**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 PRESTIN^{1} and EWALD QUAK^{2}

*Dedicated to Professor Dr. Werner Haussmann on his 60th birthday*

**Abstract.** As pointed out recently by Strichartz [5], the arclength of the graph
Γ(S*N*(f)) of the partial sums*S**N*(f) of the Fourier series of a jump function*f*
grows with the order of log*N*. 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 *L*^{1}_{2π}-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(S* _{N}*(f)) of the partial sums

*S*

*(f) of the Fourier series of a piecewise smooth function*

_{N}*f*. It turns out that in the case of jump discontinuities, the arclengths of the graphs Γ(S

*(f)) tend to infinity with logarithmic order, while for continuous piecewise*

_{N}*C*

^{1}functions the arclength of Γ(S

*N*(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 integer*N*the trigonometric interpolant
*L*_{N}*f* to a given 2π-periodic function*f* by

*L**N**f*(t) =

2N−1

X

*s=0*

*f* ^{sπ}_{N}^{}*ϕ**N* *t*−^{sπ}_{N}^{}*,*

where

*ϕ** _{N}*(t) =

_{2N}

^{1}

^{}1 + 2

*N*−1

X

*k=1*

cos*kt* + cos*N t*^{}
is a modified Dirichlet kernel.

Then*L*_{N}*f*(^{kπ}* _{N}*) =

*f*(

^{kπ}*) holds for all integer*

_{N}*k. As in the case of the Fourier*sum we can restrict our attention to the 2π-periodic jump function

*f*0(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
*f*_{0}(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 of*f*_{0} 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
*f*0 and and its translates*f**ε*(t) =*f*0(t−*ε), where 0< ε <* _{N}^{π}*.*

It turns out that the behaviour of the arclengths of the graphs Γ(L* _{N}*(f

*)) depends essentially on the choice of*

_{ε}*ε. Namely, for*

*ε*= 0 we have bounded arclength and for 0

*< ε <*

_{N}*the arclength behaves like log*

^{π}*N*. 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 of

*f*

_{0}not only stems from the fact that the jump discontinuity is at an interpolation point, but also from

(1) *f*_{0}(0) = *f*0(0−) +*f*0(0+)

2 *.*

If an arbitrary jump function does not satisfy (1), we have to add to the
interpolation polynomial a multiple of*ϕ** _{N}*, 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 FUNCTION*f*_{ε}

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^{`sπ}* _{N}* cos

^{ksπ}*= 0,*

_{N}2N−1

X

*s=0*

cos^{`sπ}* _{N}* cos

^{ksπ}*=*

_{N}*N*·(˜

*δ*

*+ ˜*

_{`,k}*δ*

*`,−k*)

*and*

2N−1

X

*s=0*

sin^{`sπ}* _{N}* sin

^{ksπ}*=*

_{N}*N*·(˜

*δ*

*−*

_{`,k}*δ*˜

*`,−k*),

*where*

*δ*˜* _{`,k}* =

1, *`*≡*k* mod 2N,
0, otherwise.

*Proof.* These identities follow directly from the identities for integer*r*

2N−1

X

*s=0*

cos^{rsπ}* _{N}* = ˜

*δ*

*·2N and*

_{r,0}2N−1

X

*s=0*

sin^{rsπ}* _{N}* = 0.

Next, we compute explicitly the interpolants for the jump functions *f**ε*. It
turns out that the interpolant to *f**ε* is equal to the interpolant to *f*_{0}, shifted
vertically by*ε/2, plus a perturbation term that is completely independent of*
*ε*6= 0.

Lemma2.2. *The trigonometric interpolantsL**N**f**ε**possess the following rep-*
*resentations*

*L*_{N}*f*_{0}(t) = _{2N}^{π}

*N*−1

X

*k=1*

cot_{2N}* ^{kπ}* sin

*kt,*

*and for* 0*< ε <* _{N}^{π}

*L*_{N}*f**ε*(t) = *L*_{N}*f*0(t)−^{π}_{2}*ϕ** _{N}*(t) +

^{ε}_{2}

= ^{ε}_{2} −_{4N}^{π}^{}1 +

*N*

X

|k|=1

0

1 +*i*cot_{2N}^{kπ}^{}*e*^{ikt}^{}*.*

*Here* ^{P}^{0} *means that the terms for*|k|=*N* *have to be multiplied by* 1/2.

*Proof.* For arbitrary 0≤*ε <* _{N}* ^{π}* we obtain

*L*_{N}*f** _{ε}*(t) =

_{2N}

^{1}

2N−1

X

*s=0*

∞

X

*`=1*

sin*`(*^{sπ}* _{N}*−ε)

*`*

1 + 2

*N*

X

*k=1*

0cos*k(t*−^{sπ}* _{N}*)

^{}

*.*

Now we simplify for 0*< k*≤*N* using Lemma 2.1

2N−1

X

*s=0*

∞

X

*`=1*

sin*`(*^{sπ}* _{N}*−ε)

*`* cos*k(t*− ^{sπ}* _{N}*) =

=

∞

X

*`=1*
1

*`*
2N−1

X

*s=0*

sin^{`sπ}* _{N}* cos

*`ε*cos

*kt*cos

^{ksπ}*+ sin*

_{N}

^{`sπ}*cos*

_{N}*`ε*sin

*kt*sin

^{ksπ}

_{N}−cos^{`sπ}* _{N}* sin

*`ε*cos

*kt*cos

^{ksπ}*−cos*

_{N}

^{`sπ}*sin*

_{N}*`εö sinkt*sin

^{ksπ}

_{N}^{}

= sin*kt*

∞

X

*`=1*
cos*`ε*

*`* *N*(˜*δ** _{`,k}*−˜

*δ*

*`,−k*)−cos

*kt*

∞

X

*`=1*
sin*`ε*

*`* *N*(˜*δ** _{`,k}*+ ˜

*δ*

*`,−k*)

= *N*sin*kt*^{}^{cos}_{k}* ^{kε}* +

∞

X

*L=1*

cos(k+2N L)ε

*k+2N L* − cos(−k+2N L)ε

−k+2N L

−Ncos*kt*^{}^{sinkε}* _{k}* +

∞

X

*L=1*

sin(k+2N L)ε

*k+2N L* +sin(−k+2N L)ε

−k+2N L

= ^{1}_{2}sin*kt*

∞

X

*L=−∞*

cos (_{2N}* ^{k}* +L)(2N ε)

*k*

2N+L −^{1}_{2}cos*kt*

∞

X

*L=−∞*

sin (_{2N}* ^{k}* +L)(2N ε)

*k*
2N+L

= ^{π}_{2} sin*kt*·cot_{2N}* ^{kπ}* −

0, if *ε*= 0,

*π*

2cos*kt,* if 0*<*2N ε <2π.

For the series representation yielding the last equality, compare [6, pp. 71,73].

In the case*k*= 0 we write

2N−1

X

*s=0*

∞

X

*`=1*

sin`(^{sπ}* _{N}*−ε)

*`* =

∞

X

*`=1*
1

*`*
2N−1

X

*s=0*

sin^{`sπ}* _{N}* cos

*`ε*−cos

^{`sπ}*·sin*

_{N}*`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 <* _{N}^{π}*.*

Summing up*k* from 0 to*N* 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: *L*16*f*0, Right: *L*32*f*0.

-3 -2 -1 1 2 3

-2 -1 1

-3 -2 -1 1 2 3

-2 -1 1 2

Fig. 2. Left: *L*16*f*0.01, Right: *L*32*f*0.01.

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(L_{N}*f**ε*) *of the interpolant* *L*_{N}*f**ε*

*remains bounded iff* *ε*= 0,*i.e.,*

length^{}Γ(L_{N}*f*_{0})^{}=O(1), *N* → ∞,
*while for* 0*< ε <* _{N}^{π}

length^{}Γ(L*N**f**ε*)^{}∼log*N,* *N* → ∞.

*Proof.* For*ε*= 0 we obtain by definition
(2) length^{}Γ(L*N**f*0)^{}=

Z *π*

−π

r

1 +^{}(L*N**f*0)^{0}(t)^{}^{2}*dt*≤2π+k(L_{N}*f*0)^{0}k_{1}*.*
Using the function

*g*0(x) =

1, if *x*= 0,

0, if |x|*>* ^{π}_{2}*,*
*x*cot*x,* otherwise,
we can write

(L*N**f*0)^{0}(t) =

*N−1*

X

*k=1*

*g*0

*kπ*
2N

·cos*kt*

= ^{1}_{2}

*N*

X

*k=−N*

*g*0

_{kπ}

2N

*e** ^{ikt}*−

^{1}

_{2}

*,*

and we can estimate with the help of Poisson’s summation formula (cf. [1, Lemma 1])

*N*

X

*k=−N*

*g*_{0}^{}_{2N}^{kπ}^{}*e*^{ik◦}

_{1}

≤ kˆ*g*_{0}k* _{L}*1(R)

*,*

where ˆ*g*_{0} is the Fourier transform of *g*_{0}. Now we can use (cf. [1, Lemma 3])
that

(3) kˆ*g*_{0}k* _{L}*1(R) ≤4

^{q}

*V*(g

_{0}

^{0})· kg

_{0}k

*1(R)*

_{L}*.*

Here it holds that kg_{0}k* _{L}*1(R) =

*π*ln 2, while for the total variation of the derivative one obtains

*V*(g

^{0}

_{0}) = 2π and hence

k(L_{N}*f*_{0})^{0}k_{1} ≤*π*+ 4π√
2 ln 2.

This proves the first part of the theorem.

Using (2) and the representation of*L**N**f**ε* from Lemma 2.2, we conclude
length(Γ(L_{N}*f** _{ε}*))∼ k

^{π}_{2}(ϕ

*)*

_{N}^{0}k

_{1}

*.*

From Bernstein’s inequality it follows easily that
k^{π}_{2}(ϕ*N*)^{0}k_{1} ≤ ^{πN}_{2} kϕ* _{N}*k

_{1}≤2 ln

*N*+

*C.*

On the other hand the lower bound fork(ϕ^{M}* _{N}*)

^{0}k

_{1}is derived analogously to the standard arguments for the Dirichlet kernel (cf. [6, p. 67]).

For the convenience of the reader we include plots of*g*_{0} and*g*_{0}^{0} to illustrate
the smoothness properties of *g*0.

-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: *g*0 withkg0k*L*^{1}(R)=*π*ln 2, Right: *g*^{0}_{0} with*V*(g_{0}^{0}) = 2π.

Moreover, let us mention that for *ε >*0
(L_{N}*f** _{ε}*)

^{0}(t) =

^{1}

_{2}

*N*

X

*k=−N*

*g*^{}_{2N}^{kπ}^{}*e** ^{ikt}*−

^{1}

_{2}

*,*

where

*g(x) =*

1, if *x*= 0,

0, if |x|*>* ^{π}_{2}*,*
*x*cot*x*−*ix* otherwise.

Then the real part of*g*is the smooth function*g*_{0}, 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*

*f*^{0} ∈*L*^{p}_{2π}
*for a certain* *p >*1. Then

(4) lim

*N*→∞length(Γ(L_{N}*f)) = length(Γ(f*)).

*Proof.* Following the ideas of Strichartz [5, Proposition 2] we estimate

|length(Γ(f))−length(Γ(L_{N}*f))| ≤ k(f*−*L*_{N}*f*)^{0}k_{1}*.*

In the next steps we need a mean of the Fourier sum which approximates in
the order of best approximation for all*L*^{p}_{2π}-spaces and reproduces polynomials.

For that reason we choose the de la Vall´ee Poussin mean
*σ*_{2N}^{N}*f(t) =* _{2π}^{1}

Z *π*

−π

*f(t*−*u)*^{}1 + 2

*N*

X

*k=1*

cos*ku*+

3N−1

X

*k=N+1*
3N−k

2N cos*ku*^{}du.

We obtain by Bernstein’s inequality

k(f −*L**N**f*)^{0}k_{1} ≤ k(f −*σ*_{2N}^{N}*f*)^{0}k_{1}+k(σ_{2N}* ^{N}* (f−

*L*

*N*

*f*))

^{0}k

_{1}

≤ *cE**N*(f^{0}*, L*^{1}_{2π}) + 2Nkσ_{2N}* ^{N}* (f −

*L*

*N*

*f*)k

_{1}

≤ *cE** _{N}*(f

^{0}

*, L*

^{1}

_{2π}) +

*cN*kf−

*L*

_{N}*f*k

_{1}

≤ *cE** _{N}*(f

^{0}

*, L*

^{p}_{2π}) +

*cN*kf−

*L*

_{N}*f*k

_{p}≤ *cE** _{N}*(f

^{0}

*, L*

^{p}_{2π}), (5)

where for the last inequality we have used a result on trigonometric Lagrange
interpolation proved in [4]. As *E**N*(f^{0}*, L*^{p}_{2π}) tends to zero for *f*^{0} ∈ *L*^{p}_{2π}, 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 case*ε >*0 by modifying the
Lagrange interpolation process. If one interpolates in 2N points one can allow
the interpolation polynomial to have a degree bigger than*N*. Let us write for
1≤*M* ≤*N* (cf. [3])

*ϕ*^{M}* _{N}*(t) =

_{2N}

^{1}

^{}1 + 2

*N*

X

*k=1*

cos*kt*+

*N+M−1*

X

*k=N−M*+1

*N*+M−k

2M cos*kt*^{}
and

*L*^{M}_{N}*f*(t) =

2N−1

X

*s=0*

*f* ^{sπ}_{N}^{}*ϕ*^{M}_{N}*t*−^{sπ}_{N}^{}*.*

Then *L*^{M}_{N}*f*(^{kπ}* _{N}*) =

*f*(

^{kπ}*) for all integer*

_{N}*k*and

*L*

^{M}

_{N}*f*is a trigonometric poly- nomial of degree less than

*N*+

*M*. Note also that

*ϕ*

*=*

_{N}*ϕ*

^{1}

*and*

_{N}*L*

_{N}*f*=

*L*

^{1}

_{N}*f.*

The particular feature of these interpolation polynomials *L*^{M}* _{N}* is the bound-
edness of the kernels

*ϕ*

^{M}*depending on the quotient*

_{N}*N/M*only. Using the well-known estimate

kϕ^{M}* _{N}*k

_{1}∼

_{πN}^{4}ln

^{2N}

_{M}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* *f*_{ε}*at the jump it holds that*

length^{}Γ(L^{M}_{N}*f**ε*)^{}=O(1), N → ∞*,*

*and for* 2π-periodic absolutely continuous functions *f, i.e.,* *f*^{0} ∈*L*^{1}_{2π}*, it holds*
*that*

*N→∞*lim length(Γ(L_{N}*f*)) = 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.