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CHEBYSHEV INTERPOLATION FOR FUNCTIONS WITHENDPOINT SINGULARITIES VIA EXPONENTIAL AND

DOUBLE-EXPONENTIAL TRANSFORMS

MARK RICHARDSON∗

Abstract. We present five theorems concerning the asymptotic convergence rates of Chebyshevinterpolation applied to functions transplanted to either a semi-infinite or an infinite interval underexponential or double-exponential transformations. This strategy is useful for approximating andcomputing with functions that are analytic apart from endpoint singularities. The use of Chebyshevpolynomials instead of the more commonly used cardinal sinc or Fourier interpolants is importantbecause it enables one to apply maps to semi-infinite intervals for functions which have only a singleendpoint singularity. In such cases, this leads to significantly improved convergence rates.

Key words. endpoint singularity, conformal map, exponential transform, double-exponentialtransform, semi-infinite interval, infinite interval, Chebyshev interpolation, sinc function, Chebfun

AMS subject classifications. 41A05, 41A17, 41A25, 65D05

1. Introduction. Polynomial interpolation in Chebyshev points is known to bea robust and rapidly convergent technique for approximating smooth functions. Theconnection between Chebyshev expansions and Fourier cosine series provides a linkto several fast evaluation and summation algorithms based upon the FFT, and forthis reason, Chebyshev interpolants are often favoured in practice over other spectralexpansion bases. An example of this is Chebfun, which uses Chebyshev interpolants toroutinely compute integrals, derivatives, roots, and solutions of differential equationswith high speed and accuracy close to machine-precision [24].

A failing of Chebyshev interpolants, however, is their inability to cope efficientlywith functions that are singular, that is, functions which are not analytic at somepoint in the domain of interest. Some strategies do exist for dealing with simpleinstances of singularity. For example, in Chebfun, the function |x| on [−1, 1] is notapproximated by a global polynomial, but rather by two individual linear pieces. Thisapproach is satisfactory in many cases, however, the difficulty is more serious when afunction is encountered that cannot be divided into analytic pieces, such as

√1− x.

A recent publication by Richardson and Trefethen [14] built upon the work ofStenger and others [2, 10, 16, 17] to construct an adaptive Chebfun-like system usingmapped cardinal sinc interpolants to approximate such functions. At the heart of themethod is a map which transplants functions f defined on x ∈ [0, 1] to functions Fon s ∈ (−∞,∞), mapping in the process any endpoint singularities to infinity. If thetransplanted function F is analytic in some infinite strip about the real axis, thenthe sinc interpolant provides a rapidly convergent approximation. Indeed, it is wellknown that under reasonable hypotheses, this technique converges at a rate C−

√n as

n→∞, where n is the number of sample points.Almost all of the literature concerning the approximation of functions with end-

point singularities via variable transformation has so far concentrated on the use ofmaps to (−∞,∞). This implicitly treats the situation where the problem functionhas possible endpoint singularities at both ends of the domain. We shall argue inthis paper that in many practical situations, it is preferable to instead use a map to asemi-infinite interval such as (−∞, 0]. To motivate this point, consider approximating

∗Oxford University Mathematical Institute, 24–29 St Giles’, Oxford OX1 3LB, UK([email protected], http://people.maths.ox.ac.uk/richardsonm/).

1

2 M. RICHARDSON

the function f(x) =√x − x on [0, 1]. Since the only singularity is at x = 0, the use

of a transformation to an infinite interval is wasteful. It is in fact much better isto use a transformation from [0, 1] to (−∞, 0]; then no “waste” is introduced on theright-hand side, since the point x = 1 is not mapped to infinity. However, since thesetransformations effect exponential decay in only one direction, sinc functions (andfor that matter, Fourier series) cease to be a viable approximation basis. Chebyshevinterpolants, on the other hand, will cope perfectly well with this decay asymmetry.

Related to this is a conclusion of [4, 14] that the use of sinc functions is notin itself a necessary component for the success of variable-transformation methods.With suitable modifications, other basis sets such as Fourier series and Chebyshevpolynomials can and should be considered. In Sincfun, the problem functions f(x)on [0, 1] were assumed to take zero endpoint values (general functions were reducedto this case by subtracting a linear function). This resulted in exponential decayof the function F(s) on (−∞,∞). Then, given a fixed precision ε, it was possibleto determine in advance an interval [−L,L], L > 0, such that |F(s)| < ε for all|s| > L. It is a simple task to approximate F on this finite interval, and any reasonableinterpolation scheme will result in exponentially convergent approximations. Sincfunctions were used in [14], but Fourier series could equally well have been used, sincethe exponential decay of F makes it a good approximation to regard it as periodic on[−L,L]. In particular, the associated Gibbs overshoot is negligible for large L.

Chebyshev polynomials could have been used too, though their use would incurthe usual asymptotic loss of a factor of π/2 in the number of sample points [7, 26].Crucially, for both Fourier series and Chebyshev polynomials, the FFT enables fastsummation and evaluation; this is not the case for the sinc basis.

There are two classes of map that we shall discuss in this paper: exponential trans-forms and double-exponential transforms. In each case, we consider maps from theproblem interval [0, 1] to both (−∞, 0] and (−∞,∞), giving a total of four transforma-tions. As we shall see, exponential maps result in transplanted functions which decayexponentially, while double-exponential maps result in a transplanted functions whichdecay double-exponentially. In general, double-exponential transformations result inmuch more rapidly convergent approximations than exponential transforms, thoughthere are caveats to this; see e.g. [21, 22]. Moreover, due to a result by Sugihara [18],at least in the infinite interval setting, we know that double-exponential formulas areoptimal in the sense that the convergence rate cannot be improved further. There istherefore little point in considering, for example, triple-exponential transformations.

Table 1 summarises four of the five convergence results that will be proven. Thefifth result is a special case of the situation involving the exponential map to (−∞, 0]for which C−n geometric convergence may be achieved in certain circumstances. Nat-urally, it will be necessary to make certain assumptions in order to justify these resultsand the associated function classes will be defined in due course.

(−∞, 0 ] (−∞,∞)

exponential C−n2/3

C−√n

double-exponential C−n/ logn C−n/ logn

Table 1.1Summary of convergence rates for Chebyshev interpolation applied to functions on [0, 1]

transplanted to either (−∞, 0 ] or (−∞,∞). The constant C > 1 is different in each case.

EXPONENTIAL AND DOUBLE-EXPONENTIAL TRANSFORMS 3

The original motivation for this paper is an early work of Boyd [3] in whichChebyshev interpolation combined with a strategy termed domain truncation wasused to approximate functions defined on infinite and semi-infinite intervals. Drawingupon the steepest descent techniques developed by Elliott [6], estimates of asymptoticChebyshev coefficients for several representative example functions were determined.In this paper, we link this domain truncation technique to the problem of approximat-ing functions on the problem domain [0, 1] by introduction of a variable transformationand provide general theorems describing the asymptotic convergence behaviour of thisstrategy for large classes of functions. In [3], an emphasis was placed upon the differ-ence between convergence rates for entire functions and for functions that are merelyanalytic. We demonstrate that it is not enough for a function to be entire; to achievethe improved C−n convergence rate, it must grow at most algebraically as |x| → ∞ inthe complex plane. Concerning double-exponential maps to a semi-infinite interval,the only other investigation that we are aware of is by Hamada in [8]. That articleproposes, in essence, the method which we shall analyse in Theorem 3.3, but for anapplication related to solving differential equations. We are not aware of any previousstudy which discusses convergence rates for this transformation.

On the infinite interval, we have two results that are familiar from what one mightcall “classic sinc”, and double-exponential quadrature theory. The C−

√n convergence

rate for exponential maps was proven for cardinal sinc functions in [16, 17] and forFourier series in [25]. The last of these papers concerned the problem of using Fourierspectral methods to approximate the derivatives of an exponentially decaying functionon the real line. Though the connection to our problem of approximating functionson [0, 1] is not explicit, the arguments presented there lead us in the right directiontoward proving the equivalent results in the Chebyshev case. The C−n/ logn resultfor double-exponential transformations to the infinite interval is also familiar and wasoriginally demonstrated in the context of trapezoidal-rule quadrature in [20]. Theclosely related result pertaining to sinc interpolation was discussed in [11], and theFourier equivalent in [25]. Both the C−

√n and C−n/ logn results are widely known in

these contexts, but as far as we are aware, have not yet been explicitly demonstratedfor Chebyshev interpolants.

This paper is structured as follows. In Section 2, we make some key definitions andestablish several notational conventions that will be needed throughout. In Section 3,we provide convergence proofs regarding the results on the semi-infinite interval. InSection 4, we do the the same for the infinite interval. In Section 5, we present nu-merical experiments which verify the results of Sections 3 and 4. Finally, in Section 6,we discuss our findings and make some concluding remarks.

2. Definitions and notation. Throughout this paper, we shall consider func-tions f(x) which are continuous on [0, 1], and analytic on at least (0, 1). Functionsdefined on more general intervals [a, b] may be trivially rescaled by use of a lineartransformation, so it will suffice for us to consider only the [0, 1] case.

The four conformal maps we shall analyse consist of two exponential transformsand two double-exponential transforms and are defined as follows:

ϕE(x) = log(x), ϕ−1E (s) = es;

ψE(x) = log(x/(1− x)), ψ−1E (s) = es/(1 + es);

ϕDE(x) = − log(1− ϕE(x)), ϕ−1DE (s) = ϕ−1E (1− e−s);ψDE(x) = sinh−1(ψE(x)/π), ψ−1DE (s) = ψ−1E (π sinh(s)).

4 M. RICHARDSON

The action of these maps upon the interval [0, 1] can be summarised by

ϕE , ϕDE : [0, 1] 7→ (−∞, 0 ];

ψE , ψDE : [0, 1] 7→ (−∞,∞).

The following mnemonic may be helpful for remembering which symbol is which: Eand DE, of course, stand for exponential and double-exponential. As for ϕ and ψ, wesee that the symbol ϕ has only one endpoint tip, whereas ψ has two. These correspondto maps onto intervals with one or two points at infinity, respectively.

Letting ϕ = ϕE, ϕDE and ψ = ψE, ψDE as appropriate, we define

F(s) =

{f(ϕ−1(s)), s ∈ (−∞, 0 ];f(ψ−1(s)), s ∈ (−∞,∞).

Thus, F(s) denotes transplanted functions in the s-variable. We shall see later thatunder reasonable continuity assumptions, F(s) approaches its limiting values eitherexponentially or double-exponentially quickly as s tends towards −∞ or ±∞.

This motivates a strategy of truncating the semi-infinite and infinite intervals to[−L, 0] and [−L,L], respectively, and approximating the transplanted functions withChebyshev interpolants. We can accomplish this by applying a further transformationwhich scales [−L, 0] and [−L,L] to the interval [−1, 1]. We therefore define

FL(y) =

{f(ϕ−1(L(y − 1)/2)), y ∈ (−∞, 1 ];

f(ψ−1(Ly)), y ∈ (−∞,∞).

These are the functions to which Chebyshev interpolation will be applied on [−1, 1].See Figure 2.1 for a schematic illustration of this process.

f(x)

F(s)

FL(y)

0 1

−L 0 −L L

−1 1 −1 1

Fig. 2.1. A function f(x) on [0, 1] (top); its transplants F(s) on both (−∞, 0] (middle, left), and(−∞,∞) (middle, right); and scaled versions of each, FL(y), corresponding to mapping [−L, 0] and[−L,L] to [−1, 1] for some arbitrary domain-size parameter L (bottom). Chebyshev interpolationwill be applied to the functions FL(y) on [−1, 1], as indicated by the solid part of the curve.

It now becomes clear that there are two sources of error to be considered. The firstis the interpolation error, corresponding to sampling FL(y) at a finite number of gridpoints on [−1, 1]; the second is the domain error, which we define as the maximum of


the two quantities |FL(−1)− f(0)| and |FL(1)− f(1)|. The overall convergence ratesof our approximations will be obtained by determining the optimal balance of thesetwo sources of error. For a related discussion, see e.g. [2, Ch. 17].

It remains to specify an approximation to the function f(x) on [0, 1]. By con-struction, for any L > 0, the three functions f(x), F(s) and FL(y) have a point-wisecorrespondence across their respective domains. However, the Chebyshev interpolantPn(y) only provides an approximation to FL(y) on [−1, 1], which corresponds to onlya sub-interval of [0, 1] in the x-variable. For the maps ϕE, ϕDE to (−∞, 0], we denotethis subinterval by [xL, 0], and for the maps ψE, ψDE to (−∞,∞), by [xL, 1−xL]. Thenumber xL is positive and, depending on the transform, decreases either exponentiallyor double-exponentially to zero as L → ∞. Outside of these intervals, we define ourapproximation to be equal to the limiting endpoint values f(0) or f(1).

Our approximation to f(x) on [0, 1] is denoted by p(n)L . In the (−∞, 0] case,

letting ϕ denote either ϕE or ϕDE, we define this to be

p(n)L (x) =

{f(0), x ∈ [0, xL);

Pn(2ϕ(x)/L+ 1), x ∈ [xL, 1];(2.1)

where xL = ϕ−1(−L). Similarly, in the (−∞,∞) case, letting ψ denote either ψE orψDE, with xL = ψ−1(−L) = ψ−1(L), we define

p(n)L (x) =

f(0), x ∈ [0, xL);Pn(ψ(x)/L), x ∈ [xL, 1− xL];

f(1), x ∈ (1− xL, 1].(2.2)

The error in these approximations will be measured by examining the quantity

||f − p(n)L ||, where || · || is the L∞ norm. Since p(n)L is defined piecewise, the total error

is the maximum of the errors over each piece. The approximation error correspondingto (2.1) is therefore

||f − p(n)L || = max{||FL − Pn||y∈[−1,1], ||f − f(0)||x∈[0,xL)

}, (2.3)

and the error corresponding to (2.2) is

||f − p(n)L || = max{||FL − Pn||y∈[−1,1],

||f − f(0)||x∈[0,xL), ||f − f(1)||x∈(1−xL,1]

}. (2.4)

In each of (2.3) and (2.4), the first term on the right-hand side represents the inter-polation error, and the remaining term or terms the domain error.

To quantify the interpolation error, we shall require the notion of a Bernsteinellipse Eµ, which, for some µ > 0, we define to be the open set in the complex planebounded by the curve {

1

2

(eµeiθ + e−µe−iθ

): θ ∈ [0, 2π]

}.

The degree n Chebyshev interpolant to FL(y) on y ∈ [−1, 1] can be written

Pn(y) =

n∑k=0

ckTk(y),

6 M. RICHARDSON

where Tk are the Chebyshev polynomials of the first kind. Pn corresponds to inter-polation at the second-kind Chebyshev points yk = cos(kπ/n), and the ck are aliasedexpansion coefficients corresponding to the function samples FL(yk).

If FL is analytic in a particular ellipse Eµ, and Mµ denotes the maximum absolutevalue of FL in Eµ, then a bound for the error in n+ 1 point Chebyshev interpolationgoing back to Bernstein in 1912 is [23, Ch. 8]

||FL − Pn||y∈[−1,1] ≤4

µMµe

−µn. (2.5)

The domain error depends upon both the function f and the conformal map. Inparticular, if f satisfies, for some real A,α > 0,

|f(x)− f0| ≤ A|x|α, x ∈ [0, 1], (2.6)

then it is straightforward to show that there exist A1, A2 > 0 such that

||f − f0||x∈[0,xL) ≤{A1e

−αL (for the map ϕE);

A2e−αeL (for the map ϕDE).

(2.7)

Similarly, if f satisfies, for some real B, β > 0,

|f(x)− f0| ≤ B|x|β ,|f(x)− f1| ≤ B|1− x|β , x ∈ [0, 1], (2.8)

then there exist B1, B2 > 0 such that

max{||f − f0||x∈[0,xL), ||f − f1||x∈(1−xL,1]

}≤{B1e

−βL (for the map ψE);

B2e−βeL (for the map ψDE).

(2.9)

We now have enough preparation to proceed to the theorems.

3. Maps to the semi-infinite interval. Typically in exponential and double-exponential mapping methods, one is concerned with a function analytic in an infinitestrip about the real axis. Such strips are the natural setting for trapezoidal rulequadrature, and for sinc and Fourier interpolation. The vast majority of past literaturehas focussed on this setting; see [10, 11, 12, 14, 15, 16, 17, 19, 20]. Given someconformal map from the infinite interval to [0, 1], the image of an infinite strip underthis transformation corresponds to a particular region of analyticity in the x-variablefrom which suitable definitions of function classes follow; see [21, 22].

Since in this paper we will be working with Chebyshev interpolants, we shall beconcerned primarily with ellipses. Nevertheless, we shall see that for the maps ψE andψDE to (−∞,∞) considered in Section 4, in order to accommodate the limiting caseL→∞, this still amounts to requiring analyticity in an infinite strip in the s-variable.On the other hand, for the maps ϕE and ϕDE to (−∞, 0], we shall require analyticityin a region which is essentially parabolic. We begin by describing this case.

Given some complex number ζ = ζ1 + iζ2, ζ1 < 0, ζ2 6= 0, we define the quasi-parabolic region Pζ in the complex s-plane to be the the union of 1) the interior of aparabola symmetric about the real axis passing through both ζ and the origin, and 2)the interior of an ellipse with foci at 2ζ1, 0 also passing through ζ. We consider thisregion to be an open set bounded by the parabola for Im(s) < ζ1, and by the ellipsefor Im(s) ≥ ζ1. See Figure 3.1.


ζ

Fig. 3.1. A visualisation of the quasi-parabolic region Pζ in the s-plane. The complex numberζ = ζ1 + iζ2 is indicated, and the two black dots at s = 2Re(ζ), 0 are the foci of the ellipse.

A formal description of this region is as follows:

P(1)ζ =

{s ∈ C : ζ22Re(s) < ζ1Im(s)2

},

P(2)ζ =

{s ∈ C : ζ22 (Re(s)− ζ1)2 + (ζ21 + ζ22 )Im(s)2 < ζ22 (ζ21 + ζ22 )

},

Pζ = P(1)ζ ∪ P

(2)ζ .

The reason for requiring a region of analyticity such as this is in order to ensure thatfor all L ≥ 2|ζ1|, the interior of every ellipse which passes through ζ and has fociat −L, 0 is contained within the quasi-parabolic region Pζ . Incidentally, there is nospecial reason why the appended region needs to be elliptical: any sufficiently largeregion containing a fixed neighbourhood of the origin could have been used. We havechosen to use an ellipse for convenience.

The transforms ϕ−1E and ϕ−1DE are both analytic functions throughout Pζ whichmap it onto the following regions in the complex x-plane:

Qζ = ϕ−1E (Pζ),Rζ = ϕ−1DE (Pζ).

These maps are not one-to-one. To make them one-to-one, one must view the imagesas infinitely sheeted Riemann surfaces wrapping around x = 0. It is in this sense, asRiemann surfaces, that we shall regard the regions Qζ and Rζ . See Figure 3.2.

s-plane

x-plane

Pζ

Qζ

Rζ

Fig. 3.2. The functions ϕ−1E and ϕ−1

DE map the quasi-parabolic region Pζ in the s-plane onto

the regions Qζ = ϕ−1E (Pζ) and Rζ = ϕ−1

DE (Pζ) in the x-plane. Qζ and Rζ are both infinitely sheetedRiemann surfaces which wrap around x = 0. We also show a sample ellipse (dashed line) and someinterpolation nodes (dots) in the s plane, and the images of each in the regions Qζ ,Rζ .

8 M. RICHARDSON

The following theorem is the more general of our two results concerning thetransformation ϕE. It describes the convergence rate of a sequence of Chebyshevinterpolants, given an assumption of analyticity in the region Qζ .

Theorem 3.1. Let f satisfy the condition (2.6) and be analytic in the region Qζfor some ζ, where it satisfies |f(x)| ≤M for some M . Let p

(n)L be the approximation

defined by (2.1) corresponding to the exponential transform ϕ = ϕE. If L is definedfor any c > 0 by L = cn2/3, then for all sufficiently large n,

||f − p(n)L || ≤ C−n2/3

for some C > 1.

Proof. Since f(x) is analytic in Qζ , the transplanted function F(s) is analyticin the quasi-parabolic region Pζ , an example of which is given in Figure 3.2. Conse-quently, given any L > 0, if we define p = 2ζ1/L+ 1, q = 2ζ2/L as the real and imagi-nary parts of the image of ζ in the y-variable, then the function FL(y) = F(L(y−1)/2)is analytic inside a Bernstein ellipse Eµ whose parameter µ is given by1

µ = sinh−1√

1

2

(p2 + q2 − 1 +

√(1− p2 − q2)2 + 4q2

). (3.1)

It can be shown that with µ defined by (3.1), there exists a constant η > 0 such thatµ > η/

√L for all L > 0. Thus, from (2.3), (2.5) and (2.7), we have

||f − p(n)L || ≤ max

{4√L

ηMµe

−ηn/√L, A1e

−αL

}.

Now for any L ≥ 2|ζ1|, with µ defined as above, the ellipse L(Eµ−1)/2 in the s-planeis entirely contained within the region Pζ . Thus Mµ may be bounded by M , themaximum absolute value of f in Qζ , which exists by assumption. We then see thatfor any c > 0, the choice L = cn2/3 gives

||f − p(n)L || ≤ max

{4M√c

ηn1/3, A1

}e−δn

2/3

,

where δ = min(η/√c, αc). This is valid for all n ≥ (2|ζ1|/c)3/2. The result now follows

with a suitable C chosen to absorb the bracketed terms.The proof of Theorem 3.1 is methodologically very similar to the proofs of The-

orems 3.3, 4.1 & 4.2. The general pattern followed in these four theorems is to fix aparticular region of analyticity in the s-plane and then deduce how rapidly the Bern-stein ellipses in the y-variable must shrink as L → ∞. For example, in the abovetheorem, we showed that µ = O(L−1/2), which implies that the choice L = O(n2/3)balances the error contributions. Theorem 3.2, however, is fundamentally differentto the other theorems in that we shall fix the size of the Bernstein ellipse in the y-variable at the outset and allow the corresponding regions of analyticity in the s- andx-variables to grow without bound as L → ∞. A Bernstein ellipse of fixed size is ofcourse necessary if one wishes to obtain true C−n geometric convergence.

Before proceeding, it will be helpful to consider some additional properties of thetransformation ϕ−1E (s) = es. For any integer k, this conformal map transplants the

1This is one possible formula for computing the elliptic coordinate parameter µ in terms of thereal and imaginary parts of some complex number p + iq which is assumed to lie on the ellipse’sboundary. The elliptic coordinate system is defined to be: p = cosh(µ) cos(θ), q = sinh(µ) sin(θ), forµ > 0, θ ∈ [0, 2π].


infinite strip (2k−1)π < |Im(s)| < (2k+1)π in the complex s-plane onto the slit planeC\(−∞, 0] in the complex x-plane. Corresponding to each k are individual sheets ofa Riemann surface in the x-plane, each separated by a branch cut along the negativereal axis. See Figure 3.3 for an illustration.

3πi

πi

−πi

−3πi

0−L1−L2

0 1

s-plane

x-plane

Fig. 3.3. For each k ∈ Z, ϕ−1E maps the infinite strips (2k − 1)π < |Im(s)| < (2k + 1)π in the

s-plane onto sheets of a Riemann surface in the x-plane separated by the branch cut (−∞, 0]. In thebottom panel is the image of the principal strip −π < |Im(s)| < π. We also show two sample ellipsesL1(Eµ − 1)/2 and L2(Eµ − 1)/2 in the s-plane and their images in the x-plane. The first ellipseremains within the principal strip and does not cross the branch cut in the x-plane; the opposite istrue for the second ellipse. Sample Chebyshev interpolation nodes corresponding to the first ellipseare also shown, along with their images.

We see from Figure 3.3 that in the limit L → ∞, we shall require analyticity ofthe function F(s) in the entire complex s-plane. However, since only infinite stripsof width 2π are mapped in one-to-one fashion onto the x-plane, we must require thatthe problem function f(x) is analytic on the entire Riemann surface in the x-plane,including on the branch cuts. This condition ensures that ellipses in the s-plane cancross over onto adjacent strips without difficulty.

Allowing these regions of analyticity in the s- and x-variables to grow withoutbound comes with an associated cost. This is in having to account for the quantityMµ in (2.5), which will now grow rapidly as L increases. It turns out that the requiredcondition upon the function F is that it should be a function of exponential type, i.e.,satisfy |F(s)| = O(exp |s|ρ) for some ρ > 0 as |s| → ∞. If this is the case, then one

can prove geometric convergence of the approximations p(n)L . However, most problem

functions f in the x-variable do not correspond to functions of exponential type inthe s-variable. One can see that this is the case by recalling that for the map ϕ−1E ,we have F(s) = f(es); thus, only functions which grow algebraically in the x-planewill correspond to functions of exponential type in the s-variable.

We state these concepts precisely in the following theorem:

Theorem 3.2. Let f satisfy the condition (2.6) and be analytically continuablealong any curve in the complex plane disjoint from x = 0. Suppose it satisfies |f(x)| =O(|x|γ) as |x| → ∞ for some γ > 0 uniformly on all such curves. Let p

(n)L be the

approximation defined by (2.1) corresponding to the exponential transform ϕ = ϕE. IfL is defined for any c > 0 by L = cn, then

||f − p(n)L || ≤ C−n for some C > 1.

10 M. RICHARDSON

Proof. From (2.3), (2.5) and (2.7), we have

||f − p(n)L || ≤ max

{4Mµ

µe−µn, A1e

−αL}. (3.2)

The quantity Mµ is defined as the maximum absolute value of the function FL(y) =f(ϕ−1E (L(y−1)/2)) on the ellipse Eµ. Our assumption on the growth rate of f impliesMµ ≤W exp (Lγ (eµ + e−µ − 2) /4) for some γ,W > 0. Thus with L = cn,

||f − p(n)L || ≤ max

{4W

µe−w(µ)n, A1e

−αcn},

where w(µ) = µ− cγ4 (eµ+e−µ−2). Using a Taylor series argument, it is straightforwardto show that for all c, γ > 0 this function takes positive values in some interval (0, µ̃),and that w attains a maximum at some point µm ∈ (0, µ̃). (For the purposes ofproving this theorem, it is not crucial to compute the maximum µ, as any value in(0, µ̃) will suffice; we do so in order to obtain as strong a bound as possible.) Usingthis value µm, we then have

||f − p(n)L || ≤ max

{4W

µm, A1

}e−δn,

where δ = min(w(µm), αc). The result now follows with a suitable choice of C.It is quite remarkable that true geometric convergence is attainable under a

domain-truncation strategy. However, we reiterate that this result only applies toa relatively small class of functions: see Section 5 for examples. We note again thatBoyd was aware of this result in a less general form [3].

Finally in this section, we consider the double-exponential transform ϕDE.Theorem 3.3. Let f satisfy the condition (2.6) and be analytic in the region Rζ

for some ζ, where it satisfies |f(x)| ≤M for some M . Let p(n)L be the approximation

defined by (2.1) corresponding to the double-exponential transform ϕ = ϕDE. If L isdefined for any c > 0 by L = log(cn), then for all sufficiently large n,

||f − p(n)L || ≤ C−n/ logn for some C > 1.

Proof. Since f(x) is analytic on Rζ , F(s) is analytic in Pζ , as in Figure 3.2. Ifwe again define µ according to (3.1), then using (2.3), (2.5) and (2.7), we have

||f − p(n)L || ≤ max

{4√L

ηMµe

−ηn/√L, A2e

−αeL}.

As before, for any L ≥ 2|ζ1|, Mµ may be bounded by the maximum absolute value off in Rζ . Choosing L = log(cn) for any c > 0 then gives

||f − p(n)L || ≤ max

{4√

log(cn)

ηMe−ηn/

√log(cn), A1e

−αcn

},

which is valid for all n ≥ 1/c. If c ≤ 1, then e−ηn/√

log(cn) ≤ e−ηn/ logn; if c > 1, then

there is a constant C1 > 1 such that e−ηn/√

log(cn) ≤ C−n/ logn1 for all n ≥ 1/c. In

either case, the result follows with a suitable choice of C.


4. Maps to the infinite interval. Next, by considering the transformations ψE

and ψDE, we verify that the C−√n and C−n/ logn results from classic sinc and double-

exponential quadrature theory also hold in the Chebyshev interpolation setting.In the proofs of the following two theorems, we shall see that in the limit L→∞,

the region of analyticity that bounds the ellipses in the s-variable is now required tobe an infinite strip, rather than a (quasi-)parabola, as was the case for the maps ϕE

and ϕDE. However, there is also a more subtle difference associated with this setting:regardless of whether or not f has any singularities, the transplanted function F(s)always does have singularities, due to those present in the transplanting maps ψ−1E

and ψ−1DE . Consequently, in contrast to the results of the previous section, there canbe no class of functions for which, say, geometric convergence is attainable.

The regions of analyticity associated with the transformations ψE and ψDE areprecisely those which appear in classic sinc and double-exponential theory, and aredefined as follows. Given some 0 < d ≤ π, let Sd be the infinite strip of half-width din the complex s-plane described by

Sd = { s ∈ C : |Im(s)| < d } .

The transforms ψ−1E and ψ−1DE are both analytic functions throughout Sd which mapit onto the following regions in the complex x-plane:

Td = ψ−1E (Sd),Ud = ψ−1DE (Sd).

There is a one-to-one correspondence between Sd and Td, but not between Sd and Ud.As before, a one-to-one correspondence exists in the latter case if Ud is regarded asan infinitely sheeted Riemann surface wrapping around both x = 0 and x = 1. Tdis simply a lens-shaped region formed by two circular arcs meeting at x = 0, 1 withhalf-angle d. See Figure 4.1.

s-plane

x-plane

Sd

Td

Ud

Fig. 4.1. The functions ψ−1E and ψ−1

DE map Sd onto Td = ψ−1E (Sd) and Ud = ψ−1

DE (Sd). Td is alens-shaped region formed by two circular arcs meeting with half-angle d. Ud is an infinitely sheetedRiemann surface wrapping around x = 0, 1. We also show a sample ellipse (dashed line) and someinterpolation nodes (dots) in the s plane and the images of each in the regions Td,Ud.

We begin with the C−√n result concerning the exponential transform ϕE.

Theorem 4.1. Let f satisfy the condition (2.8) and be analytic in the region

Td for some d > 0, where it satisfies |f(x)| ≤ M for some M . Let p(n)L be the

12 M. RICHARDSON

approximation defined by (2.2) corresponding to the exponential transform ψ = ψE. IfL is defined for any c > 0 by L = c

√n, then

||f − p(n)L || ≤ C−√n for some C > 1.


||f − p(n)L || ≤ max

{4

µMµe

−µn, B1e−βL

}.

Due to the assumed analyticity of F(s) in the strip Sd, for any L > 0, the functionFL(y) = F(Ly) is analytic inside the Bernstein ellipse Eµ with semi-minor axis lengthd/L. The corresponding ellipse parameter is

µ = log(d/L+

√d2/L2 + 1

), (4.1)

and if we define η = log(d+√d2 + 1), then µ > η/L for all L > 0.

Now, since f is bounded in Td, we may infer that Mµ ≤ M for all L > 0. Thus,setting L = c

√n for any c > 0, we have

||f − p(n)L || ≤ max

{4M

ηc√ne−η

√n/c, B1e

−βc√n

},

≤ max

{4M

ηc√n,B1

}e−δ√n,

where δ = min(η/c, βc). The result follows with a suitable choice of C.In the final theorem, we demonstrate the C−n/ logn convergence result associated

with the double-exponential map ψDE.Theorem 4.2. Let f satisfy (2.8) and be analytic in the region Ud for some

d > 0, where it satisfies |f(x)| ≤ M for some M . Let p(n)L be the approximation

defined by (2.2) corresponding to the double-exponential transform ψ = ψDE. If L isdefined for any c > 0 by L = log(cn), then for all sufficiently large n,

||f − p(n)L || ≤ C−n/ logn for some C > 1.


||f − p(n)L || ≤ max

{4

µMµe

−µn, B2e−βeL

}.

As in Theorem 4.1, for any L > 0, FL(y) is analytic inside the Bernstein ellipse Eµ,where µ is given by (4.1), and again we have µ > η/L where η = log(d +

√d2 + 1).

Also, for all L > 0, we may again bound Mµ by M .Setting L = log(cn) for any c > 0 then yields

||f − p(n)L || ≤ max

{4M

ηlog(cn)e−ηn/ log(cn), B2e

−βcn}.

If c ≤ 1, then e−ηn/ log(cn) ≤ e−ηn/ logn; if c > 1, then there is a constant C1 > 1 such

that e−ηn/ log(cn) ≤ C−n/ logn1 . In either case, the result follows with a suitable choiceof C.


5. Numerical experiments. We now consider some examples of functions de-fined on [0, 1] which may be treated by the above techniques. Consider for instance

f1(x) = x, f2(x) =√x, f3(x) = 1 + x1/4, f4(x) = x log x.

Each of these functions is free from singularities away from x = 0, and grows onlyalgebraically in the complex plane. The conditions of Theorem 3.2 are met in eachcase, and therefore using the transform ϕE and setting L = cn will lead to geometricconvergence.

In contrast to this, the following functions, for one reason or another, do not meetthe conditions of Theorem 3.2:

f5(x) = sin(x), f6(x) =√x cosx, f7(x) =

1 + x1/4

x2 − x+ 1, f8(x) =

x log x

1 + x.

The reasons are that although f5 and f6 do not have any singularities away fromx = 0, they grow faster than algebraically in the complex plane. By contrast, f7and f8 grow only algebraically, but have singularities away from x = 0. Functionsf5, f6, f7, f8 do all meet the conditions of Theorems 3.1 and 3.3, however. Thus byusing the transforms ϕE or ϕDE, and setting L = cn2/3 or L = log(cn) as appropriate,

we may expect to obtain either C−n2/3

or C−n/ logn convergence, respectively.Finally, consider

f9(x) = x1/3(1− x)2/3 + x, f10(x) =√x− x2 tanh(3x− 2).

Since these functions have singularities at both endpoints, they cannot satisfy theconditions of any of Theorems 3.1, 3.2, or 3.3. They do, however, satisfy the conditionsof Theorems 4.1 and 4.2. Thus, by using the transforms ψE or ψDE, and settingL = c

√n or L = log(cn) as appropriate, we may expect to achieve either C−

√n or

C−n/ logn convergence, respectively. We note also that as soon we are forced to workwith the maps ψE and ψDE rather than ϕE and ϕDE, issues such as growth rates inthe complex plane or the presence of additional singularities become factors that areirrelevant with regards to the overall convergence rate.

||fj − p(n)L ||

n0 10 20 30 40 50 60 70 80 90 100

10−15

10−10

10−5

100

c = 2c = 3.2

c = 0.8

c = 1

c = 1.6

c = 2.4

Fig. 5.1. Convergence of the approximations p(n)L defined by (2.1) corresponding to the expo-

nential transformation ϕE for f2(x) =√x (dots), and f6(x) =

√x cosx (circles). In each case L is

set according to L = cn, for some sample values of c, which are indicated in the plot. For f2, weobserve geometric convergence as predicted by Theorem 3.2.

14 M. RICHARDSON

There are of course also many examples of functions which are not suited toany of these classes, for example sin(1/x) and e

√x/x. We note, however, that often,

functions which blow up only algebraically at the endpoints (such as the latter ofthese two examples) may be treated in our framework after multiplication of a suitablealgebraic factor.

Let us now provide some numerical confirmation of the theoretical convergencerates. We consider first the functions f2 and f6, which are ostensibly similar, yetexhibit markedly different convergence behaviour under the transformation ϕE. Set-ting L = cn for any c > 0, we observe geometric convergence for f2, but, due to thepresence of the cosine, sub-geometric convergence for f6; see Figure 5.1.

||fj − p(n)L ||

n2/30 5 10 15 20 25 30 35 40

10−15

10−10

10−5

100

c = 1.5 c = 3 c = 0.5c = 0.8

c = 15

c = 9

Fig. 5.2. Convergence of the approximations p(n)L defined by (2.1) corresponding to the

exponential transformation ϕE for f2(x) =√x (dots), and f6(x) =

√x cosx (circles). L is

set according to L = cn2/3, for some sample values of c, which are indicated in the plot. For

each function, we observe C−n2/3

convergence as predicted by Theorem 3.1.

||fj − p(n)L ||

n/ logn2 4 6 8 10 12 14 16 18 20

10−15

10−10

10−5

100

c = 2.3 c = 6 c = 22

c = 160

c = 0.55

c = 0.4


double-exponential transformation ϕDE for f2 (dots), and f6 (circles). L is set according toL = log(cn), for some sample values of c, which are indicated in the plot. For each function,

we observe C−n/ logn convergence as predicted by Theorem 3.3.

Figure 5.1 provides an illustration of the importance of growth rates in the com-plex plane for the map ψE. We considered for illustration the functions f2 and f6,but could equally well have compared any of f1, f2, f3, f4 to any of f5, f6, f7, f8 andobserved similar effects. Particularly interesting is the function f8, which has a sin-


gularity (a simple pole) lying on the branch cut (−∞, 0]. Due to the mapping effectsof ϕE described in Figure 3.3, this is enough to ensure that ellipses in the s-variableremain bounded by a point lying on the boundary of the principal infinite strip. In

contrast to this, we see in Figure 5.2 that setting L = cn2/3 leads to C−n2/3

con-vergence for both f2 and f6. The same functions are considered again in Figure 5.3,though this time with the double-exponential transform ψDE. As expected, settingL = log(cn), leads to C−n/ logn convergence.

In Figures 5.4 and 5.5, we approximate the functions f4 and f9 under the trans-formations ψE and ψDE. Due to the singularity present at x = 1 in f9, we have usedMaple to obtain the necessary data down to the level of around 10−15. Barycen-tric interpolation [1] seems to be the most straightforward way of implementing this,since one never need worry about computing Chebyshev expansion coefficients. Allresults in the other figures were obtained using Chebfun [24], which automates theinterpolation process, but shares the limitations of double-precision arithmetic.

||fj − p(n)L ||

√n

0 5 10 15 20 25 30 35 4010

−15

10−10

10−5

100

c = 1

c = 6

c = 4.9

c = 0.8c = 3.1

c = 2.3


exponential transformation ψE for f4(x) = x log(x) (dots), and f9(x) = x1/3(1 − x)2/3 + x(circles). L is set according to L = c

√n for some sample values of c which are indicated in

the plot. For each function, we observe C−√n convergence as predicted by Theorem 4.1.

||fj − p(n)L ||

n/ logn0 10 20 30 40 50

10−15

10−10

10−5

100

c = 0.15

c = 0.08

c = 70c = 5

c = 1.5c = 0.3


dobule-exponential transformation ψDE for f4 (dots), and f9 (circles). L is set according toL = log(cn) for some sample values of c which are indicated in the plot. For each function,

we observe C−n/ logn convergence as predicted by Theorem 4.2.

16 M. RICHARDSON

We also give a brief experimental illustration of the differences between what onemight call the resolution properties of the four maps. In the sense that we intend,resolution power is the notion of the average number of points-per-wavelength (ppw)required to resolve high frequency oscillations. For example, Fourier series are knownto require at least 2 ppw before exponential convergence sets in, and Chebyshev seriesπ ppw [7]. Table 2 shows the results of a simple resolution experiment involvingthe function sin(Mx). Without drawing rigorous conclusions, it is apparent that inthis respect, Chebyshev interpolation under the maps to (−∞, 0] is much superior toChebyshev interpolation under the maps to (−∞,∞). This is still true even afterone accounts for the π/2 factor associated with using Fourier or sinc interpolants inthe (−∞,∞) case. A forthcoming publication by Richardson and Trefethen and willconduct a detailed analysis in this regard.

ϕE ϕDE ψE ψDE

M n ppw n ppw n ppw n ppw

1 84 – 46 – 443 – 99 –

10 144 90.5 64 40.2 724 455 168 106

100 501 31.5 204 12.8 2 073 130 499 31.4

1 000 3 334 20.9 1 295 8.14 12 967 81.5 3 097 19.5

10 000 29 730 18.7 11 582 7.28 118 322 74.3 27 458 17.3

Table 5.1An indication of the resolution power of the four maps. For each, the function sin(Mx)

has been approximated to machine precision by an n+1-point Chebyshev interpolant to F(s)on either [−L, 0] or [−L,L]. The parameter L is determined in each case such that |F(s)| isnegligible outside of these intervals. ppw stands for “points-per-wavelength”, which we defineto be 2πn/M . We note that L grows with M , albeit slowly; thus it is not clear if these ppwestimates converge as M → ∞. We leave a theoretical analysis of resolution issues to afuture work.

Finally, we suggest how the techniques described in this paper can used for prac-tical computation. Utlilising the black-box interpolation capabilities of Chebfun, wemerely need to define the transplanted function FL(y) for some L, which we com-pute in advance such that FL(−1) ≈ 10−16, and then rely on Chebfun to adaptivelycompute the correct number of interpolation points to use. This a priori domaindetermination technique also was utilised in the Sincfun software system describedin [14]. The Matlab code is extremely simple; here is an example involving ϕDE :

f6 = @(x) sqrt(x).*cos(x); L = 4.3; % function; domain

phiIDE = @(s) exp(1-exp(-s)); % inverse transform

Pn = chebfun(@(y) f6(phiIDE(L*(y-1)/2))); % chebfun on [-1,1]

It is then a straightforward matter to evaluate such a representation:

phiDE = @(x) -log(1-log(x)); % transform

pnL = @(x) Pn(2*phiDE(x)/L+1); % representation

Of course, one must account for the endpoint values separately in line with (2.1) sincethe chebfun Pn only provides an approximation for x ∈ [xL, 1]. Similar code can easilybe written to deal with situations involving the other three maps.


6. Discussion. Preliminary numerical experiments such as those conducted inthe previous section provided the initial stimulus for this investigation. These experi-ments seemed to suggest that variable transform methods onto a semi-infinite intervalwere often preferable to those onto an infinite interval. The purpose of this paper hasbeen to attempt to quantify this observation.

The majority of previous investigations concerning exponential or double expo-nential variable transform methods have been concerned with the process of trans-planting the problem function to an infinite interval and subsequently approximatingit there using some infinite interval basis set. If one assumes from the outset that theonly sensible way to go about such approximation is to use sinc functions or Fourierseries, then of course, the notion of transplanting functions to a semi-infinite intervaldoes not arise. This is perhaps the reason that the discussion of transformations to asemi-infinite interval is conspicuously absent from the literature.

The strategy of using Chebyshev interpolants as a viable alternative to sinc func-tions was recommended in both [4] and [14], and we wish to reinforce that positionhere. In [14], the primary reason for citing Chebyshev polynomials as a good alterna-tive was essentially ease of computation. Crucially, however, a further advantage ofChebyshev polynomials is their ability to handle semi-infinite interval maps. This isin contrast to sinc, or Fourier discretisations, which are most readily applicable in theinfinite interval case. It should be noted that a cursory inspection of the literature(see e.g. [16, 21]) may suggest that the problem of dealing with functions defined ona semi-infinite interval has been effectively addressed. However, one will find that thestrategy followed is to transplant again to an infinite interval. This, of course, doesnot really address the issue of introducing unnecessary waste into the computation onone side of the domain if there happens to be no endpoint singularity present there.

We have shown in Section 3 that one may expect genuinely superior asymptoticconvergence rates in the case of the exponential map ϕE. For the double-exponentialtransform ϕDE, although the exponent in the asymptotic convergence rate is the sameas for ψDE, the constants involved are better: numerical evidence provided in Table 2suggests that the resolution properties associated with ϕDE are superior.

Our recommendation, then, is to use the map ϕDE wherever possible. In practice,this means that if one has a function which has only a single endpoint singularity, thenthe map ϕDE should usually be used. Functions with singularities at both endpointsmust of course be dealt with with one of ψE or ψDE (at least, if one requires a singleglobal representation), and similarly we recommend the use of the transformationψDE wherever possible. The caveat “wherever possible” in both cases refers to thefact that functions can arise in practice which will not be analytic in the regions Rζor Uζ , for any ζ, corresponding to the maps ϕDE or ψDE. However, situations wherethis is the case occur infrequently. One can, if pressed, construct examples of functionsthat are not analytic in these regions: such functions tend to have an infinite numberof singularities which cluster near the wrap-around points of the Riemann surface;Tanaka et al. [21] provided examples of this in the infinite interval case. The numericalexperiments there also suggest that in such cases the convergence rate reverts back toC−√n anyway; so in practice, one probably need not worry too much about whether

or not it is safe to use ϕDE or ψDE.

An additional consideration is that that in double precision arithmetic, we usuallycannot approximate functions such as f9 and f10 anyway, even using the maps ψE, ψDE.The reason is the inability of the floating point number system to cope well withsingularities at x = 1; these matters were discussed in [14], and also previously in [17].

18 M. RICHARDSON

At x = 0, the situation is much better, since the density of floating point numbersthere is much greater [9, 13]. Though one may not always have a choice, in floatingpoint arithmetic it is always better to position the singularities at x = 0 if it ispossible to do so. This is also the reason we chose the default problem interval to be[0, 1], rather than, say, [−1, 1]. For these reasons, and in light of the demonstratedsuperiority of the semi-infinite interval maps, we believe that the infinite interval mapsψE, ψDE should be used infrequently in practice. It only makes sense to use these mapsif one is prepared to go beyond the usual floating-point arithmetic.

From a purist’s perspective, if one does happen to be approximating on a trun-cated infinite interval after transplantation using ψE or ψDE, then Fourier series arepreferential to Chebyshev polynomials, due to the π/2 saving; this point has alsobeen made previously by Boyd [5]. However, given the availability and convenience ofChebfun, unless one has an excellent reason for using Fourier series, it hardly seemsworth going to the extra effort of coding from scratch.

It is also worth noting that we have not provided any converse theorems, sowe are not able to conclude that the bounds we have provided are best possible.However, the numerical evidence does suggest that our bounds are sensible. Oneway to improve the bounds associated with ψE and ψDE would be to modify thecontinuity assumption (2.8) in such a way that differing decay rates of F as s→ ±∞are accounted for. Such steps are often taken in the sinc literature (see e.g. [10, 16,17]), but since the modifications are trivial, yet cumbersome, and do not affect theasymptotic convergence rates, we have chosen not to do so here in order to expeditethe discussion. We also mention once more the optimality result regarding ψDE dueto Sugihara [18].

Acknowledgements. The author would like to thank Nick Trefethen, AlexTownsend and Pedro Gonnet for valuable discussions and input.

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