Rev. Anal. Num´er. Th´eor. Approx., vol. 30 (2001) no. 2, pp. 219–227 ictp.acad.ro/jnaat
ON THE ARCLENGTH OF TRIGONOMETRIC INTERPOLANTS∗
J ¨URGEN PRESTIN1 and EWALD QUAK2
Dedicated to Professor Dr. Werner Haussmann on his 60th birthday
Abstract. As pointed out recently by Strichartz [5], the arclength of the graph Γ(SN(f)) of the partial sumsSN(f) of the Fourier series of a jump functionf grows with the order of logN. In this paper we discuss the behaviour of the arclengths of the graphs of trigonometric interpolants to a jump function. Here the boundedness of the arclengths depends essentially on the fact whether the jump discontinuity is at an interpolation point or not. In addition convergence results for the arclengths of interpolants to smoother functions are presented.
MSC 2000. 41A15.
1. INTRODUCTION
The famous Gibbs phenomenon of overshooting is one of the well-known disadvantages of the Fourier series approach. It is closely related to the log- arithmic growth of the L12π-norm of the Dirichlet kernel, i.e. the Lebesgue constant for the Fourier partial sum operator. It can also be seen as one of the motivations for introducing different means of Fourier sums.
In the very recent paper [5], Strichartz investigated the behaviour of the arclengths of the graphs Γ(SN(f)) of the partial sums SN(f) of the Fourier series of a piecewise smooth function f. It turns out that in the case of jump discontinuities, the arclengths of the graphs Γ(SN(f)) tend to infinity with logarithmic order, while for continuous piecewise C1 functions the arclength of Γ(SN(f)) converges to the arclength of Γ(f).
∗Research of the authors supported by the EU Research Training Network MINGLE, RTN1-1999-00212.
1Medical University of L¨ubeck, Institute of Mathematics, D–23560 L¨ubeck, Germany, e-mail: [email protected].
2SINTEF Applied Mathematics, P.O. Box 124 Blindern, N-0314 Oslo, Norway, e-mail:
It is the aim of this paper to investigate analogous questions for trigono- metric Lagrange interpolation.
Therefore we define for each positive integerNthe trigonometric interpolant LNf to a given 2π-periodic functionf by
LNf(t) =
2N−1
X
s=0
f sπNϕN t−sπN,
where
ϕN(t) = 2N1 1 + 2
N−1
X
k=1
coskt + cosN t is a modified Dirichlet kernel.
ThenLNf(kπN) =f(kπN) holds for all integerk. As in the case of the Fourier sum we can restrict our attention to the 2π-periodic jump function
f0(t) =
(π−t)/2, if 0< t≤π,
0, if t= 0,
(−π−t)/2, if −π≤t <0.
With its Fourier expansion given as f0(t) =
∞
X
`=1
sin`t
` ,
this piecewise linear function is a standard example for the Gibbs phenomenon.
The underlying idea is then to consider functions with finitely many jumps in the period interval as the sum of translates off0 and a smooth function.
Different from the case of Fourier sums, for the interpolation process it is important, however, to know whether the jump discontinuity is at an interpo- lation point or not. Therefore we distinguish between our jump test function f0 and and its translatesfε(t) =f0(t−ε), where 0< ε < Nπ.
It turns out that the behaviour of the arclengths of the graphs Γ(LN(fε)) depends essentially on the choice of ε. Namely, for ε = 0 we have bounded arclength and for 0< ε < Nπ the arclength behaves like logN. Some overshoot, however, is always present also in the case of bounded arclengths, see Figures 1 and 2. Notice that the nice behaviour of the interpolant off0not only stems from the fact that the jump discontinuity is at an interpolation point, but also from
(1) f0(0) = f0(0−) +f0(0+)
2 .
If an arbitrary jump function does not satisfy (1), we have to add to the interpolation polynomial a multiple ofϕN, which results also in an unbounded arclength (cf. the proof of Theorem 3.1).
Finally we mention that the use of modified interpolation processes can improve the behaviour of the graphs of the interpolants essentially. In this note we restrict ourselves to certain de la Vall´ee Poussin kernels, which possess interesting features for generating corresponding wavelets (cf. [2], [3]).
2. THE INTERPOLANT OF THE JUMP FUNCTIONfε
We start by stating some basic identities for discrete inner products of trigonometric functions.
Lemma 2.1. The following discrete orthogonality relations hold for all in- tegers k, `
2N−1
X
s=0
sin`sπN cosksπN = 0,
2N−1
X
s=0
cos`sπN cosksπN = N ·(˜δ`,k+ ˜δ`,−k) and
2N−1
X
s=0
sin`sπN sinksπN =N·(˜δ`,k−δ˜`,−k), where
δ˜`,k =
1, `≡k mod 2N, 0, otherwise.
Proof. These identities follow directly from the identities for integerr
2N−1
X
s=0
cosrsπN = ˜δr,0·2N and
2N−1
X
s=0
sinrsπN = 0.
Next, we compute explicitly the interpolants for the jump functions fε. It turns out that the interpolant to fε is equal to the interpolant to f0, shifted vertically byε/2, plus a perturbation term that is completely independent of ε6= 0.
Lemma2.2. The trigonometric interpolantsLNfεpossess the following rep- resentations
LNf0(t) = 2Nπ
N−1
X
k=1
cot2Nkπ sinkt,
and for 0< ε < Nπ
LNfε(t) = LNf0(t)−π2ϕN(t) +ε2
= ε2 −4Nπ 1 +
N
X
|k|=1
0
1 +icot2Nkπeikt.
Here P0 means that the terms for|k|=N have to be multiplied by 1/2.
Proof. For arbitrary 0≤ε < Nπ we obtain
LNfε(t) = 2N1
2N−1
X
s=0
∞
X
`=1
sin`(sπN−ε)
`
1 + 2
N
X
k=1
0cosk(t−sπN).
Now we simplify for 0< k≤N using Lemma 2.1
2N−1
X
s=0
∞
X
`=1
sin`(sπN−ε)
` cosk(t− sπN) =
=
∞
X
`=1 1
` 2N−1
X
s=0
sin`sπN cos`εcosktcosksπN + sin`sπN cos`εsinktsinksπN
−cos`sπN sin`εcosktcosksπN −cos`sπN sin`εö sinktsinksπN
= sinkt
∞
X
`=1 cos`ε
` N(˜δ`,k−˜δ`,−k)−coskt
∞
X
`=1 sin`ε
` N(˜δ`,k+ ˜δ`,−k)
= Nsinktcoskkε +
∞
X
L=1
cos(k+2N L)ε
k+2N L − cos(−k+2N L)ε
−k+2N L
−Ncosktsinkεk +
∞
X
L=1
sin(k+2N L)ε
k+2N L +sin(−k+2N L)ε
−k+2N L
= 12sinkt
∞
X
L=−∞
cos (2Nk +L)(2N ε)
k
2N+L −12coskt
∞
X
L=−∞
sin (2Nk +L)(2N ε)
k 2N+L
= π2 sinkt·cot2Nkπ −
0, if ε= 0,
π
2coskt, if 0<2N ε <2π.
For the series representation yielding the last equality, compare [6, pp. 71,73].
In the casek= 0 we write
2N−1
X
s=0
∞
X
`=1
sin`(sπN−ε)
` =
∞
X
`=1 1
` 2N−1
X
s=0
sin`sπN cos`ε−cos`sπN ·sin`ε
= −2N
∞
X
`=1 sin`ε
l ˜δ`,0
= −2N
∞
X
L=1
sin 2N Lε 2N L
= −
∞
X
L=1
sin 2N Lε L
=
( 0, if ε= 0,
2N ε−π
2 , if 0< ε < Nπ.
Summing upk from 0 toN we obtain the assertions of Lemma 2.2.
The different behaviour of the Lagrange interpolants is illustrated by the following figures.
-3 -2 -1 1 2 3
-1.5 -1 -0.5 0.5 1 1.5
-1.5 -1 -0.5 0.5 1 1.5
-1.5 -1 -0.5 0.5 1 1.5
Fig. 1. Left: L16f0, Right: L32f0.
-3 -2 -1 1 2 3
-2 -1 1
-3 -2 -1 1 2 3
-2 -1 1 2
Fig. 2. Left: L16f0.01, Right: L32f0.01.
3. THE ARCLENGTH OF THE INTERPOLANT
For the arclengths of the jump function interpolants we obtain the following result.
Theorem 3.1. The length of the graph Γ(LNfε) of the interpolant LNfε
remains bounded iff ε= 0,i.e.,
lengthΓ(LNf0)=O(1), N → ∞, while for 0< ε < Nπ
lengthΓ(LNfε)∼logN, N → ∞.
Proof. Forε= 0 we obtain by definition (2) lengthΓ(LNf0)=
Z π
−π
r
1 +(LNf0)0(t)2dt≤2π+k(LNf0)0k1. Using the function
g0(x) =
1, if x= 0,
0, if |x|> π2, xcotx, otherwise, we can write
(LNf0)0(t) =
N−1
X
k=1
g0
kπ 2N
·coskt
= 12
N
X
k=−N
g0
kπ
2N
eikt−12,
and we can estimate with the help of Poisson’s summation formula (cf. [1, Lemma 1])
N
X
k=−N
g02Nkπeik◦
1
≤ kˆg0kL1(R),
where ˆg0 is the Fourier transform of g0. Now we can use (cf. [1, Lemma 3]) that
(3) kˆg0kL1(R) ≤4qV(g00)· kg0kL1(R).
Here it holds that kg0kL1(R) = πln 2, while for the total variation of the derivative one obtainsV(g00) = 2π and hence
k(LNf0)0k1 ≤π+ 4π√ 2 ln 2.
This proves the first part of the theorem.
Using (2) and the representation ofLNfε from Lemma 2.2, we conclude length(Γ(LNfε))∼ kπ2(ϕN)0k1.
From Bernstein’s inequality it follows easily that kπ2(ϕN)0k1 ≤ πN2 kϕNk1≤2 lnN +C.
On the other hand the lower bound fork(ϕMN)0k1 is derived analogously to the standard arguments for the Dirichlet kernel (cf. [6, p. 67]).
For the convenience of the reader we include plots ofg0 andg00 to illustrate the smoothness properties of g0.
-2 -1 1 2
0.2 0.4 0.6 0.8 1
-2 -1 1 2
-1.5 -1 -0.5 0.5 1 1.5
Fig. 3. Left: g0 withkg0kL1(R)=πln 2, Right: g00 withV(g00) = 2π.
Moreover, let us mention that for ε >0 (LNfε)0(t) = 12
N
X
k=−N
g2Nkπeikt− 12,
where
g(x) =
1, if x= 0,
0, if |x|> π2, xcotx−ix otherwise.
Then the real part ofgis the smooth functiong0, whereas the imaginary part has jumps so that the estimate (3) does not hold.
In the following result we describe the behaviour of the arclength of the graph of the interpolant for smoother functions f.
Theorem 3.2. Let the 2π-periodic function f be sufficiently smooth in the sense that
f0 ∈Lp2π for a certain p >1. Then
(4) lim
N→∞length(Γ(LNf)) = length(Γ(f)).
Proof. Following the ideas of Strichartz [5, Proposition 2] we estimate
|length(Γ(f))−length(Γ(LNf))| ≤ k(f−LNf)0k1.
In the next steps we need a mean of the Fourier sum which approximates in the order of best approximation for allLp2π-spaces and reproduces polynomials.
For that reason we choose the de la Vall´ee Poussin mean σ2NN f(t) = 2π1
Z π
−π
f(t−u)1 + 2
N
X
k=1
cosku+
3N−1
X
k=N+1 3N−k
2N coskudu.
We obtain by Bernstein’s inequality
k(f −LNf)0k1 ≤ k(f −σ2NN f)0k1+k(σ2NN (f−LNf))0k1
≤ cEN(f0, L12π) + 2Nkσ2NN (f −LNf)k1
≤ cEN(f0, L12π) +cNkf−LNfk1
≤ cEN(f0, Lp2π) +cNkf−LNfkp
≤ cEN(f0, Lp2π), (5)
where for the last inequality we have used a result on trigonometric Lagrange interpolation proved in [4]. As EN(f0, Lp2π) tends to zero for f0 ∈ Lp2π, the
theorem is proved.
Note that specific orders of convergence in (4) can be obtained from (5) for sufficiently smooth functions by using standard Jackson type arguments for trigonometric best approximation.
One can also achieve bounded arclength in the caseε >0 by modifying the Lagrange interpolation process. If one interpolates in 2N points one can allow the interpolation polynomial to have a degree bigger thanN. Let us write for 1≤M ≤N (cf. [3])
ϕMN(t) = 2N1 1 + 2
N
X
k=1
coskt+
N+M−1
X
k=N−M+1
N+M−k
2M coskt and
LMNf(t) =
2N−1
X
s=0
f sπNϕMN t−sπN.
Then LMNf(kπN) =f(kπN) for all integer k and LMNf is a trigonometric poly- nomial of degree less thanN+M. Note also thatϕN =ϕ1N andLNf =L1Nf.
The particular feature of these interpolation polynomials LMN is the bound- edness of the kernels ϕMN depending on the quotient N/M only. Using the well-known estimate
kϕMNk1 ∼ πN4 ln2NM
and the same methods of proof as above, we obtain the following result.
Theorem 3.3. Let N/M be bounded. Then for arbitrary ε and arbitrary values of fε at the jump it holds that
lengthΓ(LMNfε)=O(1), N → ∞,
and for 2π-periodic absolutely continuous functions f, i.e., f0 ∈L12π, it holds that
N→∞lim length(Γ(LNf)) = length(Γ(f)).
REFERENCES
[1] Girgensohn, R. and Prestin, J., Lebesgue constants for an orthogonal polynomial Schauder basis, J. Comput. Anal. Appl.,2, pp. 159–175, 2000.
[2] Prestin, J. and Quak, E.,Trigonometric interpolation and wavelet decompositions, Numer. Algor.,9, pp. 293–318, 1995.
[3] Prestin, J.andSelig, K.,Interpolatory and orthonormal trigonometric wavelets, in:
Signal and Image Representation in Combined Spaces (Eds. J. Zeevi, R. Coifman), Academic Press, pp. 201–255, 1998.
[4] Xu, Y. and Prestin, J., Convergence rate for trigonometric interpolation of non- smooth functions, J. Approx. Theory,77, pp. 113–122, 1994.
[5] Strichartz, R. S. Gibbs’ phenomenon and arclength, J. Fourier Anal. Appl., 6, pp. 533–536, 2000.
[6] Zygmund, A.,Trigonometric Series, Cambridge University Press, Second Ed., 1959.
Received December 19, 2000.
