On Interpolation Approximation: Convergence rates for polynomial interpolation for functions of limited regularity^{†}^{†}thanks: This work was supported by National Science Foundation of China (No. 11371376).
Abstract
The convergence rates on polynomial interpolation in most cases are estimated by Lebesgue constants. These estimates may be overestimated for some special points of sets for functions of limited regularities. In this paper, by applying the Peano kernel theorem and Wainerman’s lemma, new formulas on the convergence rates are considered. Based upon these new estimates, it shows that the interpolation at strongly normal pointsystems can achieve the optimal convergence rate, the same as the best polynomial approximation. Furthermore, by using the asymptotics on Jacobi polynomials, the convergence rates are established for GaussJacobi, JacobiGaussLobatto or JacobiGaussRadau pointsystems. From these results, we see that the interpolations at the GaussLegendre, LegendreGaussLobatto pointsystem, or at strongly normal pointsystems, has essentially the same approximation accuracy compared with those at the two Chebyshev piontsystems, which also illustrates the equally accuracy of the Gauss and ClenshawCurtis quadrature. In addition, numerical examples illustrate the perfect coincidence with the estimates, which means the convergence rates are optimal.
olynomial interpolation, Peano kernel, convergence rate, limited regularity, strongly normal pointsystem, GaussJacobi point, JacobiGaussLobatto point, Chebyshev point.
65D05, 65D25
1 Introduction
A central problem in approximation theory is the construction of simple functions that are easily implemented on computers and approximate well a given set of functions.
There exist many investigations for the behavior of continuous functions approximated by polynomials. Weierstrass [73] in 1885 proved the well known result that every continuous function in can be uniformly approximated as closely as desired by a polynomial function. This result has both practical and theoretical relevance, especially in polynomial interpolation.
Polynomial interpolation is a fundamental tool in many areas of scientific computing. Lagrange interpolation is a classical technique for approximation of continuous functions. Let us denote by
(1) 
the distinct points in the interval and let be a function defined in the same interval. The th Lagrange interpolation polynomial of is unique and given by the formula
(2) 
where .
There is a well developed theory that quantifies the convergence or divergence of the Lagrange interpolation polynomials (Brutman [7, 8] and Trefethen [59]). Two key notions for interpolation in a given set of points are that of the Lebesgue function
(3) 
and Lebesgue constant
(4) 
which are of fundamental importance (Cheney [9], Davis [12] and Szegö [55]). The Lebesgue constant can also be interpreted as the norm of the projection operator
where is the set of polynomials of degree less than or equal to .
Based upon the Lebesgue constant, the interpolation error can be estimated by
(5) 
where is the best polynomial approximation of degree . Thus, the Lebesgue constant indicates how good the interpolant is in comparison with the best polynomial approximation .
The study of the Lebesgue constant originated more than years ago. Comprehensive reviews can be found in Brutman [8], Lubinsky [41], Trefethen [59, Chapter 15], etc. For an arbitrarily given system of points , Bernstein [2] and Faber [18] in 1914 obtained that
which, together with the boundedness principle, implies that there exists a continuous function in for which the sequence () is not uniformly convergent to in ^{1}^{1}1Grünwald [24] in 1935 and Marcinkiewicz [43] in 1937, independently, showed that even for the Chebyshev points of first kind there is a continuous function in for which the sequence is divergent everywhere in .. More precisely, Erdös [15] and Brutman [7] proved that
(6) 
where is the Euler’s constant. In particular, for equidistant pointsystem
Schönhage [52] showed that
Additionally, Trefethen and Weideman [61] established that
Then generally, the set of equally spaced points is a bad choice for Lagrange interpolation (see Runge [49]).
Whereas, for well chosen sets of points, the growth of may be extremely slow as :

The roots of Jacobi polynomial (): The asymptotic estimate of was found by Szegö [55] as
(9)
Comparing Equations (1.7), (1.8) and (1.9) with (1.6), we see that the two Chebyshev pointsystems and the Jacobi pointsystem with are nearly optimal and of order .
Nevertheless, it is worth noting that if has an absolutely continuous st derivative on for some and its th derivative is of bounded variation , Mastroianni and Szabados [42], Trefethen [59] and Xiang et al. [76] proved that
(10) 
where is at the Chebyshev points of first or second kind, which has the same asymptotic order as for the best approximation , following de la Vallée Poussin [62]. In particular, for , the error on the at the above two Chebyshev pointsystems satisfies
(see Bernstein [3] and Varga and Capenter [63]). Thus, the error estimate (1.5) by using the Lebesgue constant may be overestimated for some special points of sets for functions of limited regularities.
Moreover, it has been observed, by ClenshawCurtis [10] and O’Hara and Smith [29], that point Gauss quadrature and point ClenshawCurtis quadrature have essentially the same accuracy, which has been showed recently by Trefethen [58, 59], Brass and Petras [6] and Xiang and Bornemann [75]. Both of these two quadrature are derived from the interpolation polynomial by
based on the GaussLegender and ClenshawCurtis points, respectively. From this observation, we may conclude that the corresponding interpolation based on these two pointsystems may have the same convergence rate. However, it can not be derived from (1.5).
In this paper, we present new convergence rates of the interpolation polynomials for functions of limited regularities, based upon the famous Peano kernel theorem [45] and applying an interesting Wainerman’s lemma [72]. Suppose has an absolutely continuous st derivative on , and its th derivative is of bounded variation . We will show that
(11) 
which leads to
(12) 
The Lebesgue constant is replaced by in some sense since [62].
Particularly, from (1.12), it directly follows that the interpolation at a strongly normal pointsystem (see Fejér [19]) can achieve the optimal convergence rate as .
Furthermore, can be explicitly estimated for GaussJacobi, JacobiGaussLobatto or JacobiGaussRadau pointsystems, by using the asymptotics on Jacobi polynomials given by Szegö [55] and some results given in Kelzon [34, 35], Vértesi [66, 68], Sun [54], Prestin [46], Kvernadze [38], Vecchia et al. [69], etc., as follows

For the GaussJacobi points:

For the JacobiGaussLobatto points (the roots of ):

For the JacobiGaussRadau points

For the JacobiGaussRadau points
From the above estimates, we see that the interpolation at the GaussLegendre or at the LegendreGaussLobatto pointsystem, has essentially the same approximation accuracy compared with those at the two Chebyshev piontsystems. All of them satisfy that (for more general cases see Fig. 1.1). In addition, the convergence rate is attainable illustrated by some functions of limited regularities.
Thus, the best approximation polynomial is challenged by the interpolation polynomials at the special pointsystems showed in Fig. 1.1. Furthermore, we will see that the interpolation polynomials at the special pointsystems perform much better than the best approximation polynomial for approximation the derivatives and by , , and , respectively, illustrated by numerical examples in the final section.
It is worthy of special mention that the interpolation polynomial , at the GaussJacobi, JacobiGaussLobatto or GaussJacobiRadau pointsystem, can be efficiently evaluated by applying the second barycentric formula
which is robust in the presence of rounding errors [33] and costs overall computational complexity [4], where the nodes and the barycentric weights are computed by jacpts and the formulas given in [28, 70, 71], respectively. A Matlab routine jacpts, which uses the algorithm in [26] for the computation of these nodes and weights, can be found in Chebfun system [60]. For more details on this topic, see Salzer [50], Henrich [30], Berrut and Trefethen [4], Higham [32, 33], Glaser et al. [23], Wang and Xiang [70], Bogaert et al. [5], Hale and Trefethen [28], Hale and Townsend [26], Trefethen [59], Wang et. al. [71], etc. Matlab routines can be found in Chebfun system [60] and Xiang and He [77].
The paper is organized as follows: In section 2, we present the error of for each fixed by using the Peano representation and the bounded variation. In section 3, we introduce the interesting Wainerman’s lemma and deduce the error bound on by . We consider, in section 4, the estimates of and derive the convergence rates for the interpolation polynomial at strongly normal pointsystems, GaussJacobi, JacobiGaussLobatto and JacobiGaussRadau pointsystems, respectively, where the convergence rates and attainability are illustrated by numerical experiments.
Throughout this paper, means that there exist positive constants and such that
For simplicity, in the following we abbreviate as and as .
All the numerical results in this paper are carried out by using Matlab R2012a on a desktop (2.8 GB RAM, 2 Core2 (32 bit) processors at 2.80 GHz) with Windows XP operating system.
2 The Peano kernel theorem
There are two general methods for deriving strict error bounds (Dahlquist and Björck [11]). One applies the norms and distance formula together with the Lebesgue constants, which often overestimates the error. The other is due to the Peano kernel theorem.
Suppose a continuously linear functional that maps functions to satisfying for any and for any scalar . In addition, we assume for some , where denotes the set of polynomials with degree less than or equal to .
The Peano kernel theorem (Peano [45], see also Kowalewski [36], Schmidt [51] and Mises [44]) is the identity
(13) 
holding for all such functions , where and
(14) 
For each fixed , we consider the special functional , where is defined for by
with . is a continuously linear functional since for arbitrary , and then by the Peano theorem [45] can be represented if for as
(15) 
with
(16) 
Particularly, from (2.3) it implies
Similar to the Peano kernel for quadrature [6], the kernel for interpolation satisfies the following proposition.
(Peano representation) Let
(17) 
Then for , the Peano kernel satisfies and can be rewritten as
(18) 
From the definition of in (2.5), it is easy to verify that by using for . Furthermore, we find that
(19) 
Define and and suppose for some nonnegative integer . By (2.7), similarly we have
(20) 
Then from
we get by (2.7) and (2.8).
In the following, we consider functions of limited regularities as
(21) 
From Stein and Shakarchi [53, p. 130] and Tao [56, pp. 143145], we see that a function is absolutely continuous if and only if it takes the form for some absolutely integrable and a constant . It is obvious that such is not unique. Then in this paper, we suppose satisfies (2.9) and define
Remark 1. Here, we use the condition “, where is absolutely integrable and of bounded variation ” instead of “ is of bounded variation ” in [58, 59]. If is of bounded variation, then exists almost everywhere and (see Lang [39] and Rudin [48]). Whereas, in [58, 59] denotes an equivalent representation in the sense of almost everywhere. An example for is given in [58, 59], where is not differentiable at , but can be chosen as
then . Using the new condition, we see that can be represented as with and is unique.
Suppose satisfies (2.9), then for , we have
(22) 
Applying the Peano theorem implies that for each fixed ,
Then, directly following Brass and Petras [6], integrating by parts and using yields
Since can be written as with and are monotonically increasing, and (see Lang [39, pp. 280281]). Without loss of generality, assume is monotonically increasing. Then by the second mean value theorem of integral calculus, it follows from that there exists a such that
which leads to the desired result.
(23) 
In Lemma 2.3, letting , , representing as , and noting that , by Theorem 2.2 we have
Consequently, by Lemma 2.3 we get that
From Theorem 2.2 and Lemma 2.4 we obtain that {theorem} Suppose satisfies (2.9), then for
(24) 
3 Wainerman’s lemma
In the following, we shall focus on the estimate of .
Notice that for and
(25) 
If , we have for , and for . While for , we have for , and for . Thus, in these cases we obtain
(26) 
since for .
Suppose that for some positive integer , then for we get
(27) 
while for we have
(28) 
(Wainerman’s lemma [72]) Suppose for some positive integer , and let
and . Then it follows for that
(29) 
and for that
(30) 
where denotes the sign function. {proof} The interesting result and its proof is published in Russian in [72]. For convenience and completeness, we present the proof here.
For , from the definition of we see that
which directly leads to the desired result (3.5) for based on or , respectively.
In the following, we will show that also satisfies (3.5).
In the case : Since
then by the Rolle’s theorem it follows
for some satisfying for .
Note that is a polynomial of degree , then is a polynomial of degree , which implies that are the exact zeros of and then has the form of
(31) 
for some nonzero constant . In addition, from (3.7) has alternative sign between these roots. Then, by and , it yields
and
since is strictly increasing in and .
By the alternative property of between these roots, it deduces that for and , particularly,
and