvement over piecewise — polynomial interpolation. It uses less information of the given function, yet furnishes smoother interpolates. The plan of this chapter is as follows: In Section6.2 we shall define the spline space S _m,τ (Δ), and for a given function x (t) the spline and Lidstone — spline interpolates S _m,τ ^Δ and LS _m,2m−2 ^Δ x(t),respectively. Here, we shall also show that to acquire the bounds for ‖ D ^k(x− S _m,τ ^Δ ) ‖∞ and ‖ D ^k(x− LS _m,2m−2 ^Δ x) ‖∞ in terms of the derivatives of x(t) it is necessary to estimate several terms. While some of these terms can be estimated by the results of Chapter 5, other terms which require a different analysis for each m and τ, need to be bounded. In Sections6.3,6.4 and 6.6 respectively, we shall consider the cases m = 2, τ = 2; m = 3, τ = 4 and m = 3, τ = 3. These cases correspond to cubic in the class C ⁽²⁾ [a, b], and quintic in the classes C ⁽⁴⁾[a, b] and C ⁽³⁾[a, b] spline interpolates. For the cases m = 3, τ = 4 and m = 3, τ = 3 in Sections 6.5 and 6.7 respectively, we shall discuss the construction of approximated quintic splines, and for these interpolates we will provide explicit error bounds in L _∞ — norm. For the cubic and quintic Lidstone — spline interpolates error bounds in L _∞ — norm are obtained in Sections 6.8 and 6.9, respectively. In Section6.10 we shall extend Theorems 5.3.16 and 5.3.17 for the spline interpolate S _m,τ ^Δ x(t)