a quasi-interpolation scheme for periodic data based on multiquadric trigonometric b-splines

11
Journal of Computational and Applied Mathematics 271 (2014) 20–30 Contents lists available at ScienceDirect Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam A quasi-interpolation scheme for periodic data based on multiquadric trigonometric B-splines Wenwu Gao a,b , Zongmin Wu b,a School of Economics, Anhui University, Hefei, PR China b Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai, PR China article info Article history: Received 28 May 2013 Received in revised form 3 March 2014 MSC: 41A25 41A30 65D10 65D15 Keywords: Quasi-interpolation Periodic data Trigonometric divided differences Multiquadric trigonometric B-splines Active contours abstract Multiquadric (MQ) quasi-interpolation has been studied extensively in the literature. How- ever, the MQ quasi-interpolant is not well defined for periodic data, since its kernel (the MQ function) itself is not periodic. Note that in many applications, the data may arise from a closed curve (surface) and thus possess some kind of periodicity, for example, analysis of geodetic and meteorological data, constructing active contours, estimating the region of attraction of dynamical systems, and so on. Therefore, it is meaningful to construct a quasi- interpolant (whose kernel itself is periodic) for periodic data. Combining the construction techniques of the MQ quasi-interpolant and the trigonometric B-spline quasi-interpolant, the paper constructs such a quasi-interpolant. The quasi-interpolant couples together the periodicity of the trigonometric B-spline quasi-interpolant and the smoothness of the MQ quasi-interpolant. Moreover, both the quasi-interpolant and its derivatives are periodic. The quasi-interpolant covers the trigonometric B-spline quasi-interpolant as a special case. In addition, the error estimate shows that a proper shape parameter can be chosen such that the quasi-interpolant provides the same approximation order as a trigonometric B-spline quasi-interpolant for a periodic function. Furthermore, the quasi-interpolant gives bet- ter approximations to high-order derivatives than the trigonometric B-spline quasi- interpolant, as illustrated by numerical examples presented at the end of the paper. © 2014 Elsevier B.V. All rights reserved. Contents 1. Introduction............................................................................................................................................................................................. 21 2. Preliminaries ........................................................................................................................................................................................... 21 2.1. Trigonometric divided differences ............................................................................................................................................ 22 2.2. The trigonometric B-spline quasi-interpolation scheme ......................................................................................................... 22 3. Construction of the main quasi-interpolation scheme ......................................................................................................................... 23 4. Numerical examples ............................................................................................................................................................................... 27 5. Conclusions and discussions .................................................................................................................................................................. 29 Acknowledgments .................................................................................................................................................................................. 29 References................................................................................................................................................................................................ 29 This work is supported by SGST 12DZ 2272800. Corresponding author. E-mail addresses: [email protected] (W. Gao), [email protected] (Z. Wu). http://dx.doi.org/10.1016/j.cam.2014.03.012 0377-0427/© 2014 Elsevier B.V. All rights reserved.

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Page 1: A quasi-interpolation scheme for periodic data based on multiquadric trigonometric B-splines

Journal of Computational and Applied Mathematics 271 (2014) 20–30

Contents lists available at ScienceDirect

Journal of Computational and AppliedMathematics

journal homepage: www.elsevier.com/locate/cam

A quasi-interpolation scheme for periodic data based onmultiquadric trigonometric B-splines✩

Wenwu Gao a,b, Zongmin Wu b,∗

a School of Economics, Anhui University, Hefei, PR Chinab Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai,PR China

a r t i c l e i n f o

Article history:Received 28 May 2013Received in revised form 3 March 2014

MSC:41A2541A3065D1065D15

Keywords:Quasi-interpolationPeriodic dataTrigonometric divided differencesMultiquadric trigonometric B-splinesActive contours

a b s t r a c t

Multiquadric (MQ) quasi-interpolation has been studied extensively in the literature. How-ever, the MQ quasi-interpolant is not well defined for periodic data, since its kernel (theMQ function) itself is not periodic. Note that in many applications, the data may arise froma closed curve (surface) and thus possess some kind of periodicity, for example, analysisof geodetic and meteorological data, constructing active contours, estimating the region ofattraction of dynamical systems, and so on. Therefore, it is meaningful to construct a quasi-interpolant (whose kernel itself is periodic) for periodic data. Combining the constructiontechniques of the MQ quasi-interpolant and the trigonometric B-spline quasi-interpolant,the paper constructs such a quasi-interpolant. The quasi-interpolant couples together theperiodicity of the trigonometric B-spline quasi-interpolant and the smoothness of the MQquasi-interpolant. Moreover, both the quasi-interpolant and its derivatives are periodic.The quasi-interpolant covers the trigonometric B-spline quasi-interpolant as a special case.In addition, the error estimate shows that a proper shapeparameter can be chosen such thatthe quasi-interpolant provides the same approximation order as a trigonometric B-splinequasi-interpolant for a periodic function. Furthermore, the quasi-interpolant gives bet-ter approximations to high-order derivatives than the trigonometric B-spline quasi-interpolant, as illustrated by numerical examples presented at the end of the paper.

© 2014 Elsevier B.V. All rights reserved.

Contents

1. Introduction............................................................................................................................................................................................. 212. Preliminaries ........................................................................................................................................................................................... 21

2.1. Trigonometric divided differences ............................................................................................................................................ 222.2. The trigonometric B-spline quasi-interpolation scheme ......................................................................................................... 22

3. Construction of the main quasi-interpolation scheme......................................................................................................................... 234. Numerical examples ............................................................................................................................................................................... 275. Conclusions and discussions .................................................................................................................................................................. 29

Acknowledgments .................................................................................................................................................................................. 29References................................................................................................................................................................................................ 29

✩ This work is supported by SGST 12DZ 2272800.∗ Corresponding author.

E-mail addresses:[email protected] (W. Gao), [email protected] (Z. Wu).

http://dx.doi.org/10.1016/j.cam.2014.03.0120377-0427/© 2014 Elsevier B.V. All rights reserved.

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W. Gao, Z. Wu / Journal of Computational and Applied Mathematics 271 (2014) 20–30 21

1. Introduction

Periodic data exist widely in practical applications. Examples include reconstructing any simply connected closedcurve/surface (i.e., constructing active contours, estimating the region of attraction of dynamical systems, geometric mod-eling, etc.), fitting data collected at the Earth’s surface (i.e., the geodetic and meteorological data, the seismic data, etc.),studying the circadian rhythm of hormones, hydrogen atoms, the solar corona, and so on.

Interpolation is a common tool to deal with the periodic data. Precioso et al. [1] used B-splines to construct active con-tours for image segmentations. Giesl [2] provided a thorough discussion on estimating the region of attraction of dynam-ical systems with radial basis function interpolation. Schoenberg [3] used trigonometric B-splines to interpolate periodicdata.

But interpolation requires solving a large-scale linear system of equations, which is time consuming. Moreover, thesampling data may be noised in some cases, and interpolation cannot deal with noised data well. For improving the speedof computation and filtering the noise in the data, people usually use quasi-interpolation.

Quasi-interpolation has been widely discussed in the literature, see [4–10] and the references therein. The majoradvantage of quasi-interpolation is that it yields a solution directly without the need to solve any linear system ofequations.

Based on the MQ function proposed by Hardy [11], Beatson and Powell [12] first constructed three univariate MQ quasi-interpolants to scattered centers from a bounded interval. An improved MQ quasi-interpolant was provided in [13]. Theyalso discussed some shape preserving properties of these MQ quasi-interpolants. By introducing the definition of MQ B-splines, Beatson and Dyn [14] discussed MQ quasi-interpolation theoretically. For applications of MQ quasi-interpolation innumerical solutions of differential equations, we refer readers to [15,16] and the references therein.

The MQ quasi-interpolant is smooth, efficient and easy to compute. Moreover, it approximates high-order derivativeswell [17] and requires only computing the second-order divided differences of the MQ function, and thus is simple andstable [18].

However, the MQ quasi-interpolant is not suitable for periodic data directly, since its kernel (the MQ function) itself isnot periodic.

Using the technique of periodic extensions, a MQ quasi-interpolation scheme for periodic data was constructed in [15].But periodic extensions require boundary conditions, which may be extremely complicated in many cases. Moreover, peri-odic extensions yield unwanted high-order discontinuous points at the boundaries, and thus destroy high-order smoothnessof the approximand (see Table 6 and the red curve in Fig. 2 in Section 4 for an example).

For periodic data, it is better to construct a quasi-interpolation scheme whose kernel itself is also periodic.Lyche, Schumaker and Stanley [19] constructed a quasi-interpolation scheme based on trigonometric B-splines [20]

and studied its approximation orders for high-order derivatives. For more applications of the quasi-interpolation scheme,see [21,22] for instance.

The trigonometric B-spline quasi-interpolant is well suited for periodic data, since its basis (trigonometric B-splines)themselves are periodic, as pointed out by Lyche [23]. But, analogous to B-splines [24], the smoothness order of trigono-metric B-splines is only n − 2, where n is the order of trigonometric B-splines. This implies that we have to use high-ordertrigonometric B-splines for approximating high-order derivatives in some cases (e.g., constructing active contours, numer-ical solutions of differential equations, etc.).

Motivated by the above discussions, we construct in this paper a quasi-interpolant that couples together the periodicityof the trigonometric B-spline quasi-interpolant and the smoothness of the MQ quasi-interpolant.

The construction of the quasi-interpolant consists of three steps. We first construct a periodic kernel based on theMQ function. Then, applying trigonometric divided differences to the periodic kernel gives the multiquadric trigonometricB-splines. Finally, with these multiquadric trigonometric B-splines being the basis, the quasi-interpolant is constructed.

The quasi-interpolant preserves many fair properties of MQ quasi-interpolant such as smoothness, simplicity, efficiency,capabilities of approximating high-order derivatives and so on. Moreover, the quasi-interpolant as well as its derivatives areperiodic. The quasi-interpolant covers the trigonometric B-spline quasi-interpolant as a special case. In addition, one canchoose a proper shape parameter such that it provides the same approximation order as the trigonometric B-spline quasi-interpolant for a periodic function. Furthermore, the quasi-interpolant gives better approximations to high-order derivativesthan the trigonometric B-spline quasi-interpolant.

The paper is organized as follows.To make the paper self-contained, some preliminaries about trigonometric divided differences and the trigonometric

B-spline quasi-interpolation scheme are presented in Section 2. In Section 3, we construct the main quasi-interpolationscheme (3.3) and we derive its error estimates (see the Theorem 3.1). Numerical examples of applying the scheme in ap-proximating a periodic function, the first-order and the second-order derivatives of the function, respectively, are presentedin Section 4. Finally, conclusions and discussions are given in Section 5.

2. Preliminaries

We start by introducing the definition of trigonometric divided differences.

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22 W. Gao, Z. Wu / Journal of Computational and Applied Mathematics 271 (2014) 20–30

2.1. Trigonometric divided differences

Given a positive integer n, the space of trigonometric polynomials of order n is defined as [19]

Trn =

span

1, sin x, cos x, . . . , sin

(n − 1)x2

, cos(n − 1)x

2

, n odd,

spansin

x2, cos

x2, . . . , sin

(n − 1)x2

, cos(n − 1)x

2

, n even.

Observe that Trn is the null space of the nth-order differential operator

Qn(D) =

D(D2

+ 1)(D2+ 4) · · ·

D2

+

n − 12

2, n odd,

D2

+14

D2

+94

· · ·

D2

+

n − 12

2, n even,

where D :=ddx .

Moreover, to discretize the differential operator Qn(D), Lyche and Winther [20] introduced the nth-order trigonometricdivided differences.

Let {xj} be a 2π-periodic knots sequence with ascending order, i.e.,

0 = x0 < x1 < · · · < xj < xj+1 < · · · < xN = 2π, xkN+j = 2kπ + xj, k ∈ Z .

Then the nth-order trigonometric divided difference of a function f with respect to the space Trn as defined in [20]:

[xj, xj+1, . . . , xj+n]Trn f = 2n

Vn

sin

x2

cosx2

· · · sin(n − 1)x

2cos

(n − 1)x2

f (x)xj xj+1 · · · xj+n−2 xj+n−1 xj+n

Vn

1 sin x cos x · · · sin

nx2

cosnx2

xj xj+1 xj+2 · · · xj+n−1 xj+n

if n is even, or

[xj, xj+1, . . . , xj+n]Trn f = 2n−1

Vn

1 sin x cos x · · · sin

(n − 1)x2

cos(n − 1)x

2f (x)

xj xj+1 xj+2 · · · xj+n−2 xj+n−1 xj+n

Vn

sin

x2

cosx2

· · · sinnx2

cosnx2

xj xj+1 · · · xj+n−1 xj+n

if n is odd. Here

Vn

p0 · · · pnxj · · · xj+n

=

p0(xj) · · · p0(xj+n)...

. . ....

pn(xj) · · · pn(xj+n)

.Note that the nth-order trigonometric divided difference kills the trigonometric polynomial space Trn.

Particularly, when n = 2, it reads

[xj, xj+1, xj+2]Tr2 f = 4

sin

xj2

sinxj+1

2sin

xj+2

2

cosxj2

cosxj+1

2cos

xj+2

2f (xj) f (xj+1) f (xj+2)

1 1 1sin xj sin xj+1 sin xj+2cos xj cos xj+1 cos xj+2

. (2.1)

Thus it kills the second-order trigonometric polynomial space Tr2 = {sin x/2, cos x/2} and is a discretization of (D2+

1/4)f (xj).

2.2. The trigonometric B-spline quasi-interpolation scheme

Lyche, Schumaker and Stanley [19] constructed a quasi-interpolation scheme for periodic data. In parallel with theB-spline quasi-interpolation scheme, we call it the trigonometric B-spline quasi-interpolation scheme.

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W. Gao, Z. Wu / Journal of Computational and Applied Mathematics 271 (2014) 20–30 23

The construction consists of three steps.By truncating a trigonometric polynomial (of order n), they first constructed a periodic kernel

siny − x2

n−1

+

=

0, y < x,sin

y − x2

n−1

, y ≥ x.

As the second step, applying the nth-order trigonometric divided differences to the kernel gives the nth-order trigono-metric B-splines [20]

T nj,+(x) = [xj, xj+1, . . . , xj+n]Trn

sin

y − x2

n−1

+

.

Finally, with these trigonometric B-splines being the basis, the trigonometric B-spline quasi-interpolation scheme wasdefined as

Qn,lf (x) =

N−1j=0

sinxj+n − xj

2

li

αj,iλj,i(f )T nj,+(x). (2.2)

Here {αj,i} are defined by the formula (3.6) in [19], {λj,i} are some prescribed linear functionals, and 1 ≤ l ≤ n, where n − lis an even integer.

Particularly, when n = l = 2, αj,1 = 0, αj,2 = 1, λj,1(f ) = f (xj) and λj,2(f ) = f (xj+1), the above trigonometric B-splinequasi-interpolation scheme reads (that is the formula (7.8) in [19])

Q2,2f (x) =

Nj=1

sinxj+1 − xj−1

2f (xj)T 2

j−1,+(x), (2.3)

where

sinxj+1 − xj−1

2T 2j−1,+(x) =

(sin(x − xj+1)/2)+ − cos xj+1−xj2 (sin(x − xj)/2)+

2 sin xj+1−xj2

−cos xj−xj−1

2 (sin(x − xj)/2)+ − (sin(x − xj−1)/2)+

2 sin xj−xj−12

.

The scheme requires only computing the second-order trigonometric divided differences and thus is simple, efficientand easy to compute. Moreover, the error estimate was given as [19]

∥Q2,2f − f ∥∞ ≤ O(h2). (2.4)

Here h is the density of the centers {xj}, i.e., h = maxj{xj+1 − xj}. This implies that the quasi-interpolant provides an optimalapproximation order O(h2) with respect to the order of trigonometric B-splines. However, the quasi-interpolant is onlycontinuous and thus cannot approximate derivatives of the periodic function.

In the following section,we shall construct a quasi-interpolation scheme for periodic data. The scheme covers the trigono-metric B-spline quasi-interpolation scheme as a special case (see Remark 3.3). In addition, the scheme provides the sameapproximation order as the trigonometric B-spline quasi-interpolation scheme for periodic functions. More importantly, thescheme gives better approximations to high-order derivatives than the trigonometric B-spline one.

As an example, the papermainly considers the case n = 2. Similar to [14], the construction techniques can be generalizedto any integer n.

3. Construction of the main quasi-interpolation scheme

We begin the section with a lemma that can be viewed as a special case of Theorem 1 in [25].

Lemma 3.1. Let

T 2j,+(x) = [xj, xj+1, xj+2]Tr2

sin

y − x2

+

,

T 2j,−(x) = [xj, xj+1, xj+2]Tr2

sin

y − x2

,

and

T 2j,±(x) =

T 2j,+(x)+ T 2

j,−(x)

2,

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24 W. Gao, Z. Wu / Journal of Computational and Applied Mathematics 271 (2014) 20–30

wheresin

y − x2

=

0, y > x,

− siny − x2

, y ≤ x.

Then one has

T 2j,+(x) = T 2

j,−(x) = T 2j,±(x).

Remark 3.1. Based on the above lemma, the second-order trigonometric B-splines can be rewritten as

T 2j−1,±(x) =

12[xj−1, xj, xj+1]Tr2

sin y − x2

.Now it comes to constructing the quasi-interpolation scheme.

Analogous to the trigonometric B-spline quasi-interpolation scheme in Section 2.2, the construction of our quasi-interpolation scheme consists of three steps.

We first construct a periodic kernel.Following Hardy’s MQ function trick [11], we construct the periodic kernel as

φ(x) =

c2 + sin2 x

2. (3.1)

Moreover, in parallel with the MQ function, we call φ(x) the MQ trigonometric function. Here c is a shape parameter de-pending on the density of sampling centers.

Remark 3.2. Similar to the MQ function that smoothes away the discontinuous point x = 0 of |x|, the MQ trigonometricfunction smoothes away all the discontinuous points x = 2kπ, k ∈ Z , of | sin x/2|.

As the second step, applying the second-order trigonometric divided differences to the kernel gives the MQ trigonometricB-splines

ψj(x) =12[xj−1, xj, xj+1]Tr2φ(x − t) = 2

sin

xj−1

2sin

xj2

sinxj+1

2

cosxj−1

2cos

xj2

cosxj+1

2φ(x − xj−1) φ(x − xj) φ(x − xj+1)

1 1 1sin xj−1 sin xj sin xj+1cos xj−1 cos xj cos xj+1

. (3.2)

Note that the second-order trigonometric divided differences [xj−1, xj, xj+1]Tr2 are taken with respect to the variable t .Finally, taking these MQ trigonometric B-splines {ψj(x)} as the basis, we construct an ansatz

Qf (x) =

Nj=1

sinxj+1 − xj−1

2f (xj)ψj(x), (3.3)

where 0 = x0 < · · · < xN = 2π < xN+1 = 2π + x1.With some simple calculations, the equality holds:

sinxj+1 − xj−1

2ψj(x) =

φ(x − xj+1)− cos xj+1−xj2 φ(x − xj)

2 sin xj+1−xj2

−cos xj−xj−1

2 φ(x − xj)− φ(x − xj−1)

2 sin xj−xj−12

. (3.4)

The error of the scheme can be captured in the following theorem.

Theorem 3.1. Let f be a C2 continuous function with period 2π . Denote by h the density of the centers {xj}, that is, h = maxj{xj+1 − xj}. Let c be an h-dependent shape parameter. Then the error of the quasi-interpolation scheme (3.3) is given as

∥Qf − f ∥∞ ≤ O(h2)+ O(ch)+ O(c2| ln h|). (3.5)

Proof. Following the approach of [13], we split ∥Qf − f ∥∞ into two parts, i.e., ∥Qf −Q2,2f ∥∞ and ∥Q2,2f − f ∥∞. In addition,noting that

∥Qf − f ∥∞ ≤ ∥Qf − Q2,2f ∥∞ + ∥Q2,2f − f ∥∞, (3.6)

∥Q2,2f − f ∥∞ ≤ O(h2), (3.7)

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W. Gao, Z. Wu / Journal of Computational and Applied Mathematics 271 (2014) 20–30 25

one needs only to bound the first part

∥Qf − Q2,2f ∥∞.

In view of the periodicity of f (x), φ(x), {xj}, and sin x, we have

Nj=1

f (xj)

φ(x − xj+1)− cos xj+1−xj

2 φ(x − xj)

2 sin xj+1−xj2

−cos xj−xj−1

2 φ(x − xj)− φ(x − xj−1)

2 sin xj−xj−12

=

N+1j=2

f (xj−1)φ(x − xj)

2 sin xj−xj−12

Nj=1

f (xj)

cos xj+1−xj

2

2 sin xj+1−xj2

+cos xj−xj−1

2

2 sin xj−xj−12

φ(x − xj)+

N−1j=0

f (xj+1)φ(x − xj)

2 sin xj+1−xj2

=

Nj=1

f (xj−1)

2 sin xj−xj−12

cos xj+1−xj

2

2 sin xj+1−xj2

+cos xj−xj−1

2

2 sin xj−xj−12

f (xj)+

f (xj+1)

2 sin xj+1−xj2

φ(x − xj).

This together with the definition of the second-order trigonometric divided difference (2.1) give

2Qf (x) =

Nj=1

sinxj+1 − xj−1

2φ(x − xj)[xj−1, xj, xj+1]Tr2 f .

Similarly, we have

2Q2,2f (x) =

Nj=1

sinxj+1 − xj−1

2

sin x − xj2

[xj−1, xj, xj+1]Tr2 f .

These in turn give

2(Qf (x)− Q2,2f (x)) =

Nj=1

sinxj+1 − xj−1

2

φ(x − xj)−

sin x − xj2

[xj−1, xj, xj+1]Tr2 f .

Thus it reduces to getting the bound of I(x) defined as

I(x) =

Nj=1

sinxj+1 − xj−1

2

c2 + sin2 x − xj

2−

sin x − xj2

.

Sincec2 + sin2 x − xj

2−

sin x − xj2

≤ c,c2 + sin2 x − xj

2−

sin x − xj2

≤c2

2sin x−xj

2

, sinx − xj

2= 0,

the following inequalities hold:

I(x) ≤ c

|x−xj|≤h

sinxj+1 − xj−1

2+ c

2π−h≤|x−xj|≤2π

sinxj+1 − xj−1

2+

c2

2

h≤|x−xj|≤2π−h

sin xj+1−xj−12sin x−xj2

≤ 2c

|x−xj|≤h

sinxj+1 − xj−1

2+

c2

2

h≤|x−xj|≤2π−h

sin xj+1−xj−12sin x−xj2

.Besides, the inequalities

xj+1 − xj−1 ≤ |xj+1 − x| + |x − xj−1|

≤ |xj+1 − xj| + 2|x − xj| + |xj − xj−1|

≤ 2h + 2|x − xj|

give

sinxj+1 − xj−1

2≤

xj+1 − xj−1

2≤ h + |x − xj|.

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26 W. Gao, Z. Wu / Journal of Computational and Applied Mathematics 271 (2014) 20–30

These lead to the inequality

I(x) ≤ O(ch)+c2

2

h≤|x−xj|≤2π−h

xj+1−xj−12sin x−xj

2

. (3.8)

Furthermore, observing that the above summation is a discretization of the definite integral

c2h≤|x−t|≤2π−h

1sin x−t2

dt,we have

c2

2

h≤|x−xj|≤2π−h

xj+1−xj−12sin x−xj

2

≤ c2h≤|x−xj|≤2π−h

1sin x−t2

dt + O(c2h). (3.9)

Now it remains to bound the definite integral

c2h≤|x−t|≤2π−h

1sin x−t2

dt.Applications of the change of variables to the definite integral yield

c2h≤|x−t|≤2π−h

1sin x−t2

dt = 4c2h/2≤x≤π−h/2

1sin x

dx

= 4c2ln1 + cos

h2

− ln

1 − cos

h2

≤ O(c2| ln h|).

This together with inequalities (3.8), (3.9), lead to the inequality

I(x) ≤ O(ch)+ O(c2| ln h|)+ O(c2h).

Thus we have

∥Qf − Q2,2f ∥∞ ≤ O(ch)+ O(c2| ln h|). (3.10)

Finally, the theorem follows from inequalities (3.6), (3.7) and (3.10). �

Remark 3.3. Particularly, when the shape parameter c is zero, the MQ trigonometric function φ(x) in the expression (3.1)is just | sin x/2|, the kernel of the second-order trigonometric B-spline quasi-interpolant (3.3). This implies that our quasi-interpolant covers the second-order trigonometric B-spline quasi-interpolant as a special case. In addition, in this case, theerror (3.5) is just the error (2.4), i.e.,

∥Q2,2f − f ∥∞ ≤ O(h2).

However, since the kernel | sin x/2| is only C0 continuous, our quasi-interpolant in this special case is also C0 continuousand thus cannot approximate derivatives of the function (see the red curves in Fig. 1(b) and (c) in Section 4, respectively).

Remark 3.4. From the error (3.5), i.e.,

∥Qf − f ∥∞ ≤ O(h2)+ O(ch)+ O(c2| ln h|),

we can find that for any shape parameter c satisfying the inequalities

0 ≤ c ≤ O

h

√| ln h|

, (3.11)

the error (3.5) keeps

∥Qf − f ∥∞ ≤ O(h2)+ O

h2

√| ln h|

+ O(h2).

This implies that although our quasi-interpolant sacrifices a little approximation accuracy, its approximation order is stillthe same as the one of the second-order trigonometric B-spline quasi-interpolant.

More importantly, our quasi-interpolant provides more choices of the shape parameter than the second-order trigono-metric B-spline quasi-interpolant (a special case with c = 0). Besides, the quasi-interpolant gives better approximations tohigh-order derivatives and the approximation accuracy increases as c approaches O( h

√| ln h| ).

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W. Gao, Z. Wu / Journal of Computational and Applied Mathematics 271 (2014) 20–30 27

(a) Approximants of the function. (b) Approximants of the first-orderderivative.

(c) Approximants of the second-orderderivative.

Fig. 1. Approximantswith different choices of the shape parameter. (For interpretation of the references to color in this figure legend, the reader is referredto the web version of this article.)

Particularly, when c = O( h√

| ln h| ), the quasi-interpolant gives the best approximations to high-order derivatives (seethe cyan curves in Figs 1(b) and (c) in Section 4, respectively). Therefore, a proper shape parameter can be chosen (i.e.c = O( h

√| ln h| )), such that the quasi-interpolant provides the same approximation order as the second-order trigonometric

B-spline quasi-interpolant for the function. Moreover, in such a case, the quasi-interpolant also provides excellent approx-imations to high-order derivatives.

Note that such a choice of the shape parameter can give a MQ quasi-interpolant with fair approximations to high-orderderivatives, as pointed out in [17].

Up to now, we have constructed a quasi-interpolation scheme for periodic data based on MQ trigonometric B-splines.

4. Numerical examples

In this section, we shall compare our quasi-interpolation with trigonometric B-spline quasi-interpolation andMQ quasi-interpolation for approximating the periodic function as well as its derivatives, respectively.

As an example, we choose the periodic function f (x) = esin x, x ∈ [0, 2π). The first-order derivative is esin x cos x, whilethe second-order derivative is esin x(cos2 x− sin x). We use the quasi-interpolant (3.3), its first-order derivative and second-order derivative to approximate them, respectively. The first-order derivative and the second-order derivative of the quasi-interpolant are:

(Qf (x))′ =

Nj=1

sinxj+1 − xj−1

2f (xj)ψ ′

j (x)

and

(Qf (x))′′ =

Nj=1

sinxj+1 − xj−1

2f (xj)ψ ′′

j (x).

Here ψ ′

j (x), ψ′′

j (x) are the second-order trigonometric divided differences of φ′(x), φ′′(x), respectively.In the sequel, we assume that the centers are equidistant, that is,

xi =2πN, i = 0, . . . ,N.

We first demonstrate the effect of the shape parameter on the performance of our quasi-interpolant.For N = 16, Fig. 1 illustrates the approximations of the periodic function and its derivatives by our quasi-interpolant

with different shape parameters.In the three figures, the red, green, magenta and cyan curves corresponding to the approximants with the shape param-

eter c = 0, c = π/26−m, m = 0, 1, 2, respectively, while the blue curves are the approximands.

Remark 4.1. The figures illustrate vividly that larger values of the shape parameter yield smoother quasi-interpolants withbetter approximations to derivatives, while smaller ones yield more accurate quasi-interpolants (for the function) withworse smoothness. Particularly, when c is zero, the quasi-interpolant is just the second-order trigonometric B-spline quasi-interpolant, and thus gives the best approximation to the function (see the red curve in Fig. 1(a)). However, in this case, thequasi-interpolant cannot approximate the derivatives of the function (see the red curves in Fig. 1(b) and (c), respectively).

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28 W. Gao, Z. Wu / Journal of Computational and Applied Mathematics 271 (2014) 20–30

Table 1Approximation errors of approximating the function.

N L∞-norm Posteriori error estimate

32 0.009664 0.0022 2.1255

128 4.8902e−004 2.1695256 1.0997e−004 2.1554512 2.4169e−005 2.1859

Table 2Approximation errors of approximating the first-order derivative.

N L∞-norm Posteriori error estimate

32 0.072264 0.0263 1.4569

128 0.0089 1.5632256 0.0026 1.7753512 7.8327e−004 1.7309

Table 3Approximation errors for approximating the second-order derivative.

N L∞-norm Posteriori error estimate

32 0.620164 0.2500 1.3106

128 0.0735 1.7661256 0.0208 1.8212512 0.0091 1.1926

Table 4Approximation errors of approximating the function.

N L∞-norm Posteriori error estimate

32 0.012864 0.0033 1.9556

128 8.1632e−004 2.0153256 2.0134e−004 2.0195512 4.9780e−005 2.0160

Table 5Approximation errors of approximating the first-order derivative.

N L∞-norm Posteriori error estimate

32 0.117964 0.0503 1.2289

128 0.0259 0.9575256 0.0136 0.9293512 0.0074 0.8780

Now we get posteriori error estimates of the quasi-interpolant (3.3) for approximating the function and its derivatives.For the definition of the posteriori error estimate, see Babuska [26].

The function and its derivatives are approximated by the quasi-interpolant under different sampling densities (i.e., N =

2k, k = 5, 6, 7, 8, 9), respectively. Approximation errors and posteriori error estimates are presented in Tables 1–3.

Remark 4.2. The posteriori error estimate of our quasi-interpolant for approximating the function is about O(h2.2), whichis better than its priori counterpart O(h2). This is expected since the priori error estimate considers the general case, whilethe posteriori one is just for a special example. Meanwhile, the posteriori error estimates for approximating the first-orderand second-order derivatives are about O(h1.6), O(h1.4), respectively. These demonstrate that our quasi-interpolant alsoapproximates derivatives well.

As the last part of the section, we compare our quasi-interpolant with the MQ quasi-interpolant for periodic data con-structed in [15].

Tables 4–6 are numerical results of approximating the function and its derivatives with the MQ quasi-interpolant [15].

Remark 4.3. The posteriori error estimate of the MQ quasi-interpolant [15] for approximating the function is aboutO(h2), while the one for approximating its first-order derivative is about O(h0.95). However, the quasi-interpolant cannot

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W. Gao, Z. Wu / Journal of Computational and Applied Mathematics 271 (2014) 20–30 29

Table 6Approximation errors of approximating the second-order derivative.

N L∞-norm Posteriori error estimate

32 0.9147 Nay64 0.8271 Nay

128 0.7252 Nay256 0.6441 Nay512 0.5910 Nay

Fig. 2. Approximants of the second-order derivative by our quasi-interpolation and MQ quasi-interpolation in [15]. (For interpretation of the referencesto color in this figure legend, the reader is referred to the web version of this article.)

approximate the second-order derivative at the boundaries (see Table 6 and the red curve in Fig. 2). This, in some way,also verifies that periodic extensions yield unwanted high-order discontinuous points at the boundaries, and thus destroyhigh-order smoothness of the function.

Remark 4.4. Numerical results show that our quasi-interpolant performs better than the MQ quasi-interpolant [15] forapproximating both the periodic function and its derivatives.Moreover, our quasi-interpolant still approximates the second-order derivative well (see Table 3 and the green curve in Fig. 2), while, on the contrary, theMQ quasi-interpolant [15] cannotapproximate it at the boundaries (see Table 6 and the red curve in Fig. 2). In addition, both our quasi-interpolant and all itsderivatives are periodic.

5. Conclusions and discussions

We have constructed a quasi-interpolant for periodic data and derived its error estimates. The quasi-interpolant pre-serves many appealing properties of the MQ quasi-interpolant (such as simplicity, efficiency, capabilities of approximatinghigh-order derivatives, and so on). Moreover, the quasi-interpolant as well as its derivatives are all periodic. The quasi-interpolant covers the trigonometric B-spline quasi-interpolant as a special case. In addition, a proper shape parameter canbe chosen such that the quasi-interpolant provides the same approximation order as the trigonometric B-spline one for thefunction. More importantly, in this case, the quasi-interpolant also approximates derivatives well.

Acknowledgments

We wish to express our great gratitude to the referees for their valuable comments and suggestions.

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