Reconstruction of a function from Hermite–Birkhoff data · Reconstruction of a function from Hermite–Birkhoff data Francesco Dell’Accio Filomena Di Tommaso Kai Hormann Abstract

Reconstruction of a function from Hermite–Birkhoff dataFrancesco Dell’Accio · Filomena Di Tommaso · Kai Hormann

Abstract

Birkhoff (or lacunary) interpolation is an extension of polynomial interpolationthat appears when observation gives irregular information about some functionand its derivatives. A Birkhoff interpolation problem is not always solvable evenin the appropriate polynomial or rational space. In this paper we split up theinitial problem in subproblems having a unique polynomial solution and usemultinode rational basis functions in order to obtain a global interpolant.

Citation Info

JournalApplied Mathematics and Computation

Volume318, February 2018

Pages51–69

1 Birkhoff interpolation

In 1906, G. D. Birkhoff [4] studied the problem related to lacunary interpolation, that is, interpolation whichappears whenever observation gives irregular information about a function and its derivatives. Few yearslater, in 1931, G. Pólya [22] gave a notable contribution by introducing certain algebraic inequalities that mustbe satisfied by the interpolation scheme to be regular, that is, solvable for each choice of pairwise distinctnodes and associated interpolation data. These papers received little attention until I. J. Schoenberg revivedinterest on the subject in 1966 [24], when he provided a generalization of the Pólya’s theorem which givesa necessary condition to the existence of the solution. Lacunary or Birkhoff interpolation, in polynomialspace, radically differs from Lagrange or Hermite interpolation in both its problems and its methods [20].

More precisely, let X = x1, x2, . . . , xn be a set of pairwise distinct real numbers, for which we assumethat x1 < x2 < · · ·< xn . In the problem of interpolation of given data fi , j = f ( j )(xi ), i = 1, . . . , n , j ∈Ji ⊂N, bya polynomial p of appropriate degree,

p ( j )(xi ) = fi , j

we mainly distinguish between Hermite interpolation and Birkhoff interpolation. We have a Hermite interpol-ation problem if, for each i , the indices j in the set Ji form an unbroken sequence, that is, Ji = 0, 1, . . . , ji ,a Birkhoff interpolation problem otherwise. In contrast to Hermite interpolation, a Birkhoff interpolationproblem does not always have a unique solution or, even worse, does not have a solution. It is, however,convenient to consider Hermite interpolation to be a special case of lacunary interpolation and to deal withHermite–Birkhoff interpolation.

For instance, there is no quadratic polynomial p (x ) = a x 2+ b x + c such that

p (−1) = p (1) = 0, p ′(0) = 1. (1)

In this case we can try to enlarge the space of possible solutions by considering rational functions

p

q

( j )

(xi ) = fi , j ,

instead of polynomials, hoping that the problem is solvable in the larger space. In [19], the univariate Birkhoffrational interpolation problem is investigated. First, The Birkhoff rational interpolation problem is convertedinto a polynomial system solving problem. Then the polynomial system is solved by means of Gröbner basisand thus the solution of the Birkhoff rational interpolation is obtained. However, by easy calculations, wecan see that the problem

p

q

(−1) =

p

q

(1) = 0,

p

q

′(0) = 1 (2)

does not have a solution in the space of rational functions of the form

p

q

(x ) =a x + b

x + c,

1

domain decomposition idea

no solution unique solution

Figure 1: The unsolvable Hermite–Birkhoff problem (on the left) is split up into two solvable subproblems (on the right).We denote by a ball the available data and by a cross the not-available data. The first line relates to f (xi ), the second oneto f ′(xi ).

neither in the space of rational functions of the form

p

q

(x ) =a

x 2+ b x + c,

which are appropriate spaces to consider if we are looking for a unique solution of (2). The interest in thiskind of interpolation lies in the fact that, in recent years, many scholars applied the Birkhoff interpolation innumerically solving boundary value problems or initial-value problems and rational functions sometimesare superior to polynomials with the same interpolation data because they can achieve more accurateapproximations with the same amount of computation (see [26] and the references therein).

In this paper we propose to split up the unsolvable problems in two or more solvable subproblems,as shown in Figure 1 for the particular case in (1), whose solutions can be blended together. Here weconsider the case of multinode basis functions [17] as blending functions. To this end we consider a coveringF = F1, F2, . . . , Fm of X by subsets Fk ⊂ X , such that, for each k = 1, . . . , m , the corresponding Hermite–Birkhoff interpolation subproblem p ( j )(xi ) = fi , j , xi ∈ Fk , j ∈Ji has a unique solution, and we associate toeach Fk , k = 1, . . . , m , a multinode basis function (see Section 2). The latter are then used in combinationwith the local Hermite–Birkhoff polynomials that interpolate the data associated to Fk (see Sections 3 and 4)and the approximation order of the combination is studied (see Section 6.1). Finally, we provide numericalexperiments which confirm the theoretical results on the approximation order and show a good accuracy ofapproximation (see Section 6).

2 Multinode basis functions

Let us consider a covering F = F1, F2, . . . , Fm of X by its not empty subsets Fk ⊂ X , that is,

m⋃

k=1

Fk = X , Fk 6= ;, for each k = 1, . . . , m . (3)

The multinode basis functions with respect to the covering F are defined by

Bµ,k (x ) =

∏

xi∈Fk

|x − xi |−µ

m∑

l=1

∏

xi∈Fl

|x − xi |−µ, k = 1, . . . , m , (4)

where µ > 0 is a parameter that determines the differentiability class of the basis functions and controlsthe range of influence of the data values, in a sense that we specify later in Section 6 (see Figure 2). Themultinode basis functions (4) are non-negative and form a partition of unity, that is,

m∑

k=1

Bµ,k (x ) = 1, (5)

but instead of being cardinal they satisfy the following properties.

2

B2,1(x )B4,1(x )

0.3

0.2

0.1

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.00.0 0.2 0.4 0.8 1.00.6

B2,3(x )B4,3(x )

0.3

0.2

0.1

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.00.0 0.2 0.4 0.8 1.00.6

B2,2(x )B4,2(x )

0.3

0.2

0.1

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.00.0 0.2 0.4 0.8 1.00.6

B2,4(x )B4,4(x )

0.3

0.2

0.1

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.00.0 0.2 0.4 0.8 1.00.6

Figure 2: Multinode basis functions B2, j (x ) and B4, j (x ) for the subsets Fk displayed with black balls.

Proposition 1. The multinode basis functions in (4) vanish at all nodes x j that are not in Fk ,

Bµ,k (x j ) = 0, µ> 0, (6)

for any k = 1, . . . , m and x j /∈ Fk , and∑

k∈Ki

Bµ,k (xi ) = 1, µ> 0, (7)

whereKi = l ∈ 1, . . . , m : xi ∈ Fl 6= ; (8)

is the set of indices of all subsets of F that contain xi .

Proof. If we multiply both the numerator and the denominator of (4) with |x − x j |µ, then

Bµ,k (x ) =Ck (x )

m∑

l=1Cl (x )

,

where

Cl (x ) = |x − x j |µ∏

xi∈Fl

1

|x − xi |µ, l = 1, . . . , m .

Then, Cl (x j ) = 0 if and only if l /∈ K j , and (6) follows because k /∈ K j . Equality (7) follows by (6) and thepartition of unity property (5).

Proposition 2. For µ> 0 even integer, the multinode basis functions (4) are rational and have no real poles,otherwise their class of differentiability is µ−1 for µ odd integer and [µ], the largest integer not greater than µ,in all remaining cases. Moreover, all derivatives of order ` > 0 vanish at all nodes x j that are not in Fk ,

B (`)µ,k (x j ) = 0, (9)

3

for any k = 1, . . . , m and x j /∈ Fk and∑

k∈Ki

B (`)µ,k (xi ) = 0, µ> 1. (10)

Proof. If µ> 1, then Cl (x ) is differentiable at x j , and (9) follows because

B ′µ,k (x ) =C ′k (x )

m∑

l=1Cl (x ) +Ck (x )

m∑

k=1C ′l (x )

m∑

l=1Cl (x )

2

and C ′k (x j ) = 0 for x j /∈ Fk . The procedure can be iterated until C (`)l (x ) exists at x j . Equation (10) follows bydifferentiating both sides of (5) and by using Equation (9).

Proposition 3. The multinode basis functions (4) may be written in the following form

Bµ,k (x ) =

∏

xi /∈Fk

|x − xi |µ

m∑

l=1

∏

xi /∈Fl

|x − xi |µ. (11)

Proof. Expression (11) is obtainable from (4) by elementary calculations.

3 Multinode local Hermite–Birkhoff interpolation

Let us consider the Hermite–Birkhoff interpolation problem

p ( j )[ f ](xi ) = f ( j )(xi ), i = 1, . . . , n , j ∈Ji , (12)

and let us assume that for each k = 1, . . . , m , the Hermite–Birkhoff interpolation subproblems

P ( j )k [ f ](xi ) = f ( j )(xi ), xi ∈ Fk , j ∈Ji (13)

have a unique solution in their appropriate polynomial spacesPqkx , where qk =

∑

xi∈Fk#(Ji )−1. In order to test

the solvability of the problems (12) and (13), one can use the necessary Pólya’s condition [20] or the sufficientcondition given by the Atkinson–Sharma theorem [5] and, in case of existence, the solution can be computedby solving a linear system of qk linear equations in qk unknowns. In fact, there are no explicit expressions forthe polynomials Pk (x ), even though many algorithms for calculating them are known [18, 21, 23].

The use of determinants allows also to give an explicit expression for the remainder term

Rk [ f ](x ) = f (x )−Pk [ f ](x ),

which is obtained in [4] by using the Peano’s kernel theorem [15] and from which a bound can be computed.Nonetheless, this formulation, for its extreme generality is not useful for our purpose. Therefore we state thefollowing

Theorem 1. Let us suppose, for each k = 1, . . . , m, that Fk = xk1, xk2

, . . . , xklwith xk1

< xk2< · · ·< xkl

. Let usset qmax =maxk qk and let Ω be a closed interval containing X . If f ∈C qmax+1(Ω), then

| f (x )−Pk [ f ](x )| ≤ (1+ ‖Pk‖∞)

∏

xi∈Fk|x − xi |#(Ji )

(qk +1)!| f (qk+1)(ξk )|, (14)

where min(x , xk1)<ξk <max(x , xkl

).

Proof. Since we require the uniqueness of the solution of the local Hermite–Birkhoff interpolation prob-lem (13), the operator Pk is a projector [8] on the polynomial space Pqk

x , that is, it reproduces polynomials up

4

to the degree qk . Let denote by Hk [ f ](x ) the Hermite interpolation polynomial on the nodes of Fk with #(Ji )interpolation conditions on each node xi ∈ Fk . Then we have

| f (x )−Pk [ f ](x )| ≤ | f (x )−Hk [ f ](x )|+ |Hk [ f ](x )−Pk [ f ](x )|≤ | f (x )−Hk [ f ](x )|+ |Pk Hk [ f ](x )−Pk [ f ](x )|≤ (1+ ‖Pk‖) | f (x )−Hk [ f ](x )|.

(15)

On the other hand, it is well known a Cauchy representation for the remainder term in Hermite interpola-tion [15],

f (x )−Hk [ f ](x ) =

∏

xi∈Fk(x − xi )

#(Ji )

(qk +1)!f (qk+1)(ξk ), (16)


). Inequality (14) follows by taking the modulus of both sides of (16) andby substituting it in (15).

4 Multinode global interpolation operator

As soon as we have provided a solution for all local Hermite–Birkhoff interpolation problems, we define themultinode global interpolation operator by

Mµ[ f ,F ](x ) =m∑

k=1

Bµ,k (x )Pk [ f ](x ), (17)

where Pk [ f ](x ) is the polynomial solution of the Hermite–Birkhoff interpolation problem on Fk . The operatorMµ[ f ,F ] has remarkable properties. Firstly, it reproduces polynomials up to a certain degree as specified inthe following proposition.

Proposition 4. The operator Mµ[ f ,F ] reproduces polynomials up to the degree qmin =mink qk .

Proof. Since the Hermite–Birkhoff interpolation subproblems (13) have a unique solution in the polynomialspaces Pqk

x , Pk [ f ](x ) reproduces polynomials up to degree qk for k = 1, . . . , m . Moreover, the basis functionsBµ,k (x ) are a partition of unity and therefore, for each p ∈Pqmin

x ,

Mµ[p ,F ](x ) =m∑

k=1

Bµ,k (x )Pk [p ](x ) =m∑

k=1

Bµ,k (x )p (x ) = p (x ).

Remark 1. If the initial problem has a unique polynomial solution, by setting F = X , Mµ[ f ,F ](x ) coincideswith that polynomial solution.

Secondly, the operator Mµ[ f ,F ] is an interpolation operator as specified in the following theorem.

Theorem 2. The operator Mµ[ f ,F ] interpolates the functional data

Mµ[ f ,F ](xi ) = f (xi ), for each i , such that 0 ∈Ji (18)

and if F is a partition of X (i.e., Fα ∩ Fβ = ; for each α 6=β ), then the operator Mµ[ f ,F ] interpolates all dataused in its definition, that is,

M ( j )µ [ f ,F ](xi ) = f ( j )(xi ), for each k = 1, . . . , m , xi ∈ Fk , j ∈Ji .

Proof. Let xi ∈ X such that 0 ∈Ji . Then we have

Mµ[ f ,F ](xi ) =m∑

k=1

Bµ,k (xi )Pk [ f ](xi ) =m∑

k=1

Bµ,k (xi ) f (xi ) = f (xi ).

Let ` ∈ 1, . . . , qmin and µ> qmin+1. By differentiating ` times (17) we get, by the Leibniz rule for differenti-ation,

M (`)µ [ f ,F ](x ) =

m∑

k=1

∑

ι=0

`

ι

B (`−ι)µ,k (x )P(ι)

k [ f ](x ) =m∑

k=1

`−1∑

ι=0

`

ι

B (`−ι)µ,k (x )P(ι)

k [ f ](x ) +m∑

k=1

Bµ,k (x )P(`)

k [ f ](x ).

5

Let xi ∈ X and ` ∈Ji . Since F is a partition of X then Ki = κ. By properties (6) and (7) we have

m∑

k=1

Bµ,k (xi )P(`)

k [ f ](xi ) =m∑

k=1k 6=κ

Bµ,k (xi )P(`)

k [ f ](xi ) +Bµ,κ(xi )P(`)κ [ f ](xi ) = f (`)(xi ).

On the other hand, by properties (9) and (10) we have

m∑

k=1

B (`−ι)µ,k (xi )P(ι)

k [ f ](xi ) =m∑

k=1k 6=κ


k [ f ](xi ) +B (`−ι)µ,κ (xi )P(ι)κ [ f ](xi ) = 0

for each ι = 0, . . . ,`−1.

Let us observe that the operator Mµ[ f ,F ] could not interpolate all derivative data at some xκ if #(Kκ)> 1 andthe sequence of indices in Jκ is broken.

Example 1. For example, let us assume

#(Kκ) = 2, Fα ∩ Fβ = xκ, Jκ = 0, 2, . . . ,`−1,` ,`≥ 2

andB (`−1)µ,α (xκ)P

′α[ f ](xκ) +B (`−1)

µ,β (xκ)P′α[ f ](xκ) 6= 0.

We note thatP ′α[ f ](xκ) 6= P ′β [ f ](xκ),

by property (10). FromP (`)α [ f ](xκ) = P (`)β [ f ](xκ) = f (`)(xκ),

by properties (6) and (7), it easily follows that

m∑

k=1

Bµ,k (xκ)P(`)

k [ f ](xκ) = Bµ,α(xκ) f(`)(xκ) +Bµ,β (xκ) f

(`)(xκ) = f (`)(xκ).

On the other hand,

m∑

k=1

`−1∑

ι=0

`

ι

B (`−ι)µ,k (xκ)P(ι)

k [ f ](xκ) =`−1∑

ι=0

`

ι

B (`−ι)µ,α (xκ)P(ι)α [ f ](xκ) +B (`−ι)µ,β (xκ)P

(ι)α [ f ](xκ)

by property (9). Let us fix our attention to the right hand side of the previous equation. For each ι ∈Jκ, weget

B (`−ι)µ,α (xκ)P(ι)α [ f ](xκ) +B (`−ι)µ,β (xκ)P

(ι)α [ f ](xκ) =

B (`−ι)µ,α (xκ) +B (`−ι)µ,β (xκ)

f (ι)(xκ) = 0

by property (10), butB (`−1)µ,α (xκ)P

′α[ f ](xκ) +B (`−1)

µ,β (xκ)P′α[ f ](xκ) 6= 0

and consequentlyM (`)µ [ f ,F ](xκ) 6= f (`)(xκ).

In order to avoid this trouble, we proceed as follows. For each κ= 1, . . . , n let νκ = #(Kκ) and Fα1, . . . , Fανκ

the subsets of X which contain xκ. As above, let us denote by Pα1[ f ], . . . , Pανκ [ f ] the polynomial solutions of

the Hermite–Birkhoff interpolation problems on Fα1, . . . , Fανκ , respectively. For all j = 0, 1, . . . , max(Jκ), we set

f ( j )(xκ) =1

νκ

P ( j )α1[ f ](xκ) + · · ·+P ( j )ανκ

[ f ](xκ)

(19)

and we note thatf ( j )(xκ) = f ( j )(xκ), (20)

as soon as j ∈Jκ. For each k = 1, . . . , m , we call the Hermite interpolation problem

P ( j )k [ f ](xi ) = f ( j )(xi ), xi ∈ Fk , j = 0, 1, . . . , max(Ji ) (21)

6

a Hermitian completion of the Hermite–Birkhoff interpolation problem (13). It is well known that eachinterpolation problem (21) has a unique solution Pk [ f ](x ) in the polynomial space Pdk

x , where dk = #(Fk )+∑

xi∈Fkmax(Ji )−1, for which there are explicit formulas in Lagrange or Newton form (see [3, 11, 25] and the

references therein). Nevertheless, as soon as νk > 1 and at least two among P ( j )α1[ f ](xκ), . . . , P ( j )ανκ [ f ](xκ) are

different, we get qk < dk ; in this case, if p ∈Pqkx is a generic polynomial, then Pk [p ] is different from p , since,

by (19), we have completed the lacunary data using solutions of several interpolation problems. In fact wehave

qk = dex(Pk [·]) = minj=0,1,...,max(Ji )

dex(Pα j[·])

and the proof is straightforward. Consequently, qk ≤ qk . Results on the bound for the remainder term

Rk [ f ](x ) = f (x )− Pk [ f ](x )

can be obtained similarly to Theorem 1, taking into account that dex(Pk [·]) = qk .

Theorem 3. Let us suppose, for each k = 1, . . . , m, that Fk = xk1, xk2

, . . . , xklwith xk1

< xk2< · · ·< xkl

. Let usset qmax =maxk qk and let Ω be a closed interval containing X . If f ∈C qmax+1(Ω), then

| f (x )− Pk [ f ](x )| ≤ (1+ ‖Pk‖∞)

∏

xi∈Fk(x − xi )

#(Ji )−#(Si )

(qk +1)!| f (qk+1)(ξk )|,


) and Si ⊂Ji , i = 1, . . . , n, such that∑

xi∈Fk#(Ji )−

∑

xi∈Fk#(Si ) = qk +1.

Proof. Since dex(Pk [·]) = qk , the operator Pk is a projector [8] on the polynomial space P qkx , that is, it repro-

duces polynomials up to the degree qk . Let us denote by Hk [ f ](x ) the Hermite interpolation polynomial onthe nodes of Fk with #(Ji )−#(Si ) interpolation conditions on each node xi ∈ Fk . Then we have

| f (x )− Pk [ f ](x )| ≤ | f (x )− Hk [ f ](x )|+ |Hk [ f ](x )− Pk [ f ](x )|≤ | f (x )− Hk [ f ](x )|+ |Pk Hk [ f ](x )− Pk [ f ](x )|≤ (1+ ‖Pk‖) | f (x )− Hk [ f ](x )|.

Therefore,

f (x )− Hk [ f ](x ) =

∏

xi∈Fk(x − xi )

#(Ji )−#(Si )

(qk +1)!f (qk+1)(ξk ),


).

Example 2. Let X = −1,0,1 with interpolation conditions as in (1) or Figure 1. In this case, F1 = −1,0,F2 = 0, 1,

P1[ f ](x ) = f (−1) + (1+ x ) f ′(0),

P2[ f ](x ) = f (1) + (−1+ x ) f ′(0),

and

P1[ f ](x ) =f (−1) + f (1)

2+ f ′(0)x +3

f (−1)− f (1)2

+ f ′(0)

x 2+

f (−1) + f (1) +2 f ′(0)

x 3,

P2[ f ](x ) =f (−1) + f (1)

2+ f ′(0)x +3

f (1)− f (−1)2

− f ′(0)

x 2+

f (−1)− f (1) +2 f ′(0)

x 3.

Let f (x ) = sin(x ). In Figure 3 we show the polynomials P1[ f ] and P2[ f ], and their Hermitian completions P1[ f ]and P2[ f ], respectively. In Figure 4 we show the absolute values of the errors e [ f ,F ](x ) = f (x )−M2[ f ,F ](x )and e [ f ,F ](x ) = f (x )− M2[ f ,F ](x ) in the interval [−1, 1], where

M2[ f ,F ](x ) = B2,1(x )P1[ f ](x ) +B2,2(x )P2[ f ](x )

andM2[ f ,F ](x ) = B2,1(x )P1[ f ](x ) +B2,2(x )P2[ f ](x ).

As we can see, at least in this case, the inequality |e [ f ,F ](x )| ≤ |e [ f ,F ](x )| holds for all x ∈ [−1, 1].

7

−1.0

− 0.5

1.0

0.5

−1.0 − 0.5 1.00.5

−1.0

− 0.5

1.0

0.5

−1.0 − 0.5 1.00.5

Figure 3: Polynomials P1[ f ] (solid) and P2[ f ] (dashed), on the left, and their Hermitian completions P1[ f ] and P2[ f ],respectively, on the right.

0.2

0.1

−1.0 − 0.5 1.00.50.0

Figure 4: Comparison between the absolute values of the errors e [ f ,F ](x ) (solid) and e [ f ,F ](x ) (dashed) for the functionf (x ) = sin(x ) in the interval [−1, 1].

We set

Mµ[ f ,F ](x ) =m∑

k=1

Bµ,k (x )Pk [ f ](x ).

The operator Mµ[·,F ] preserves the reproducing polynomial property of Mµ[·,F ] and, in addition, interpol-ates all derivative data. In fact we have

Proposition 5. The operator Mµ[ f ,F ]

(a) reproduces polynomials up to the degree qmin =mink qk ;

(b) interpolates all data used in its definition, that is,

M ( j )µ [ f ,F ](xi ) = f ( j )(xi ), for each k = 1, . . . , m , xi ∈ Fk , j ∈Ji .

Proof.

(a) Let p ∈Pqminx . Since the Hermite–Birkhoff interpolation subproblems (13) have a unique solution in the

polynomial spaces Pqkx , we have Pk [p ] = p for each k = 1, . . . , m . Setting (19) now becomes

p ( j )(xk ) =1

νk(p ( j )(xk ) + · · ·+p ( j )(xk )) = p ( j )(xk ),

and consequently the Hermitian completion of the Hermite–Birkhoff interpolation problem (13),

P ( j )k [p ](xi ) = p ( j )(xk ), xi ∈ Fk , j = 0, 1, . . . , max(Ji ),

has the solution Pk [p ] = p , since qk ≤ dk . The statement then follows, because the basis functions Bµ,k (x )are a partition of unity.

(b) Let ` ∈ 0, 1, . . . , q and µ> q +1. By differentiating ` times (17), we get, by the Leibniz rule for differenti-ation,

M (`)µ [ f ,F ](x ) =

m∑

k=1

∑

ι=0

`

ι

B (`−ι)µ,k (x )P(ι)

k [ f ](x )

=m∑

k=1

`−1∑

ι=0

`

ι

B (`−ι)µ,k (x )P(ι)

k [ f ](x ) +m∑

k=1

Bµ,k (x )P(`)

k [ f ](x ).

8

Let xi ∈ X and Fα1, . . . , Fανi

be the subsets of X which contain xi and ` ∈Ji . By properties (6) and (7), wehave

m∑

k=1

Bµ,k (xi )P(`)

k [ f ](xi ) =m∑

k=1k /∈α1,...,ανi

Bµ,k (xi )P(`)

k [ f ](xi ) +m∑

k=1k∈α1,...,ανi

Bµ,k (xi )P(`)

k [ f ](xi )

=m∑

k=1k 6=κ

0+ f (`)(xi )m∑


Bµ,k (xi )

= f (`)(xi ),

since (21) and (20). On the other hand, for each ι = 0, . . . ,`−1, we have

m∑

k=1


k [ f ](xi ) =m∑

k=1k /∈α1,...,ανi


k [ f ](xi ) +m∑



k [ f ](xi )

=m∑

k=1k 6=κ

0+ f (`)(xi )m∑


B (`−ι)µ,k (xi )

= 0,

since (9), (10), and (21).

In the next sections we discuss the approximation order of the operators Mµ[ f ,F ] and Mµ[ f ,F ], and we willcompare their accuracies of approximation in several interesting cases.

5 The approximation order

Let us now study the approximation order of the operators Mµ[ f ,F ] and Mµ[ f ,F ]. To this end let ‖·‖ be themaximum norm and Rρ(y ) = x ∈R : |x − y | ≤ρ be the closed interval with centre y and size 2ρ. We define

h = infρ > 0 :∀x ∈Ω ∃Fk ∈F : Rρ(x )∩ Fk 6= ; (22)

andL = infl > 0 :∀Fk ∈F ∃x ∈Ω : Fk ⊂Rl h (x ), (23)

M = supx∈Ω

#Fk ∈F : Rh (x )∩ Fk 6= ;, (24)

N =maxi

#Fk ∈F : xi ∈ Fk =maxi

#Ki . (25)

A small value of h corresponds to a rather uniform distribution of nodes, but does not exclude the presenceof large Fk (note that Lh is half the diameter of the largest Fk ). A large value of M corresponds to clustered Fk

and N is the maximum number of Fk with at least a common node (N = 1 means that F is a partition of X ).Moreover, we set

Nk =∑

xi∈Fk

#(Ji ) = qk +1,

and

CFmax=max

k#(Fk ), CFmin

=mink

#(Fk ),

CNmax=max

kNk , CNmin

=mink

Nk .

Theorem 4. Let Ω be an interval that contains X , f ∈ C qmax+1(Ω), µ > 1+CNmaxCFmin

, and #(Fk ) = const for each k .

Then,| f (x )−Mµ[ f ,F ](x )| ≤C M h CNminφmax

for any x ∈Ω, with C a positive constant which depends on Fk and µ, andφmax =max j=0,...,qmax‖ f ( j )‖∞.

9

Proof. For x ∈Ω letQh (y ) = x ∈R : y −h < x ≤ y +h

be the half-open interval with centre y and size 2h . Let Tj = Tj ,h (x ) be the half-open annulus with centre xand radius 2h j defined by

Tj =Qh (x −2h j )∪Qh (x +2h j ), j = 0, . . . , N .

Note that T0 =Qh (x ) and that

Ω⊂N⋃

j=0

Tj , N =

diam(Ω)2h

+1.

By settings (22) and (24) we have1≤ #(X ∩Tj ,h )≤ 2M .

T0 =Qh (x ) contains at least one node by setting (22); therefore, for each Fk with at least one node in T0, wehave

∏

xi∈Fk

|x − xi | ≤ (Lh )#(Fk ), (26)

because of 23. Let us consider now the sets Fk with at least one node in T1 and no nodes in T0; we have

h #(Fk ) ≤∏

xi∈Fk

|x − xi | ≤

(L +3)h#(Fk ),

while for the sets Fk with at least one node in T2 and no nodes in T1 ∪T0 we have

(3h )#(Fk ) ≤∏

xi∈Fk

|x − xi | ≤

(L +5)h#(Fk ),

and in general, for the sets Fk with at least one node in Tj and no nodes in Tj−1 ∪ · · · ∪T0,

(2 j −1)h#(Fk ) ≤

∏

xi∈Fk

|x − xi | ≤

(L +2 j +1)h#(Fk ). (27)

Let us now turn to the approximation error

e (x ) = | f (x )−Mµ[ f ,F ](x )|

of the multinode global operator at x . By (17) and the fact that the basis functions Bk ,µ(x ) are non-negativeand form a partition of unity,

e (x )≤

m∑

k=1

Bk ,µ(x ) f (x )−m∑

k=1

Bk ,µ(x )Pk [ f ](x )

≤m∑

k=1

| f (x )−Pk [ f ](x )|Bk ,µ(x ).

Using Proposition 1 and (4), we then get

e (x )≤m∑

k=1

(1+ ‖Pk‖)

∏


(qk +1)!| f (qk+1)(ξk )|

∏

xi∈Fk|x − xi |−µ

∑ml=1

∏

x j∈Fl|x − x j |−µ

. (28)

Let Fkmin∈F be the subset, such that

∏

xi∈Fkmin

|x − xi |=mink

∏

xi∈Fk

|x − xi |.

Then,∏

xi∈Fkmin

|x − xi | ≤ (Lh )#(Fk0),

since at least one set Fk0has a node in T0. Finally, we have to bound, for each k = 1, . . . , m ,

∏

xi∈Fk

|x − xi |#(Ji ).

10

For each Fk with at least one node in T0, we have∏

xi∈Fk

|x − xi |#(Ji ) ≤∏

xi∈Fk

(Lh )#(Ji ) ≤ (Lh )Nk ,

and for each Fk with no nodes in T0 ∪ · · · ∪Tj−1 and at least one node in Tj ,∏

xi∈Fk

|x − xi |#(Ji ) ≤∏

xi∈Fk

(L +2 j +1)h#(Ji ) ≤

(L +2 j +1)hNk .

Then we get

e (x )≤m∑

k=1

(1+ ‖Pk‖)

∏


(qk +1)!| f (qk+1)(ξk )|

∏


∏

xi∈Fkmin|x − xi |−µ

≤∏

xi∈Fkmin

|x − xi |µm∑

k=1

(1+ ‖Pk‖)

∏


(qk +1)!| f (qk+1)(ξk )|

∏

xi∈Fk

|x − xi |−µ.

We denote byI j the set of subsets Fk with at least one node in Tj and no nodes in T0∪· · ·∪Tj−1. By construction,⋃N

j=0 I j =F and⋂N

j=0 I j = ;. Moreover, if we let

Pmax =maxk‖Pk‖,

φmax = maxj=0,...,qmax

‖ f ( j )‖∞,

then by bounding (28) we have

e (x )≤(1+Pmax)(qmin+1)!

φmax

∏

xi∈Fkmin

|x − xi |µm∑

k=1

∏

xi∈Fk

|x − xi |#(Ji )∏

xi∈Fk

|x − xi |−µ

≤(1+Pmax)(qmin+1)!

φmax

∏

xi∈Fkmin

|x − xi |µ

N∑

j=0

∑

Fk∈I j

∏

xi∈Fk

|x − xi |#(Ji )∏

xi∈Fk

|x − xi |−µ


φmax

∑

Fk∈I0

∏

xi∈Fk

|x − xi |#(Ji )

∏

xi∈Fkmin|x − xi |

∏

xi∈Fk|x − xi |

µ

+N∑

j=1

∑

Fk∈I j

∏

xi∈Fk

|x − xi |#(Ji )

∏


∏

xi∈Fk|x − xi |

µ!

.

Since∏


∏

xi∈Fk|x − xi |

≤ 1

for the subsets Fk ∈ I0 and∏


∏

xi∈Fk|x − xi |

≤(Lh )#(Fk0

)

(2 j −1)h#(Fk )

=L #(Fk0

)

(2 j −1)#(Fk )h #(Fk0

)−#(Fk )

for Fk ∈ I j , j ≥ 1, then


φmax

∑

Fk∈I0

∏

xi∈Fk

|x − xi |#(Ji )+N∑

j=1

∑

Fk∈I j

∏

xi∈Fk

|x − xi |#(Ji )

L #(Fk0)

(2 j −1)#(Fk )h #(Fk0

)−#(Fk )

µ


φmax

∑

Fk∈I0

(Lh )Nk +N∑

j=1

∑

Fk∈I j

(L +2 j +1)hNk

L #(Fk0)

(2 j −1)#(Fk )h #(Fk0

)−#(Fk )

µ


φmax

∑

Fk∈I0

L Nk h Nk +N∑

j=1

∑

Fk∈I j

(L +2 j +1)Nk h Nk

L #(Fk0)

(2 j −1)#(Fk )h #(Fk0

)−#(Fk )

µ


φmaxh CNmin

∑

Fk∈I0

L CNmax +N∑

j=1

∑

Fk∈I j

(L +2 j +1)CNmax

L CFmax

(2 j −1)CFmin

µ

h (#(Fk0)−#(Fk ))µ

.

11

Let us assume that #(Fk ) = const for each k . Then,


φmaxh CNmin

M L CNmax +MN∑

j=1

(L +2 j +1)CNmax

L CFmax

(2 j −1)CFmin

µ


φmaxh CNmin M

L CNmax +

L CFmaxµ

N∑

j=1

(L +2 j +1)CNmax

(2 j −1)µCFmin

.

Let us consider the series

∞∑

j=1

(L +2 j +1)CNmax

(2 j −1)µCFmin≈∞∑

j=1

(2 j )CNmax

(2 j )µCFmin=∞∑

j=1

1

(2 j )µCFmin−CNmax

,

which converges for µCFmin−CNmax

> 1. Thus, for µ>1+CNmax

CFmin, the operator Mµ[ f ,F ] has approximation order

O (h CNmin ).

Let us now study the approximation order of the operator Mµ[ f ,F ]. To this end we set

Nk =∑

xi∈Fk

#(Ji )−∑

xi∈Fk

#(Si ) = qk +1

andCNmax

=maxk

Nk , CNmin=min

kNk .

Theorem 5. Let Ω be an interval that contains X , f ∈ C qmax+1(Ω), µ >1+C

eNmaxCFmin

and #(Fk ) = const for each k .

Then,| f (x )− Mµ[ f ,F ](x )| ≤C M h CNminφmax

for any x ∈Ω, with C a positive constant which depends on Fk and µ.

Proof. Lete (x ) = | f (x )− Mµ[ f ,F ](x )|.

By (17) and the fact that the basis functions Bk ,µ(x ) are non-negative and form a partition of unity,

e (x )≤

m∑

k=1

Bk ,µ(x ) f (x )−m∑

k=1

Bk ,µ(x )Pk [ f ](x )

≤m∑

k=1

| f (x )− Pk [ f ](x )|Bk ,µ(x ).

Using Theorem 3 and (4), we then get

e (x )≤m∑

k=1

(1+ ‖Pk‖∞)

∏


∏

xi∈Fk|x − xi |#(Si )

·| f (qk+1)(ξk )|(qk +1)!

·

∏


∑ml=1

∏

x j∈Fl|x − x j |−µ

. (29)

Let Fkmin∈F be the subset such that

∏

xi∈Fkmin

|x − xi |=mink

∏

xi∈Fk

|x − xi |.

Then,∏

xi∈Fkmin

|x − xi | ≤ (Lh )#(Fk0),

since at least one set Fk0has a node in T0. Finally, we have to bound, for each k = 1, . . . , m ,

∏

xi∈Fk

|x − xi |#(Ji ).

For each Fk with at least one node in T0, we have∏

xi∈Fk

|x − xi |#(Ji ) ≤∏

xi∈Fk

(Lh )#(Ji ) ≤ (Lh )Nk ,

12

and for each Fk with no nodes in T0 ∪ · · · ∪Tj−1 and at least one node in Tj ,

∏

xi∈Fk

|x − xi |#(Ji ) ≤∏

xi∈Fk

(L +2 j +1)h#(Ji ) ≤

(L +2 j +1)hNk .

Then we get

e (x )≤m∑

k=1

(1+ ‖Pk‖∞)

∏


∏

xi∈Fk|x − xi |#(Si )

·| f (qk+1)(ξk )|(qk +1)!

·

∏


∏

xi∈Fkmin|x − xi |−µ

≤∏

xi∈Fkmin

|x − xi |µm∑

k=1

(1+ ‖Pk‖∞)

∏


(qk +1)!| f (qk+1)(ξk )|

∏

xi∈Fk

|x − xi |−µ−#(Si ).

We denote byI j the set of subsets Fk with at least one node in Tj and no nodes in T0∪· · ·∪Tj−1. By construction,⋃N

j=0 I j =F and⋂N

j=0 I j = ;. Moreover, if we let

Pmax =maxk‖Pk‖,

φmax = maxj=0,...,qmax

‖ f ( j )‖∞,

then by bounding (29) we have

e (x )≤(1+ Pmax)(qmin+1)!

φmax

∏

xi∈Fkmin

|x − xi |µm∑

k=1

∏

xi∈Fk

|x − xi |#(Ji )−#(Si )∏

xi∈Fk

|x − xi |−µ

≤(1+ Pmax)(qmin+1)!

φmax

∏

xi∈Fkmin

|x − xi |µ

N∑

j=0

∑

Fk∈I j

∏

xi∈Fk

|x − xi |#(Ji )−#(Si )∏

xi∈Fk

|x − xi |−µ


φmax

∑

Fk∈I0

∏

xi∈Fk

|x − xi |#(Ji )−#(Si )

∏


∏

xi∈Fk|x − xi |

µ

+N∑

j=1

∑

Fk∈I j

∏

xi∈Fk

|x − xi |#(Ji )−#(Si )

∏


∏

xi∈Fk|x − xi |

µ!

.

Since∏


∏

xi∈Fk|x − xi |

≤ 1

for the subsets Fk ∈ I0 and∏


∏

xi∈Fk|x − xi |

≤(Lh )#(Fk0

)

(2 j −1)h#(Fk )

=L #(Fk0

)

(2 j −1)#(Fk )h #(Fk0

)−#(Fk )

for Fk ∈ I j , j ≥ 1, then


φmax

∑

Fk∈I0

∏

xi∈Fk

|x − xi |#(Ji )−#(Si )+N∑

j=1

∑

Fk∈I j

∏

xi∈Fk

|x − xi |#(Ji )−#(Si )

L #(Fk0)

(2 j −1)#(Fk )h #(Fk0

)−#(Fk )

µ


φmax

∑

Fk∈I0

(Lh )Nk +N∑

j=1

∑

Fk∈I j

(L +2 j +1)hNk

L #(Fk0)

(2 j −1)#(Fk )h #(Fk0

)−#(Fk )

µ


φmax

∑

Fk∈I0

L Nk h Nk +N∑

j=1

∑

Fk∈I j

(L +2 j +1)Nk h Nk

L #(Fk0)

(2 j −1)#(Fk )h #(Fk0

)−#(Fk )

µ


φmaxh CNmin

∑

Fk∈I0

L CNmax +N∑

j=1

∑

Fk∈I j

(L +2 j +1)CNmax

L CFmax

(2 j −1)CFmin

µ

h (#(Fk0)−#(Fk ))µ

.

13

1.6

10.80.60.40.20

1.41.21.00.80.60.40.2

exponential ( f1)

10.80.60.40.20

sphere ( f4)

−4.28

−4.10−4.12−4.14−4.16−4.18−4.20−4.22−4.24−4.26

0.22

10.80.60.40.20

0.160.140.120.100.080.060.04

saddle ( f2)

0.200.18

0.40

10.80.60.40.20

0.350.300.250.200.150.100.05

gentle ( f5)

0.9

10.80.60.40.20

0.60.50.40.30.20.10.0

cliff ( f3)

0.70.8

10.80.60.40.20

0.350.300.250.200.150.100.05

steep ( f6)

0.00

Figure 5: Test functions used in our numerical experiments. The definitions of the functions can be found in [6].

Let us assume that #(Fk ) = const for each k . Then,


φmaxh CNmin

M L CNmax +MN∑

j=1

(L +2 j +1)CNmax

L CFmax

(2 j −1)CFmin

µ


φmaxh CNmin M

L CNmax +

L CFmaxµ

N∑

j=1

(L +2 j +1)CNmax

(2 j −1)µCFmin

.

Let us consider the series

∞∑

j=1

(L +2 j +1)CNmax

(2 j −1)µCFmin≈∞∑

j=1

(2 j )CNmax

(2 j )µCFmin=∞∑

j=1

1

(2 j )µCFmin−CNmax

,

which converges for µCFmin−CNmax

> 1. Thus, for µ>1+CNmax

CFmin, the operator Mµ[ f ,F ] has approximation order

O (h CNmin ).

6 Numerical results

To numerically test the approximation order of the multinode rational interpolation operators predicted byTheorem 4, we carried out a series of experiments with different sets of equispaced nodes on [0, 1] and testfunctions (see Figure 5). We report these results in Section 6.1. In Section 6.2 we present numerical resultson the approximation accuracy of the multinode rational interpolation operators.

6.1 Approximation order

Our first series of experiments numerically test the theoretical result on the approximation order in Theorem 4.With this aim we consider different coverings F of the nodeset X with an increasing number of subsetsFk (see Figure 6). For each of the six test functions fi we constructed the multinode rational interpolantMµ[ fi ,F ](x ), and we determined the maximum approximation error emax by evaluating | fi (x )−M4[ fi ,F ](x )|at 100, 000 random points x ∈ [0, 1] and recording the maximum value.

For the first experiment, the subsets are generated as follows:

1. we fix 3 equispaced points on the interval [0,1] and associate to them the Birkhoff data f (0), f ′(1/2),f (1). We consider the subsets F1 = 0, 1/2 and F2 = 1/2, 1;

2. at each step we halve the distance between two successive nodes by adding equispaced points withassociated Birkhoff data as shown in Figure 6.

14

10

10

10

F1 F2 F3 F4 F5 F6 F7 F8

F1 F2 F3 F4

F1 F2step 1

step 2

step 3

Figure 6: Three of the ten coverings F used in our numer-ical experiments with n = 3 and #(F ) = 2 (top), n = 5 and#(F ) = 4 (middle), n = 9 and #(F ) = 6 (bottom).

n m h

3 2 1/25 4 1/49 8 1/8

17 16 1/1633 32 1/3265 64 1/64

129 128 1/128257 256 1/256513 512 1/512

1025 1024 1/1024

Table 1: Starting from n = 3 equispaced nodes on theunit interval, we generate coverings F with n nodes, msubsets Fk , and interval width h (compare Figure 6).

100

10−1

10−2

10−3

10−4

10−5

10−6

10−7

10010−110−210−310−4

exponential

100

10−1

10−2

10−3

10−4

10−5

10−6

10−7

10010−110−210−310−4

sphere

100

10−1

10−2

10−3

10−4

10−5

10−6

10−7

10010−110−210−310−4

saddle

100

10−1

10−2

10−3

10−4

10−5

10−6

10−7

10010−110−210−310−4

gentle

100

10−1

10−2

10−3

10−4

10−5

10−6

10−7

10010−110−210−310−4

cliff

100

10−1

10−2

10−3

10−4

10−5

10−6

10−7

10010−110−210−310−4

steep

Figure 7: Log-log-plot of the approximation error emax over the interval width for the six test functions in Figure 5. Asreference, the dashed line indicates a perfect quadratic trend.

Table 1 lists the number of nodes and subsets, as well as the interval width h for the ten sets of nodes. In thiscase, CNmin

= 2, CNmax= 2, and CFmax

= 2. Figure 7 clearly demonstrates the quadratic approximation order ofthe operator M4[ fi ,F ].

For the second experiment, the subsets are generated as follows:

1. we fix 4 equispaced points on the interval [0,1] and associate to them the Birkhoff data f (0), f ′(1/3),f ′′(1/3), f (2/3), f ′(1), f ′′(1). In this case we consider the partition of X composed by the subsetsF1 = 0, 1/3 and F2 = 2/3, 1;

2. at each step we divide by three the distance between two successive nodes by adding equispacedpoints with associated Birkhoff data as shown in Figure 8.

Table 2 lists the number of nodes and subsets, as well as the interval width h for the ten sets of nodes. In thiscase, CNmin

= 3, CNmax= 3, and CFmax

= 2. Figure 9 clearly demonstrates the cubic approximation order of theoperator M4[ f ,F ].

6.2 Approximation accuracy

To test the effectiveness of the multinode rational interpolation operators, we compare them with thecorresponding combined Shepard operators. The numerical results are obtained by locally considering the

15

10

10

10

F1 F2 F3 F4

F1 F2step 1

step 2

step 3F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 F14

F5

Figure 8: Three of the ten coverings F used in our numer-ical experiments with n = 4 and #(F ) = 2 (top), n = 10 and#(F ) = 5 (middle), n = 28 and #(F ) = 14 (bottom).

n m h

4 2 1/310 5 1/928 14 1/2782 41 1/81

244 122 1/243730 365 1/729

2188 1094 1/21876562 3281 1/6561

19684 9842 1/1968359050 29525 1/59049

Table 2: Starting from n = 4 equispaced nodes on theunit interval, we generate coverings F with n nodes, msubsets Fk , and interval width h (compare Figure 8).

105

10−10

10−15

10−5

10010−110−210−310−5

exponential

10−4

100

100

10−10

10−15

10−5

10010−110−210−310−5

sphere

10−4

100

10−10

10−15

10−5

10010−110−210−310−5

saddle

10−4

100

10−10

10−15

10−5

10010−110−210−310−5

gentle

10−4

100

10−10

10−15

10−5

10010−110−210−310−5

cliff

10−4

100

10−10

10−15

10−5

10010−110−210−310−5

steep

10−4

Figure 9: Log-log-plot of the approximation error emax over the interval width for the six test functions in Figure 5. Asreference, the dashed line indicates a perfect cubic trend.

famous cases of Hermite osculatory, Lidstone and Abel–Goncharov interpolation conditions on n nodes. Wesolve the local problems by cubic interpolating polynomials on two points with intersecting and subsequentsubsets Fk . In Tables 3 and 4 we denote by:

1. S H4 the Shepard–Hermite operator and M H

4 the multinode operator combined with Hermite polyno-mials of degree 3;

2. S L4 the Shepard–Lidstone operator and M L

4 the multinode operator combined with Lidstone polyno-mials of degree 3.

We applied all six operators to the six test functions in Figure 5, using a grid of n equispaced points in [0, 1].For M4[ fi ,F ], we considered both intersecting Fk and disjoint Fk . Tables 3 and 4 list the maximum error emax,the average error emean, and the mean square error eMS. The pointwise errors ei were determined in absolutevalue at ne = 1001 points in [0, 1], and the errors were calculated by the formulas

emax = max1≤i≤ne

ei , emean =1

ne

ne∑

i=1

ei , eMS =

√

√

√

∑ne

i=1 e 2i

ne.

The results show that the multinode global operator Mµ[ f ,F ] is comparable to the Shepard interpolationmethods.

16

S H4 M H

4 M H4

intersecting Fk F partition

emax 8.9291e-05 2.7545e-05 1.1463e-04f1 emean 1.6655e-05 5.6201e-06 1.3466e-05

eMS 2.8505e-05 8.6650e-06 2.7513e-05emax 4.5433e-06 9.9506e-07 6.1703e-06

f2 emean 4.4964e-07 1.4991e-07 3.8711e-07eMS 9.3732e-07 2.5303e-07 9.8004e-07emax 1.4480e-04 3.1294e-05 1.6588e-04




f6 emean 2.3947e-06 7.9910e-07 2.0757e-06eMS 4.2773e-06 1.2302e-06 4.4285e-06

Table 3: Comparison of the interpolation operators ap-plied to the six test functions in Figure 5 using 33 equis-paced interpolation nodes in [0, 1] in the case of Hermite-type data.

S L4 M L

4 M L4

intersecting Fk F partition

emax 1.1326e-04 1.6864e-04 1.5177e-04f1 emean 2.7617e-05 3.1229e-05 3.4990e-05

eMS 3.7231e-05 4.7020e-05 4.7786e-05emax 3.3075e-06 5.2914e-06 5.5451e-06





f6 emean 3.8300e-06 4.2435e-06 4.7761e-06eMS 5.5007e-06 6.4193e-06 6.6181e-06

Table 4: Comparison of the interpolation operators ap-plied to the six test functions in Figure 5 using 33 equis-paced interpolation nodes in [0, 1] in the case of Lidstone-type data.

7 Conclusions

In this paper we propose to split up a univariate unsolvable Hermite–Birkhoff interpolation problem in two ormore solvable subproblems and to blend together the local solutions by using multinode basis functions [17]as blending functions. Numerical experiments are provided, which show a good accuracy of approximationand confirm the theoretical results on the approximation order discussed in the paper. It would be of interestto extend this approach to the case ofR2, the sphere S 2, and other manifolds, taking into account the resultsof previously published papers on this topics (see, for example, [1, 2, 7, 9, 10, 12, 13, 14, 16] and the referencestherein).

Acknowledgements

This research was supported by the INDAM-GNCS project 2016 and by a research fellowship from the CentroUniversitario Cattolico.

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Reconstruction of a function from Hermite–Birkhoff data · Reconstruction of a function from Hermite–Birkhoff data Francesco Dell’Accio Filomena Di Tommaso Kai Hormann Abstract