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The goal of this article is to present the ideas which allow us to use semi- classical Jacobi polynomials in the spline quadrature, assuming the conjecture is true. It is not a mathematically rigorous proof. Proofs done with the assistance of a computer algebra system, which confirm the conjecture for low continuity classes, are not included.

This article is organized as follows. In section 2, the two types of semi- classical orthogonal Jacobi polynomials necessary for this article are defined. In section 3, for the two introduced types of semi-classical orthogonal Jacobi polynomials the so-called defect of a polynomial is calculated. This is the dif- ference between a weighted sum and the integral over this polynomial. Then in section 2, introduced deformation of the Jacobi polynomials by Dirac δ’s allows to express this defect in a simple manner. The calculation of a defect for the type Q () semi-classical Jacobi polynomials with the formulae of previous section is only possible for the right half of a spline with a support of 2 subintervals. Therefore in section 4 a reflection map is introduced that allows to calculate the defect also for the left half of a support 2 spline in section 5. The conjecture appears in this section. In section 5 the rational recursion map is defined. This allows to proceed from one subinterval to the next subinterval. Finally, in section 6, to handle the general case of nonuniform subintervals, stretching and the assigned stretching map are defined.

The formulae in the appendices C and D are not used in this article. They are just added to show how these formulae look like in the case of continuity classes c =0, 1 when the conjecture is true.

2. Definition of the 2 types of semi-classical Jacobi orthogonal poly- nomials

In the literature the word ”semi-classical” is used for polynomials obtained by modifying the weight function of the classical or- thogonal polynomials in a certain not specified manner. The weight function 2 4 e−x −ax e.g. results in one kind of semi-classical Hermite orthogonal poly- nomials Hn(a, x), which for a = 0 is the classical Hermite polynomial. For the semi-classical orthogonal polynomials defined in here, we modify the weight function of the Jacobi polynomials by adding Dirac δ’s and its derivations at one or at the two ends of a subinterval. This gives us exactly the necessary degrees of freedom for these polynomials.

Let c be the continuity class of the polynomial splines throughout the whole article. To make the formulae not too overloaded in some cases I omit indices

2 (c) (c+1,c+1) in c. I write e.g. Qn(l, x) instead of Qn (l, x), Jn(x) instead of Pn (x) or ...

Now the vector space D = Rc+1 is defined, the symbol D like D(irac) or (D)istribution. This is the space of coefficients of the Dirac δ’s and its derivations in the following weight functions. So we have l = (l0,l1,...,lc−1,lc) ∈ D, the bold l like (l)eft when the Dirac’s are located at the left side of the interval [−1, +1] at −1. In the same manner r like (r)ight when the Dirac’s are located at +1 To define the orthogonal polynomials Qn(l, x) and Mn(l, r, x) define their weight functions:

c (i) WQ(l, x)= wQ(x) 1+ liδ (x + 1) (1) i=0 ! X c+1 wQ(x)=(1 − x) c c (i) i (i) WM (l, r, x)= wM (x) 1+ liδ (x +1)+ (−1) riδ (x − 1) (2) i=0 i=0 ! X X wM (x)=1

For l = 0 the weight function WQ(0, x) = wQ(x) is the weight function of the (c+1,0) Jacobi polynomials Pn (x), for l, r = 0 the weight function WM (0, 0, x)= (0,0) wM (x) = 1 is the weight function of the Jacobi polynomials Pn (x), the Pn(x). The Qn(l, x) and Mn(l, r, x) are semi-classical orthogonal Jacobi polynomials for the continuity classes c = 0, 1 as already introduced by:

- H.L. Krall 1940 [7] with a Dirac δ at only 1 interval end - T.H. Koornwinder 1984 [6] with 2 Dirac δ’s at both interval ends - J. Arvesu et al 2002 [2] with additional derivations of Dirac’s

In newer publications e.g. [5], see formulae (1.4) ... (1.6) the here defined semi-classical orthogonal polynomials belong to the Krall-Jacobi families.

These orthogonal polynomials are normalized. (c+1,0) (0,0) Qn(0, x) = Pn (x) and Mn(0, 0, x) = Pn (x) = Pn(x), the Legendre polynomials. These polynomials don’t have factors like e.g. (1+al1 +bl0r1 +... ) i.e. independent from x. Such a factor would give us normalization for l, r =0 too. In my definition (2) I have chosen different signs for the odd i-th deriva- tions of the Dirac δ’s. With this definition the Mn(l, r, x) fulfills the following symmetry with l, r interchanged, x reflected at 0:

n Mn(l, r, x) = (−1) Mn(r, l, −x)

3 With respect to the weights (1) and (2) the scalar products of the corre- sponding orthogonal polynomials are defined as:

+1

SQ(m, n, l, x)= WQ(l, x)Qm(l, x)Qn(l, x)dx (3) −Z1 +1

SM (m, n, l, r, x)= WM (l, r, x)Mm(l, r, x)Mn(l, r, x)dx (4) −Z1

By definition for m 6= n these scalar products are 0, for m = n a parameter is omitted: SM (m, l, r, x) is written for SM (m, m, l, r, x)

3. The fundamental theorem for applied to the semi-classical orthogonal Jacobi polynomials

The fundamental theorem states, see [9]: When the zeros xi of an orthogonal polynomial Pn(x) with respect to the weight function ω(x) are taken as nodes (how to determine the weights wi see the FT) of a Gaussian quadrature rule (xi, wi) the rule is exact for deg(f) ≤ 2n − 1.

+1 n ω(x)f(x)dx = wif(xi) i=1 −Z1 X Looking at the proof of this fundamental theorem it can be seen, that it is also valid when the weight function is a distribution. Our two weight functions defined in section 2 are distributions.

3.1. The fundamental theorem applied to the semi-classical one-sided Q ()

Applying the fundamental theorem to the orthogonal polynomials Qn(l, x) and the corresponding weight function WQ(l, x) we get the following result: The n nodes xi are the roots of Qn(l, x), the weights are:

∗ an SQ(n − 1, l, xi) wi = ′ an−1 Qn(l, xi) Qn−1(l, xi)

n the scalar product SQ(n, l, x) see (3), an is the coefficient of x in Qn(l, x) The equation connecting integral and the weighted sum is:

+1 n ∗ WQ(l, x)h(x)dx = wi h(xi) i=1 −Z1 X for deg(h) ≤ 2n − 1

4 c+1 We set now h(x)= g(x)/wQ(x)= g(x)/(1 − x) and until the rest of this section we assume the following condition is fulfilled:

(1 − x)c+1 | g(x), deg(g) ≤ 2n + c (5) where a | b means: a is a divisor of b, equivalently g(k)(+1) = 0 for k ≤ c, this is the right half of a spline with support of 2 intervals With this new g(x) we get:

+1 c n (i) ∗ c+1 1+ liδ (x + 1) g(x)dx = wi /(1 − xi) g(xi) ! i=1 −Z1 Xl=0 X ∗ c+1 We define a new weight by wi = wi /(1 − xi) and get our final formula for the weights:

an SQ(n − 1, l, xi) wi = ′ c+1 (6) an−1 Qn(l, xi) Qn−1(l, xi)(1 − xi)

Now with this new weights wi and bringing the Dirac δ’s to the rhs and with the following definition for the Dirac δ’s

δ(k)(x + a)f(x)dx = (−1)kf (k)(−a) (7) Z f (k)(x) being the k-th derivation of f(x), and f (0)(x)= f(x) Now the defect D(g) as it is called here for a function g(x), the difference between the weighted sum and integral over g(x) is introduced:

n +1 c i (i) D(g)= wi g(xi) − g(x)= (−1) li g (−1) (8) i=1 i=0 X −Z1 X under the conditions (5) When g(x) is a spline with a support of 1 interval then the rhs is 0. Then (xi, wi) is an exact quadrature rule for splines S(x) with a support of 1 interval

n +1 D(S)= wiS(xi) − S(x)=0 i=1 X −Z1 3.2. The fundamental theorem applied to the semi-classical two-sided M ()

Applying the fundamental theorem to the orthogonal polynomials Mn(l, r, x) and the corresponding weight function WM (l, r, x) we get the following result: The n nodes xi are the roots of Mn(r, x), the weights are:

an SM (n − 1, l, r, xi) wi = ′ (9) an−1 Mn(l, r, xi) Mn−1(l, r, xi)

5 n the scalar product SM (n, l, r, x) see (4), an is the coefficient of x in Mn(l, r, x) The equation connecting integral and the weighted sum is:

+1 n WM (l, r, x)g(x)dx = wig(xi) i=1 −Z1 X for deg(g) ≤ 2n − 1 Now the defect D(g) as it is called here for a function g(x), the difference between sum and integral over g(x) is introduced, with the definition (7) of the Dirac δ’s we get:

n +1 c c i (i) (i) D(g)= wi g(xi) − g(x)= (−1) li g (−1)+ ri g (+1) (10) i=1 i=0 i=0 X −Z1 X X under the condition deg(g) ≤ 2n − 1 When g(x) is a spline with a support of 1 interval then the rhs is 0. Then (xi, wi) is an exact quadrature rule for splines S(x) with a support of 1 interval

n +1 D(S)= wiS(xi) − S(x)=0 i=1 X −Z1

4. The reflected one-sided Q (), the reflection map and a conjecture

With the formula (8) and a restriction only in the degree of a polynomial g(x) we can calculate the defect of the polynomial for a subinterval of type M(). For a subinterval of type Q () the situation is different. The condition (5) in the formula (8) for the defect requires g(k)(+1) = 0 for k ≤ c. But we need the defect for polynomials with g(k)(+1) 6= 0 and now g(k)(−1) = 0 too. This is necessary to calculate the defect of all splines with support 2 intervals. Therefore a goal of this section is, to find for the polynomial Qn(l, -x) another vector lR (the subscript R means reflected) with the property:

Qn(l, -x)= αQn(lR, x)

The factor α is allowed (and has to be allowed) because we are only interested in the roots of the Qn(l, x) and a factor doesn’t change anything. In a previous section the vector space D = Rc+1 was defined, the space of coefficients of the Dirac δ’s in a weight function for orthogonal polynomials. Now the projective space J = RPc+1 ist defined, the symbol J like J (acobi), the space of coefficients in a sum of symmetric Jacobi polynomials. So we have j = (j0 : j1 : ··· : jc : jc+1) ∈ J. The projective space is the appropriate space, because the roots of the so defined polynomial do not depend an a factor and because the weights (6) are

6 homogenous of degree 0 in this polynomial the whole quadrature rule is invariant under this scaling. The semi-classical Jacobi polynomials Qn(l, x) can be expressed by a sum (c+1,c+1) of symmetric Jacobi polynomials Jn(x) = Pn (x), the coefficients jn,i(l) are polynomials in the coefficients of the vector l and n too

c+1 Qn(l, x)= jn,i(l)Jn−i(x) (11) i=0 X Though not used in this section, to be complete (with different coefficients jn,i now depending on 2 parameters):

2c+3 Mn(l, r, x)= jn,i(l, r)Jn−i(x) i=0 X The jn,i(l) in (11) define a polynomial map:

fn : D → J

l = (l0,...,lc) 7→ (jn,0(l): ··· : jn,c+1(l)) (12)

The spaces D and J have the same dimension c + 1, the inverse of the map f is defined and the ln,i are rational in j:

−1 fn : J → D j = (j0 : ··· : jc+1) 7→ (ln,0(j),...,ln,c(j)) (13)

In the following lower case letters are used for maps in J and the correspond- ing upper case letters for maps in D The following map is an involution, the roots of the corresponding polynomi- als are just reflected at 0. The coefficients of the odd degree symmetric Jacobi polynomials are mapped to the negative coefficients (the signs here for even c).

r : J → J

(j0 : j1 : ··· : jc : jc+1) 7→ (+j0 : −j1 : ··· :+jc : −jc+1) (14)

Now this reflection map r in J is transformed by the map fn into an involu- tion in D and we get the desired lR

−1 lR = Rn(l)= fn (r (fn(l)))

−1 Rn = fn ◦ r ◦ fn (15) where f ◦ g is the composition of maps f (g ()), because of the associativity the parenthesis can be omited. We know more about the transformed rational reflection map Rn: with Γ and Ni polynomials in n and l

c+1−i Rn : li 7→ Ni(n, l)/Γ (n, l) for 0 ≤ i ≤ c (16)

7 The composition of maps (15) can be visualized by this commutative dia- gram: J −−−−→r J

−1 fn fn (17) x  D −−−−→ D  Rn y The conjecture:

For a general continuity class c the polynomials Qn(l, x) can be written as a sum of symmetric orthogonal Jacobi polynomials (11) and define a polynomial map (12). The inverse of this map (13) exists and is rational. The transformed reflection map Rn (15) has the form (16). This last part of the conjecture is not relevant for this article. The current state of the conjecture can be found in appendix A. For a connection between the conjectured reflection map Rn and Cremona groups see appendix B.

5. Connecting two subintervals of type Q (), the recursion map

Here a so called recursion map Recn is defined. This is a map that transforms the vector l of the subinterval s to the vector Recn (l) of the next subinterval s+1, so that the quadrature rule is exact for all spline functions with a support of these 2 intervals and so for all splines when the degree d of the quadrature rule is d ≥ 2c + 1. I repeat here (8) for the defect = difference between weighted sum and integral: n +1 c i (i) D(g)= wi g(xi) − g(x)= (−1) li g (−1) i=1 i=0 X −Z1 X under the condition (5) that g(x) is the right side of a spline i.e. g(k)(+1) = 0 for 0 ≤ k ≤ c The condition (5) is fulfilled for the right side gr(x) of a spline with a support of 2 intervals, we get for the defect of this side:

c i (i) D(gr)= (−1) li gr (−1) i=0 X

Now we want to apply this (8) to get the defect of the left side gl(x) of a spline with support of 2 intervals and to determine the still unknown recursion map Recn for which the sum of the left and right defects is 0. At first we reflect the Qn(l, x) and gl(x) at 0. As a result the nodes and weights of the corresponding quadrature rule are reflected too. This does not change the defect. The parameter lR of this reflected Q () is given by (15), the reflected gl(x) fulfills

8 (i) the condition (5), the reflection multiplies the odd derivations gl (+1) by −1. So the defect for this side is:

c (i) D(gl)= lR,i gl (+1) i=0 X th where lR,i is the i component of the vector lR Because the 2 interval spline gl(x),gr(x) is of countinuity class c:

(i) (i) gl (+1) = gr (−1) (18) The sum of the defects of both sides of the spline is with the condition above:

c i (i) D(gl)+ D(gr)=0= lR,i + (−1) lRec,i gl (−1) i=0 X  th where lRec,i is the i component of the vector Recn(l) belonging the next subin- (i) terval. Because this equation has to be valid for arbitrary gl (−1) we have for all 0 ≤ i ≤ c: i 0= lR,i + (−1) lRec,i (19) The following involution C like (C)onnection is defined, this name because it makes that the sum of the defects in a subinterval s and the following s +1 is 0 and the quadrature rule is exact.

C : D → D

(l0,l1,...,lc−1,lc) 7→ (−l0, +l1,..., −lc−1, +lc) (20)

Now we get the recursion map with (15), the equation (19) is fulfilled:

−1 Recn = C ◦ Rn = C ◦ (fn ◦ r ◦ fn) (21)

This composition of maps (21) can be visualized by this diagram:

J −−−−→r J

−1 fn fn x  D D −−−−→ D  y C

Recn : D −→ D | {z } A recursion map can also be given in the J domain now named recn (with lower case initial letter):

−1 recn = cn ◦ r = (fn ◦ C ◦ fn ) ◦ r

9 Again this composition of maps can be visualized by a diagram:

recn : J −→ J J −−−−→r JJ

z }| −1 { fn fn  x D −−−−→ D y C  Here a short summary of the results in the previous sections: For 2 consecutive uniform subintervals of type Q,Q the nodes and weights are defined by the polynomials Qn(l, x) and Qm(Recn(l), x). The so given quadrature rule is exact for splines with a support of 1 or 2 subintervals, when the spline degree d is ≤ min(2n + c, 2m + c).

More details without a proof, the details are neccesary to get a rule for the complete interval: For 3 consecutive uniform subintervals of type Q, M, Q- (Q- here denotes Q reflected at x = 0) the nodes and weights are defined by the polynomials Qn(l, x), Mm(Recn(l),Reco(r), x) and Qo(r, -x). The so given quadrature rule is exact for splines with a support of 1 or 2 subintervals, when the spline degree d is ≤ min(2n + c, 2m − 1, 2o + c). The first 2 uniform subintervals at the left boundary of the interval [a,b] are defined by Qn(0, x) and Qm(Recn(0), x).

6. Nonuniform subintervals and stretching

To allow nonuniform subintervals an extended recursion map including stretched intervals is given. Define the stretching factor λs for a subinterval s as the ratio of subsequent subinterval lengths : λs = Ls/Ls−1 for s ≥ 2, define λ1 =1 The derivation of the extended recursion map RecSn now works similar to the proof in the previous section. There are 2 differences, the first: the right side of a spline with a support of 2 subintervals is now the stretched spline gλ(x). gr(x) is now gλ(x) scaled in x direction to the default interval length of 2. The second difference: because the gl(x) and now the stretched gλ(x) are continuous at the common interval ends condition (18) is different and a factor λi appears: i (i) (i) λ gl (+1) = gr (−1) The sum of the defects of both sides of the spline is because the defect of a stretched interval acquires the factor λ:

c i+1 i (i) D(gl)+ λD(gr)=0= lR,i + λ (−1) lRecS,i gl (−1) i=0 X 

10 i+1 i 0= lR,i + λ (−1) lRecS,i

The stretching map Sλ like (S)tretching is defined as:

Sλ : D → D 1 2 c c+1 (l0,l1,...,lc−1,lc) 7→ (l0/λ ,l1/λ ,...,lc−1/λ ,lc/λ ) (22)

Now finally we get the recursion map with stretching:

−1 RecSn = Sλ ◦ Recn = Sλ ◦ C ◦ (fn ◦ r ◦ fn) (23) Again this composition of maps (23) can be visualized by a diagram:

J −−−−→r J

−1 fn fn x  D D −−−−→ D −−−−→ D  y C Sλ

RecSn : D −→ D | {z } At the end a short summary of all defined maps sumarized in a diagram:

(14) r c sn,λ J −−−−→ J −−−−→n J −−−−→ J

−1 −1 (12) fn fn fn (13) fn x  x  D −−−−→ D −−−−−→ D −−−−−→ D  Rn y (20) C  (22) Sλ y

D is the vector space of the parameters l of Qn(l, x), J is the projective space of (c+1,c+1) the coefficents in the sum of symmetric Jacobi polynomials Pn (x), both spaces of dimension c +1. The reflection map r, the connection map C and the stretching map Sλ are simple monomial transformations and do not depend on n. The four maps r, Rn, cn and C are involutions.

7. Conclusions

With this approach, many of the questions that arose in the concluding re- marks of section 5 could be answered [1]. Even if the conjecture in section 4 is not proven for general continuity class, with the help of a computer algebra system it should be possible to determine the recursion map symbolically for the continuity classes c =2, 3. And this one could answer questions about the exis- tence of optimal uniform quadrature rules on the real line in these cases. We find a particularly interesting question: For uniform optimal rules in the closed in- terval and c = 3, is the convergence quadratic for all n as it is the case for c = 1? Appendices

11 A. The current state of the conjecture in setion 4.

- for continuity class c ≥ 3: Formulae (11) and (16) are conjectured.

- for continuity class c = 2: Formulae (11) and (16) were proven by a computer algebra system = CAS for certain values of the degree n up to n = 8. But the reflection map was not calculated for a general n as variable as e.g. in appendix D.

- for continuity class c = 1: Formulae (11) and (16) are proven by a CAS.

- for continuity class c = 0: In an article of Chihara [3] a proof of (11) can be found, because the map is linear fractional in this case formula (16) follows. Of course this can also be proven by a CAS.

B. A connection between the reflection map Rn and Cremona groups

The map Rn is birational and homogenized it is a group element in the c+1 Cremona group Bir PC , the group of birational maps in the projective space CPc+1, see [4]. R depends on the parameter n ∈ N . Because R is rational n  + n in n we can take also n ∈ R \{ ... }. So we get a 1-parameter involution in the Cremona group. The connection map cn also represents a 1-parameter involution in the Cremona group. Maybe this kind of involutions or the map fn (12) are known in the theory of Cremona groups. Then we can find something about the conjecture above. Maybe involutions or group elements of the form (16) are known in the theory of Cremona groups. If this kind of involution is not known, maybe this 1-parameter Ra in a ∈ R is interesting in the theory of Cremona groups. 1 The explicit formulae for the involution Rn in Bir PC you can find in appendix C and for R in Bir P2 in appendix D. n C   C. The formulae for continuity class c = 0

In this case the vector space D and the projective space J are 1-dimensional. Let l = (l0), r = (r0) ∈ D and j = (j0 : j1) ∈ J. Here we express the Qn(l, x) still as a sum over orthogonal and not as in (11) as sum over the orthogonal symmetric Jacobi polynomials. But these polynomials differ just by a factor depending only on (c+1,c+1) (3/2+c) n: Pn (x)= α(n) Cn (x).

12 The Gegenbauer polynomials are used, because when these formulae were created, the connection of quadrature rules for splines to the Jacobi polynomials was unknown to me. Gegenbauer polynomials with negative index are defined as 0. The one-sided semi-classical orthogonal Jacobi polynomial Qn(l, x):

F (n)=1+ n(n + 1) l0

(3/2) (3/2) F (n) Cn (x)+ F (n + 1) C − (x) Q (l, x)= n 1 n n +1 −1 Via formula (11) this defines the maps fn and fn , see (12) and (13). 2n(n + 1)F 2(n) wi = ′ n(n + 1)Qn(l, xi) Qn−1(l, xi)(1 − xi)

The (linear fractional in l) recursion map Recn (l) without stretching:

Γ(n) = (n + 1)(1 + n(n + 2) l0)

1 + (n + 1)2 l l 7→ 0 0 (n + 1)Γ(n)

This map and the reflection map Rn = C ◦ Recn, for C see (20), are not only defined for n ∈ N+ but for n ∈ R \ {−1 }

The two-sided semi-classical orthogonal Jacobi polynomial Mn(l, r, x):

2 H(n)=1+ n (l0 + r0 + (n − 1)(n + 1) l0r0) (3/2) (3/2) H(n) Cn (x) − H(n + 1) Cn−2 (x) (3/2) M (l, r, x)= + (l − r ) C − (x) n 2n +1 0 0 n 1 2H2(n) wi = ′ nMn(l, r, xi) Mn−1(l, r, xi)

In [1] in section 3.2 for even n a positive, optimal and 1-periodic quadrature rule for the real line is presented. It is left the reader as exercise to derive this quadrature rule with the formulae given here. In a first step get the fixed point lF of the recursion map and then calculate the nodes as roots of Qn(lF , x) and the weights with the formula above.

The connection map cn (j), the transformed C, see (20) is a linear, degree 1 involution in the projective space J, i.e. an element of Bir (PC):

j0 7→ (n + 1) j0 − n j1

j1 7→ (n + 2) j0 − (n + 1) j1

13 D. Some formulae for continuity class c = 1 In this case the vector space D and the projective space J are 2-dimensional. Let l = (l0,l1), r = (r0, r1) ∈ D and j = (j0 : j1 : j2) ∈ J. The one-sided semi-classical orthogonal Jacobi polynomial Qn(l, x):

E(n)=1+(n + 1)(n + 3)(l0 +3n(n + 3) l1(2 − (n − 1)(n + 1)(n + 2)(n + 4) l1)) 2 2 2 F (n)=1+ n(n + 2)(l0 + 6(n +2n − 1) l1 − 3(n − 1)n(n + 1) (n + 2)(n + 3) l1) (5/2) (5/2) (5/2) 6F (n) C (x) 6E(n) C − (x) 6F (n + 1) C − (x) Q (l, x)= n + n 1 + n 2 n (n + 2)(2n + 3) (n + 1)(n + 2) (n + 1)(2n + 3) −1 Via formula (11) this defines the maps fn and fn , see (12) and (13). 8(n + 1)F 2(n) wi = ′ 2 n(n + 2)Qn(l, xi) Qn−1(l, xi)(1 − xi)

The recursion map Recn (l) without stretching: 2 2 G0(n) = 4(2n +6n +3)+ n(n + 3)((11n + 33n + 16) l0 4 3 2 + 24(2n + 12n + 17n − 3n − 4) l1 2 2 +3n(n + 1)(n + 2)(n + 3)(−4(n + 1)(n + 2)(2n +6n − 5) l1 2 − 3(n − 1)n(n + 1)(n + 2)(n + 3)(n + 4) l0l1 2 2 + 2(3n +9n − 6) l0l1 + l0))

G1(n)=1 − 3n(n + 1)(n + 2)(n + 3) l1

2 Γ(n) = (n + 1)(n + 2)(1 + n(n + 3) l0 +6n(n + 3)(n +3n − 1) l1 2 2 2 − 3(n − 1)n (n + 1)(n + 2)(n + 3) (n + 4) l1)

E(n)G (n) l 7→ −l + 0 0 0 (6Γ(n))2 E(n)G (n) l 7→ l + 1 1 1 3(n + 1)(n + 2)Γ(n)

This map and the reflection map Rn = C ◦ Recn, for C see (20), are not only defined for n ∈ N+ but for n ∈ R \ {−1, −2 }

The connection map cn (j), the transformed C, see(20) is a quadratic, degree 2 involution in the projective space J:

B(n)= − (n + 3) j0 + n j2 4 3 2 2 2 2 2 ∆(n) = (n + 3)(2n + 18n + 49n + 48n + 18) j0 + n (n + 3)(2n +6n + 1) j1 2 2 2 2 n (n − 1)(2n +2n − 3) j2 − 2n(n + 2)(n + 3)(2n +8n + 3) j0 j1 4 3 2 2 2 2n(2n + 12n + 25n + 15n − 9) j0 j2 − 2n (n + 1)(2n +4n − 3) j1 j2

14 j0 7→ n ∆

j1 7→ (2n +3)(∆+6B ((2n + 3) j0 − n j1) 2 j2 7→ (n +3)∆ − 6(2n + 3) B

2 For the classification of quadratic involutions in the Cremona group Bir PC see [4] section 4. and 5.  Exercise for the reader: Find the two positive, optimal and 1-periodic quadrature rules given in [1] in sections 4.1 and 4.2 for odd n as fixed points of this recursion map. Show that among the 5 fixed points the rational fixed point with positive l0 component is a quadratic attractor. To avoid errors if you want to enter these formulae into your computer algebra system you can do copy and paste with appendix B of [8]. In the above referenced article also all the details (e.g. optimal 1-parameter rules and ”1/2-rules” ...) for optimal C0 and C1 quadrature rules can be found.

References

[1] R. Ait-Haddou, H. Ruhland, Asymptotically optimal quadrature rules for uniform splines over the real line. Numerical Algorithms 86, 1189-1223 (2021), https://doi.org/10.1007/s11075-020-00929-2. [2] J. Arves´u, F. Marcell´an, R. Alvarez-Nodarse,´ On a Modification of the Jacobi Linear Functional: Asymptotic Properties and Zeros of the Cor- responding Orthogonal Polynomials, Acta Applicandae Mathematicae 71: 127-158, 2002. [3] T.S. Chihara. Orthogonal polynomials and measures with end point masses, Rocky Mountain Journal of Mathematics, Volume 15, Number 3, Summer 1985, 705-719. [4] J. D´eserti, Some Properties of the Cremona Group. Preprint, arXiv:1108.4030v6 [math.AG] 15 Apr 2012, https://arxiv.org/abs/1108.4030. [5] A.J. Dur´an, M.D. de la Iglesia, Bispectral Jacobi type poly- nomials. Preprint, arXiv:2012.07618v1 [math.CA] 14 Dec 2020, https://arxiv.org/abs/2012.07618. [6] T.H. Koornwinder, Orthogonal polynomials with weight function (1 − x)α(1 + x)β + Mδ(x +1)+ Nδ(x − 1), Canad. Math. Bull. 27 (1984), 205-214. [7] H.L. Krall, On orthogonal polynomials satisfying a certain fourth order differential equation, The Pennsylvania State College Studies, No. 6, 1940, 1-24.

15 [8] H. Ruhland, Quadrature rules for C0 and C1 splines, a recipe. Preprint, arXiv:1908.06199v1 [math.NA] 16 Aug 2019, https://arxiv.org/abs/1908.06199. [9] Wikipedia, Gaussian Quadrature, section 3.1 Fundamental Theorem, https://en.wikipedia.org/wiki/Gaussian_quadrature#Fundamental_theorem (accessed 2021-03-22).

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