generator, multiquadric generator, quasi-interpolation and multiquadric quasi-interpolation

Appl. Math. J. Chinese Univ.2011, 26(4): 390-400

Generator, multiquadric generator, quasi-interpolation

and multiquadric quasi-interpolation

WU Zong-min MA Li-min

Abstract. The aim of this survey paper is to propose a new concept “generator”. In fact,

generator is a single function that can generate the basis as well as the whole function space. It

is a more fundamental concept than basis. Various properties of generator are also discussed.

Moreover, a special generator named multiquadric function is introduced. Based on the multi-

quadric generator, the multiquadric quasi-interpolation scheme is constructed, and furthermore,

the properties of this kind of quasi-interpolation are discussed to show its better capacity and

stability in approximating the high order derivatives.

§1 Generator of the function space

What is applied mathematics? We have not found a standard definition in the references andwill not give its definition here too. However, at least, the words “applied mathematics”containtwo parts, one is the mathematical part and the other is the applications part. The key point ofapplied mathematics is the interface or the interaction between the two parts. The data are themedia of the interaction via the interface. Therefore, numerical approximation (or functionsrepresentation) based on the given data is the key point in this research field. In mathematics,approximation theory (or functions representation theory) is concerned with how underlyingfunctions f(x) can be approximated or represented by the linear combination of some simplefunctions. One usually takes a nested function space, for example, the nested polynomial space· · · ⊂ Pn ⊂ Pn+1 ⊂ · · · , and tries to find the projection of the underlying function f(x) on thefunction space Pn based on the given data. The projections f∗

n(x) are interpreted as the bestapproximations of the underlying function f(x) . What is the task of numerical approximation?It is getting an approximation f∗

n(x) based on the given data (mostly the linear functional of theunderlying function f(x)), furthermore, calculating the linear functional of f∗

n(x) to simulate the

Received: 2011-04-13.MR Subject Classification: 26B40, 41A25, 47B32, 65D25.Keywords: Generator, function representation, approximation, numerical differentiation, multiquadric,

quasi-interpolation.Digital Object Identifier(DOI): 10.1007/s11766-011-2812-5.Supported by the 973program-2006CB303102, SGST 09DZ 2272900 and NSFC No.11026089.

WU Zong-min, MA Li-min. Generator, Multiquadric Quasi-interpolation 391

linear functional of f(x). A typical example is the divided difference: given data f(x1), f(x2),one uses a linear function f∗(x) = x2−x

x2−x1f(x1)+ x−x1

x2−x1f(x2) to simulate the underlying function

by interpolating the data and then takes the first order derivative of f∗(x) to simulate the firstorder derivative of the function f(x). In fact, it is a well-known and basic algorithm which iscalled divided difference of order one: f ′(x) ∼ f(x2)−f(x1)

x2−x1in numerical approximation.

Since approximation is the projection (e.g. in Euclidean space, Hilbert space, Sobolevspace), we should take a basis of the function space in advance. To calculate the projection,an orthonormal basis is preferred. Otherwise, if we can get a dual basis, an affine basis is alsoallowed.

Function is usually approximated by the linear combination of basis functions. The keyfeature of approximation is using simpler basis functions to approximate the underlying (com-plicated) function.

In mathematics, what is the basis of the function space under consideration? The basis isthe smallest set, which can be combined to represent all elements in the function space. Findingout the basis for different purpose of applications is a basic problem in mathematics. Even forpolynomial space, there exist a lot of bases under the name of great persons such as Bernstein,Bessel, Hermite, Jacobi, Lagrange, Laguerre, Legendre, Newton, Tchebychev, etc. Basis canbe observed as “building blocks”of the function space. Finding out the building blocks of thefunction space is the basic problem in mathematics. Such problem happens almost everywhere.For example, finding out the building blocks of matter is the basic problem in chemistry, findingout the building blocks of organism is the basic problem in biology, etc.

Compare our problem with the problem in chemistry. As we all know, matter is composed ofatoms. Actually, everything in the Universe can be represented by linear combination of atomsin chemistry just as the function can be represented by the linear combination of basis in math-ematics. That is to say, atoms are the basis of chemistry. Are the atoms constructed by morebasic pieces? Sure, Chemists tell us that atoms are composed of pieces like neutrons, electrons,and protons. These raise a similar problem in mathematics: Can the basis in mathematics berepresented by more basic things?

In most cases, the answer is YES! In fact, the linear combination of the shifts of xn generatesthe linear space of polynomials of degree less than or equal to n. Moreover, any n+ 1 pairwisedistinct shifts of xn is a basis of the space Pn. Therefore xn is a more basic function and playsa more basic role in Pn than any other functions. Pn is constructed by the linear combinationof shifts of the single function xn. This phenomena happens in biology too. The cell copiesitself (shifts) and stacks the copies together (linear combination) to produce an organ. Using apopular word, function xn contains all the “DNA”of the polynomial function space Pn. Now,we would like to present the definition of such function.

Definition 1.1. Suppose S is a given linear functions space, function G(x) ∈ S (G(x) ∈ S isnot necessary, in this case, G(x) will generate a function space larger than S) is a generatorof the function space S provided that one of the following conditions holds

1. For some pairwise distinct points {xj}, S ⊂ {G(x− x1), G(x − x2), · · · },

392 Appl. Math. J. Chinese Univ. Vol. 26, No. 4

2. S ⊂ {G(x), G′(x), · · · }.

Remark 1.1. We would like to give an example to show it more briefly, xn generates allpolynomials whose degree is less than or equal to n. At the same time, the polynomials of odd(or even) degree which is less than or equal to n constitute a function space too. For suchfunction space, we could not take any function from this (odd or even degree) polynomial spaceas a generator. From now on, we will concentrate our attentions on such function space S thatthe generator G(x) ∈ S and all the shifts {G(x− ·)} just generate the space S exactly.

We point out that the dimension of the space S can be finite or infinite. We declare againthat generator which contains all the “DNA”of the space is the most fundamental function inthe function space.

Following are some examples of generators.

• xn is the generator of polynomial space Pn.

• 2 sin(x) − sin(2x) is the generator of trigonometric polynomial space.

• 2ex − e2x is the generator of exponential polynomial space.

• |x|2k+1 is the generator of polynomial spline space.

• |x|2k+1ex sin2(x) is the generator of Tchebychev spline space. It will be discussed in detailsin the following section.

• Gaussian function e−x2is mostly used in the applications related to probability.

§2 Properties of the generator

This section discusses the properties of the generator. As we have declared above, for thegenerator xn of the polynomial space Pn, more than n+ 1 shifts of xn are linearly dependent.Since the limits of the divided differences (linear combination of the shifts of the function) arethe derivatives, we have

Theorem 2.1. For univariate problems, a generator can only generate finite dimensional spaceif and only if the generator is a solution of homogeneous linear ordinary differential equationwith constant coefficients.

For multivariate problems, i.e. x ∈ Rd, if G(x) can only generate a finite dimensional space,then G(x) must be a solution of homogeneous partial differential equation with constant coeffi-cients, conversely, a solution of homogeneous partial differential equation with constant coeffi-cients may serve as a generator to generate an infinite dimensional space (e.g. ex2−y2

cos(2xy)is a harmonic function satisfying the Laplace equation, however, it generates an infinite dimen-sional space).


The following are some examples of homogeneous linear ordinary differential equations andthe corresponding solution spaces. For these function spaces, we can find a generator that onlygenerates finite dimensional space S which is exact the solution space of the ordinary differentialequation.

• Dng(x) = 0, Polynomial,

• (D2 + c2I)g(x) = 0, Trigonometric polynomial,

• (D2 − c2I)g(x) = 0, Exponential polynomial,

• Dn(D4 − c4I)g(x) = 0, Subspace of the algebra of above functions.

The subspace of the algebra of the polynomial, the trigonometric polynomial and the expo-nential polynomial contains almost all functions which we used in applications.

Considering the generator is not unique ((x−c)n is a generator of polynomial space Pn too),we would like to propose a definition of standard generator.

Definition 2.1. For the solution space of P (D)G(x) = 0, if G(x) satisfies

G(k)(0) = δk,n−1,

then G(x) is a generator [23] and is called the standard generator of the solution space.

From the definition we know that the standard generator is a special generator of thesolution space. The following theorem may help us get an explicit representation of the standardgenerator.

Theorem 2.2. Suppose that the standard generator of P (D)G(x) = 0 is G(x) and {λj} arethe roots of the characteristic equation P (λ) = 0, then the standard generator G(x) can berepresented in divided difference that

G(x) = [λ1, · · · , λn]eλx.

The proof can be found in [23]. The result still holds, if P (λ) = 0 possesses multiple roots.

From the approximation point of view, if a generator can only generate finite dimensionalspace, its approximation capacity is limited. In previous discussions, we used the shifts ofgenerator to generate the function space. A more generalized case is to use shifts and scales. Itis easy to verify that the scales will not help the polynomial generate a larger space. However,

Theorem 2.3. Any non-polynomial generator will certainly generate an infinite dimensionalspace and can approximate almost all functions by using the linear combination of shifts andscales (see [4] for details).

Wavelet method uses the shifts (integer) and scales (dilation) of the generator. In mostcases, one prefers to use shifts only (People can refer to the concept of shift invariant functionspace).


Theorem 2.4. Using the linear combination of the shifts only, the generator can generate orapproximate almost all functions if and only if its Fourier transform (the generalized Fouriertransform) does not equal to zero almost everywhere [16,17].

Remark 2.1. We would like to remind the reader that if the Fourier transform of the generatorequals to zero in some interval, then the linear combination of the shifts of such function willgenerate infinite dimensional space, however, it can not approximate all functions. Becausethe Fourier transform of the linear combination of the shifts of such generator will equal tozero, where the Fourier transform of the generator itself equals to zero. Therefore the functionspace can not approximate the function, whose Fourier transform does not equal to zero in thisinterval.

An interesting problem is to construct some examples, that the generator which is basedon a simple function can approximate almost all functions by using shifts only. Biologists mayhelp us with this problem.

What is the reason of species diversity in biology? The answer is that a new kind of speciescomes out via gene mutation.

Beginning with an example again, we change x into |x| which possesses a “gene mutation”atzero. Based on the new generator, we can construct functions via linear combination of theshifts:

Λj(x) =(|x− xj+1| − |x− xj |)

2(xj+1 − xj)− (|x − xj | − |x− xj−1|)

2(xj − xj−1),

which are the well known Euclid’s Hat functions for piecewise linear interpolation. The piecewiselinear interpolation can approximate all continuous functions.

Theorem 2.5. Any standard generator showed above with “gene mutation”at zero that G(x) →sign(x)G(x)/2 will generate an infinite dimensional space which is called Tchebychev splinefunction space and can approximate almost all functions (see [23]).

Theorem 2.6. For Tschebycheffian spline function space generated by such generator, parallelworks such as the Taylor’s expansion, The Roll’s Lemma, the dual basis, the interpolation andthe approximation, the B-spline form, the Energy minimization, the subdivisions algorithm andthe wavelets decomposition have been done in [23].

For multivariate problems, the most simple generator is perhaps the radial generator, whichgenerates the radial basis function space. Given an univariate function φ : R+ → R, we candefine a multivariate generator Φ(x) = φ(‖x‖). The linear combination of the shifts {Φ(x−xj)}is called the radial basis function space (see [15–17]). In [18], a series of compactly supportedgenerators are constructed by only one piece of polynomial. We can construct the dual basisfor the given Hilbert, Sobolev or Besov norm. Moreover, we can design Measure or Norm toform an orthogonal basis of reproducing kernel Hilbert space (see [14, 15] for details). In thefollowing section we will introduce a special radial generator named multiquadric function.


§3 Multiquadric functions and quasi-interpolation

The most simple generator is perhaps the multiquadric function. In 1968, Hardy [6] intro-duced the multiquadric (MQ) method to design aircraft for Boeing Co..

The multiquadric generator is defined as follows:

G(x) =√c2 + ‖x‖2,

where x ∈ Rd. Hardy(1988) [7] showed a lot of applications of multiquadric function in hissurvey paper. In fact, from 1968 to 1988 multiquadric method means multiquadric interpolationmethod. In Franke’s survey paper [6], the multiquadric method was rated as one of the bestmethods among 29 scattered data interpolation schemes based on accuracy, stability, efficiency,memory requirement, and ease of implementation. However there were only numerical testswithout theoretical proof. Because multiquadric interpolation method requires to solve a largelinear system of equations (the coefficient matrix is usually ill-conditioned), multiquadric quasi-interpolation method caught the attentions of many researchers.

The quasi-interpolation method is widely used in approximation theory. The first exampleof quasi-interpolation is perhaps the Bernstein polynomial

f(x) ∼∑

f(j

n)Bn

j (x), 0 ≤ j ≤ n.

A more popular quasi-interpolation is the B-spline which is widely used in CAGD

f(x) ∼∑

f(jh)B(x

h− j).

Another well known quasi-interpolation is based on Shannon’s sampling theorem in signal pro-cessing

f(x) =∑

f(jh)sinc(x

h− j)

for bandlimited function f(x). Bernstein’s polynomial basis, B-splines basis and sinc functionsare some pre-given functions. Quasi-interpolation scheme does not require to solve any linearsystem of equations and it yields the solution directly. For the B-spline and Shannon method,we would like to remind the readers that the constructed approximant in quasi-interpolationscheme is the linear combination of the shifts of the given single generator.

To introduce the quasi-interpolation, we should also recall the Schoenberg’s model (1951)

f∗(x) =∑

f(jh)φ(x

h− j).

A lot of related works respected to shift-invariant space have been done by de Boor, DeVoreand Ron.

The key feature of the approximation order of the Schoenberg’s model is the Strang-Fixconditions: The scheme is an approximation of order k and possesses polynomial reproducingproperty of order k, if and only if the Fourier transform of φ satisfies

φ̂(ω) = 1 + O(ωk),φ̂(ω + 2πj) = O(ωk),

as ω tends to zero. The Strang-Fix conditions are valid for multivariate problems too. Jia etc.even proved that there always exists a function (now we call it generator) satisfies Strang-Fixconditions for shift-invariant spaces.


As showed above, the simplest generator is the radial function. Could we construct thequasi-interpolation with radial generators? The answer is negative: there is no radial functionwhose finite linear combination of shifts satisfying Strang-Fix conditions (see [20]). Howeverwe could cite from [22] the following theorem

Theorem 3.1. Given function φ, if∫φ(x)dx �= 0, then we can find Δj, which is the coefficients

of Gaussian quadrature, such that

f∗(x) =∑

f(xj)φ(x − xj

h)Δj

is an approximation. Δj depend only on the geometry of the distribution of the sampling pointsxj of the data locally. The approximation order depends on the continuity of φ and f , and theapproximation order of the Gaussian quadrature coefficients Δj.

For the popularity of multiquadric quasi-interpolation method in the applications, we wouldlike to present some related results.

Beatson and Powell [1] constructed three kinds of multiquadric quasi-interpolation scheme.They also applied MQ quasi interpolation in the movie “The Lord of the Rings III”, whichwon eleven Oscars awards. [19] generalized Beatson and Powell’s results and constructed anew multiquadric quasi-interpolation scheme. They showed the scheme is shape preservingand constants reproducing, moreover, they gave the error estimates. Beatson and Dyn [2]discussed the multiquadric B-splines theoretically. Ling [10] used multilevel method to improvethe accuracy of scheme of [19]. [5] constructed the linear reproducing scheme and periodicscheme. [24] proposed a scheme possessing high order polynomial reproducing property.

For using multiquadric quasi-interpolation to solve partial differential equations, readers arereferred to: [9] found the cracks for boundary testing and the boundary layer problems. [5] [21]solved nonlinear PDEs such as Burgers equation. [13] solved sine-Gordon equation, as well asdetecting the coefficients in some special inverse problems.

§4 High order derivatives of multiquadric quasi-interpolation

Solving partial differential equation requires to approximate the high order derivatives ofthe functions. In approximation theory, the classical error bound with respected to derivativesusually takes the following form:

‖f∗(k)(x) − f (k)(x)‖ < O(hn−k). (1)

From the formula we know that the error bound of the highest order derivative plays themost important role in the total error estimates when solving partial differential equationsnumerically. This means, the terms of lower order derivative are somewhat over-approximatedand the terms of higher order derivative are under-approximated. We would like the errorbound takes the form

‖f∗(k)(x) − f (k)(x)‖ < O(hl), (2)that is to say, for all k < K, the derivatives possess almost the same error bound.


The basic multiquadric quasi-interpolation scheme by Beatson and Powell is as follows:

f∗(x) =∑

f(xj)ψj(x),

whereψj(x) =

φj+1(x) − φj(x)2(xj+1 − xj)

− φj(x) − φj−1(x)2(xj − xj−1)

,

φ(x) =√c2 + ‖x‖2, φj(x) = φ(x − xj).

The scheme is for unbounded interval. By adding some boundary terms [1,5,11,19], it can alsohandle the problems defined on bounded interval.

Now we concentrate ourselves on the multiquadric quasi-interpolation scheme:

(Lf)(x) =∞∑

j=−∞f(xj)ψj(x), (3)

where ψj(x) are defined as above. [3, 19] provide us the following theorem.

Theorem 4.1. Let f(x) be a twice differentiable function, such that ‖f ′(x)‖∞ and ‖f ′′

(x)‖∞are bounded. Then the inequalities

‖(Lf)(x) − f(x)‖∞ ≤ O(h2)

and‖(Lf)

′(x) − f

′(x)‖∞ ≤ O(h)

hold, provided c is small enough.

From the experiences in approximation theory and the formula (1) above, people could notexpect that the high order derivatives of the multiquadric quasi-interpolation converge to thecorresponding derivatives of the approximand. Actually, a lot of papers used the scheme tosolve partial differential equations and got good numerical results without caring about theconvergence of the high order derivatives. Recently [11] got the following result

Theorem 4.2. If f(x) ∈ C(k+1)(R) and f (j)(x) is bounded by a polynomial of degree k+2− j,then

|(Lf)(k)(x) − f (k)(x)| ≤ O(h2

k+1 )holds, provided c = O(h

1k+1 ).

These allow us to approximate each term of the partial differential equations in almost thesame approximation order.

§5 Stability of multiquadric quasi-interpolation to approximate the

high order derivatives

In applications, the function values {f(xj); j = 0, 1, · · · , N} usually can not be obtainedexactly, because of the inaccuracy of measurement or the accuracy of computer. Assume that,we can only get a noised data {f∗(xj) = f(xj)+ξj}, where {ξj} are pairwise independent whitenoise with the variance E(ξ2j ) = σ2.

For comparison, we discuss first the stability of the well-known divided difference method toapproximate the high order derivatives. From approximation theory, the divided difference is


the first coefficient of the interpolatory polynomial. Using the Lagrange interpolation formula,we can get an explicit symmetric representation of the divided difference that

[x0, x1, · · · , xn]f =n∑

k=0

f(xk)∏

j �=k

(xk − xj). (4)

Theorem 5.1. Let f ∈ Cn[a, b] and let {xj ; j = 0, 1, · · · , n} be a set of distinct points of [a, b].Then there exists ξ in the smallest interval containing the points {xj ; j = 0, 1, · · · , n}, such that

[x0, x1, · · · , xn]f = f (n)(ξ)/n!. (5)

From the above equation and theorem, one can get

Theorem 5.2. If {xj ; j = 0, · · · , n} are uniformly distributed and H denotes the step length,and if |x− ξ| ∼ O(H), then

E[(Dnf∗)(n)(x) − f (n)(x)]2 ≤ O

(σ2

H2n

)+ O(H2),

where Dn is the local polynomial interpolation and (Dnf∗)(n)(x)/n! is the divided difference on

[x0, x1, · · · , xn].

The details of the proof can be found in [12]. For non-uniform distributed sampling pointsof the data, the results will get much worse.

Remark 5.1. This Theorem tells us, to make sure that the error caused by the instability ofthe divided difference (the first term) is dominated by the theoretical error (the second term),the σ2 should satisfies

σ2D ≤ O(H2n+2).

This means for a 64-bit computer, the computer possesses only less than 20 significant decimaldigit (σ2

D > 10−40), then we can use the divided difference of order 5 for H = 10−2 and theresult is valid only with 2 decimal digit. In most cases of applications, σ2

D > 10−10, then thedivided differences of order >5 are nonsense even for H = 1/10. The result is not valid evenwith 1 significant decimal digit.

Theorem 5.3. For the multiquadric quasi-interpolation operator L, let c = O(h1

n+1 ), then

E[(Lf∗)(n)(x) − f (n)(x)]2 ≤ O(σ2

hn

n+1) + O(h

4n+1 )

holds.

This theorem is proved in [12].

Remark 5.2. By setting h2

n+1 = H to obtain the same theoretical approximation order asdivided differences, the multiquadric quasi-interpolation scheme only requires that

σ2L ≤ O(H

n2 +2).

That means multiquadric quasi-interpolation scheme is much stabler than divided differencemethod

σ2D << σ2

L.


Compared with the discussion of divided difference, roughly speaking, the significant decimaldigit of multiquadric quasi-interpolation is about 4 times of the significant decimal digit ofdivided difference.(e.g. if divided difference possesses only 2 significant decimal digit, while themultiquadric quasi-interpolation still possesses about 8 significant decimal digit.)

From the numerical results showed in [12], the errors of the divided differences alwayspossess more larger amplitudes as well as the frequencies than that of the multiquadric quasi-interpolation.

§6 Conclusions

This paper is concerned with a new concept “generator”for function space. Generator con-tains all “DNA”of the function space. Moreover, some properties of generator are discussed.Furthermore, a special generator named multiquadric is discussed in details. Actually, mul-tiquadric function as one of the radial basis functions is widely used in applications. One ofthe most important applications is to estimate or simulate the high order derivatives by usingmultiquadric quasi-interpolation method. The advantage of this method is that it does notrequire to solve large scale system of equations. It provides a new meshless numerical methodfor PDEs. Moreover, the stability of this method is shown by comparing it with the classicaldivided difference method.

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Shanghai Key Laboratory for Contemporary Applied Mathematics.

School of Mathematical Sciences, Fudan University, Shanghai 200433, China.

Email: [email protected]

generator, multiquadric generator, quasi-interpolation and multiquadric quasi-interpolation

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