On Interpolation Approximation: Convergence rates for polynomial interpolation for functions of limited regularityThis work was supported by National Science Foundation of China (No. 11371376).

# 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).

Shuhuang Xiang School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, P. R. China.
###### 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 Gauss-Jacobi, Jacobi-Gauss-Lobatto or Jacobi-Gauss-Radau pointsystems. From these results, we see that the interpolations at the Gauss-Legendre, Legendre-Gauss-Lobatto 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 Clenshaw-Curtis quadrature. In addition, numerical examples illustrate the perfect coincidence with the estimates, which means the convergence rates are optimal.

p

olynomial interpolation, Peano kernel, convergence rate, limited regularity, strongly normal pointsystem, Gauss-Jacobi point, Jacobi-Gauss-Lobatto point, Chebyshev point.

{AMS}

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) −1≤x(n)n

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) Ln[f]=n∑k=1f(x(n)k)ℓ(n)k(x),ℓ(n)k(x)=ωn(x)ω′n(x(n)k)(x−x(n)k),

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) λn(x)=n∑k=1∣∣ℓ(n)k(x)∣∣

and Lebesgue constant

 (4) Λn=maxx∈[−1,1]λn(x),

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

 Λn=supf∥Ln[f]∥∞∥f∥∞,

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) ∥∥Ln[f]−f∥∥∞≤(1+Λn)∥∥p∗n−1−f∥∥∞,

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

 Λn≥112logn,

which, together with the boundedness principle, implies that there exists a continuous function in for which the sequence () is not uniformly convergent to in 111Grünwald [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) Λn≥2πlogn+C\, for some constant C (\@@cite[ci% te]{[\@@bibref{}{Erdos1961}{}{}]});Λn≥2π(γ0+log4π)+2πlogn(\@@cite[cite][\@@bibrefBrutman1978]),

where is the Euler’s constant. In particular, for equidistant pointsystem

 {x(n)k=−1+2kn−1}n−1k=0,

Schönhage [52] showed that

 Λn∼2ne(log(n−1)+γ0)(n−1),n→∞.

Additionally, Trefethen and Weideman [61] established that

 2n−3(n−1)2≤Λn≤2n+2n−1,n≥0.

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 :

• Chebyshev pointsystem of first kind : An asymptotic estimate of was given by Bernstein [1] as

 (7) Λn(Tn)∼2πlogn,n→∞,

which is improved by Ehlich and Zeller [14], Rivlin [47] and Brutman [7] as

 2π(γ0+log4π)+2πlogn<Λn(Tn)≤1+2πlogn,n=1,2,….
• Chebyshev pointsystem of second kind (also called Chebyshev extreme or Clenshaw-Curtis points [58]): Ehlich and Zeller [14] proved that

 (8) Λn(Un)={Λn−1(Tn−1),n=2,4,6,…Λn−1(Tn−1)−αn,0≤αn<1(n−1)2,n=3,5,7,….
• The roots of Jacobi polynomial (): The asymptotic estimate of was found by Szegö [55] as

 (9) Λn(Jn)=⎧⎨⎩O(nγ+12),γ>−12O(logn),γ≤−12,γ=max{α,β}.

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) ∥f−Ln[f]∥∞=O(n−k),

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

 ∥f−Ln[f]∥∞≤4π(n−1)

(see [59, 76]), while

 ∥f−p∗n−1∥∞∼βn,0.2801685<β<0.2801734

(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 Clenshaw-Curtis [10] and O’Hara and Smith [29], that -point Gauss quadrature and -point Clenshaw-Curtis 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

 Qn[f]=∫1−1Ln[f](x)dx,

based on the Gauss-Legender and Clenshaw-Curtis 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) ∥f−Ln[f]∥∞≤πrVar(f(r))(n−1)(n−2)⋯(n−r)max1≤j≤n∥ℓ(n)j∥∞,

 (12) ∥f−Ln[f]∥∞=O(n−rmax1≤j≤n∥ℓ(n)j∥∞).

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 Gauss-Jacobi, Jacobi-Gauss-Lobatto or Jacobi-Gauss-Radau 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 Gauss-Jacobi points:

 max1≤j≤n∥ℓ(n)j∥∞=O(nmax{γ−12,0}),γ=max{α,β}.
• For the Jacobi-Gauss-Lobatto points (the roots of ):

 max1≤j≤n∥ℓ(n)j∥∞=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩O(n−min{0,α+12,β+12}),−1<α,β≤32O(n−min{0,α+12,2+α−β,52−β}),−1<α≤32,β>32O(n−min{0,β+12,2+β−α,52−α}),α>32,−1<β≤32O(n−min{0,2+α−β,2+β−α,52−α,52−β}),α,β>32.

 max0≤j≤n−1∥ℓ(n)j∥∞=⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩O(n−min{0,α+12,α−β}),−1<α≤12O(n−min{0,12−β,52−α,α−β}),α>12.

 max1≤j≤n∥ℓ(n)j∥∞=⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩O(n−min{0,β+12,β−α}),−1<β≤12O(n−min{0,12−α,52−β,β−α}),β>12.

From the above estimates, we see that the interpolation at the Gauss-Legendre or at the Legendre-Gauss-Lobatto 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 Gauss-Jacobi, Jacobi-Gauss-Lobatto or Gauss-Jacobi-Radau pointsystem, can be efficiently evaluated by applying the second barycentric formula

 Ln[f](x)=∑nj=1λjx−xjf(xj)∑nj=1λjx−xj,

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, Gauss-Jacobi, Jacobi-Gauss-Lobatto and Jacobi-Gauss-Radau 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

 C1B≤A≤C2B.

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) L[f]=∫1−1f(r)(t)Kr(t)dt

holding for all such functions , where and

 (14) (x−t)r−1+={(x−t)r−1,x≥t0,x

For each fixed , we consider the special functional , where is defined for by

 En[f](x)=f(x)−n∑j=1f(xj)ℓj(x)=f(x)−Ln[f](x)

with . is a continuously linear functional since for arbitrary , and then by the Peano theorem [45] can be represented if for as

 (15) En[f](x)=∫1−1f(r)(t)Kr(t)dt

with

 (16) Kr(t)=1(r−1)!(x−t)r−1+−1(r−1)!n∑j=1(xj−t)r−1+ℓj(x).

Particularly, from (2.3) it implies

 |En[f](x)|≤∥f(r)∥∞∫1−1|Kr(t)|dt≤2∥f(r)∥∞∥Kr∥∞.

Similar to the Peano kernel for quadrature [6], the kernel for interpolation satisfies the following proposition.

{proposition}

(Peano representation) Let

 (17) Ks(t)=1(s−1)!(x−t)s−1+−1(s−1)!n∑j=1(xj−t)s−1+ℓj(x),s=1,2,….

Then for , the Peano kernel satisfies and can be rewritten as

 (18) Ks(u)=∫1uKs−1(t)dt,s=2,3,….
{proof}

From the definition of in (2.5), it is easy to verify that by using for . Furthermore, we find that

 (19) 1(s−2)!∫1u(x−t)s−1+dt=⎧⎪⎨⎪⎩0,u>x1(s−2)!∫xu(x−t)s−1dt=1(s−1)!(x−u)s−1,u≤x=1(s−1)!(x−u)s−1+.

Define and and suppose for some nonnegative integer . By (2.7), similarly we have

 (20) 1(s−2)!n∑j=1∫1u(xj−t)s−2+ℓj(x)dt=1(s−2)!m∑j=1ℓj(x){0,u>xj∫xju(xj−t)s−2dt,u≤xj=1(s−1)!n∑j=1(xj−u)s−1+ℓj(x).

Then from

 ∫1uKs−1(t)dt=1(s−2)!∫1u(x−t)s−2+dt−1(s−2)!n∑j=1∫1u(xj−t)s−2+ℓj(x)dt,

we get by (2.7) and (2.8).

In the following, we consider functions of limited regularities as

 (21) \emph{Suppose that f(t) has an absolutely % continuous (r−1)st derivative f(r−1) on [−1,1]}for some r≥1 with f(r−1)(t)=f(r−1)(−1)+∫t−1g(y)dy% , \emph{where g is absolutely}\emph{integrable and of bounded variation Var(g)<∞ on [−1,1].}

From Stein and Shakarchi [53, p. 130] and Tao [56, pp. 143-145], 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

 Vr=inf{Var(g)∣∣f(r−1)(t)=f(r−1)(−1)+∫t−1g(y)dy\, for all t∈[−1,1] with g being absolutely integrable and of bounded variation}.

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

 f′(t)=⎧⎪⎨⎪⎩1,t>0c,t=0−1,t<0,

then . Using the new condition, we see that can be represented as with and is unique.

{theorem}

Suppose satisfies (2.9), then for , we have

 (22) ∥En[f]∥∞≤Vr∥Kr+1∥∞.
{proof}

Applying the Peano theorem implies that for each fixed ,

 En[f](x)=∫1−1f(s)(t)Ks(t)dt,s=1,2,…,r−1.

Then, directly following Brass and Petras [6], integrating by parts and using yields

 En[f](x)=∫1−1f(r−1)(t)Kr−1(t)dt=∫1−1g(t)Kr(t)dt.

Since can be written as with and are monotonically increasing, and (see Lang [39, pp. 280-281]). 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

 En[f](x)=g(−1)∫ξ−1Kr(t)dt+g(1)∫1ξKr(t)dt=(g(1)−g(−1))Kr+1(ξ)=Var(g)Kr+1(ξ),

which leads to the desired result.

{lemma}

[6, Lemma 5.7.1] Assume that

Then, for every positive integer and every , there is a satisfying

 qu(y)≥tu(y)for all y∈[−1,1]

and

 ∫1−1[tu(y)−qu(y)]w(y)dy≥−πℓ+1sup−1≤t≤1w(t)√1−t2.
{lemma}
 (23) |Ks+1(u)|≤πn−s+1sup−1≤t≤1|Ks(t)|.
{proof}

In Lemma 2.3, letting , , representing as , and noting that , by Theorem 2.2 we have

 0=En[pn−1]=∫1−1p(s)n−1(t)Ks(t)dt=∫1−1qu(t)Ks(t)dt.

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) ∥En[f]∥∞≤πrVr(n−1)(n−2)⋯(n−r)∥K1∥∞.

## 3 Wainerman’s lemma

In the following, we shall focus on the estimate of .

Notice that for and

 (25) K1(u)=(x−u)0+−n∑j=1(xj−u)0+ℓj(x).

If , we have for , and for . While for , we have for , and for . Thus, in these cases we obtain

 (26) |K1(u)|≤1≤max1≤j≤n∥ℓj∥∞

since for .

Suppose that for some positive integer , then for we get

 (27) K1(u)=1−n∑j=1(xj−u)0+ℓj(x)=1−m∑j=1ℓj(x)=n∑j=m+1ℓj(x),

while for we have

 (28) K1(u)=−n∑j=1(xj−u)0+ℓj(x)=−m∑j=1ℓj(x).
{lemma}

(Wainerman’s lemma [72]) Suppose for some positive integer , and let

 ak(u)=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩k∑j=1ℓj(u),k=1,2,…,mn∑j=kℓj(u),k=m+1,m+2,…,n

and . Then it follows for that

 (29) sgn(ak(u))=sgn(ℓk(u))={(−1)m−k,k=1,2,…,m(−1)k−m−1,k=m+1,m+2,…,n

and for that

 (30) |ak(u)|≤|ℓk(u)|,k=1,2,…,n,

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

 sgn(ℓk(u))=sgn((u−x1)⋯(u−xk−1)(u−xk+1)⋯(u−xn)(xk−x1)⋯(xk−xk−1)(xk−xk+1)⋯(xk−xn))=(−1)1−ksgn((u−x1)⋯(u−xk−1)(u−xk+1)⋯(u−xn)),

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

 ak(xj)=k∑i=1ℓi(xj)={[]ll1,j=1,2,…,k0,j=k+1,k+2,…,n,

then by the Rolle’s theorem it follows

 a′k(yj)=0

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) a′k(t)=C(t−y1)⋯(t−yk−1)(t−yk+1)⋯(t−yn−1)

for some non-zero constant . In addition, from (3.7) has alternative sign between these roots. Then, by and , it yields

 a′k(t)>0,t∈(yk+1,yk−1)

and

 sgn(ak(t))=1,t∈(xk+1,xk)⊂(yk+1,yk−1)

since is strictly increasing in and .

By the alternative property of between these roots, it deduces that for and , particularly,

 sgn(a′k(t))=1,t∈(yk+1,xk+1)⊂(yk+1,yk−1)

and

 sgn(a′k(t))=−1,t∈(xk+2,y