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## A Moment Matrix Approach to Multivariable Cubature

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360 |
Approximate Calculation of Multiple Integrals,
- Stroud
- 1971
(Show Context)
Citation Context ...t nodes possible in a cubature rule of prescribed degree, and the construction of rules with the fewest nodes possible (cf., [C1], [Mo1] – [Mo3], [My1] – [My3], [My5], [My6], [P2], [R], [S], [Str1] – =-=[Str4]-=-, [SX], [X1] – [X3]). In [R], Radon introduced the technique of constructing minimal cubature rules whose nodes are common zeros of multivariable orthogonal polynomials. In the 1960s and 1970s this ap... |

83 | Solution of the truncated complex moment problem with flat data,
- Curto, Fialkow
- 1996
(Show Context)
Citation Context ...he moment matrix M(bm2 c)[µ] that we associate to µ. This approach emerges naturally from a recent study of multivariable truncated moment problems by R. Curto and the first-named author (cf., [CF1], =-=[CF2]-=-, [CF3]); for terminology and notation concerning moment matrices, see below and Section 2. Suppose µ (as above) is square positive, i. e., if f ∈ Pdbm2 c and f 6= 0, then ∫ |f |2 dµ > 0. For this cas... |

76 | Recursiveness, positivity, and truncated moment problems,
- Curto, Fialkow
- 1991
(Show Context)
Citation Context ...es of the moment matrix M(bm2 c)[µ] that we associate to µ. This approach emerges naturally from a recent study of multivariable truncated moment problems by R. Curto and the first-named author (cf., =-=[CF1]-=-, [CF2], [CF3]); for terminology and notation concerning moment matrices, see below and Section 2. Suppose µ (as above) is square positive, i. e., if f ∈ Pdbm2 c and f 6= 0, then ∫ |f |2 dµ > 0. For t... |

59 | Monomial cubature rules since Stroud: a compilation – Part 2, - Cools - 1999 |

59 | The truncated complex k-moment problem
- Curto, Fialkow
(Show Context)
Citation Context ...ription of the 5-node (minimal) cubature rules of degree 4 for arclength measure on the parabolic arc y = x2, 0 ≤ x ≤ 1, where symmetry is not available; further, techniques from the K-moment problem =-=[CF6]-=- were used to characterize which of these rules are supported inside the arc. As noted above, Radon [R] pioneered the technique of constructing cubature rules supported on the common zeros of orthogon... |

58 |
Computing cubature formulas: the science behind the art,
- Cools
- 1997
(Show Context)
Citation Context ...Two recurrent themes in cubature literature are the estimation of the fewest nodes possible in a cubature rule of prescribed degree, and the construction of rules with the fewest nodes possible (cf., =-=[C1]-=-, [Mo1] – [Mo3], [My1] – [My3], [My5], [My6], [P2], [R], [S], [Str1] – [Str4], [SX], [X1] – [X3]). In [R], Radon introduced the technique of constructing minimal cubature rules whose nodes are common ... |

43 |
An Introduction to Numerical Methods and Analysis
- Epperson
- 2002
(Show Context)
Citation Context ...ith A = A∗ ∈ L(H1), C = C∗ ∈ L(H2). Theorem 2.5 concerns the case A = M(n)(β), Ã = M(n+ 1)(β̃), so we need a characterization of the case when Ã ≥ 0 and rank Ã = rankA. Proposition 2.6. (cf. [Smu] =-=[Epp]-=-) Suppose Ã is as in (2.6). Then Ã ≥ 0 if and only if A ≥ 0 and there exists W ∈ L(H1,H2) such that B = AW and C ≥ W ∗AW . In this case, W ∗AW is independent of W satisfying B = AW , and when H is f... |

41 | Truncated K-moment problems in several variables,
- Curto, Fialkow
- 2005
(Show Context)
Citation Context ...xistence of a sequence of rank increasing positive extensions M(n+ 1), . . . ,M(n+ k), followed by a flat extension M(n+ k+ 1); although a number of concrete existence theorems are known (cf. [CF2] – =-=[CF7]-=-, [F3]), much remains to be learned about moment matrix extensions. 2. Moment matrices Let Cdr [z, z̄] denote the space of polynomials with complex coefficients in the indeterminates z ≡ (z1, . . . , ... |

41 | The complex moment problem and subnormality: A polar decomposition approach, - Stochel, Szafraniec - 1998 |

31 |
Construction of algebraic cubature rules using polynomial ideal theory,
- Morrow, Patterson
- 1978
(Show Context)
Citation Context ...en and Cools obtained concrete criteria for the existence of rules attaining Möller’s bound and showed, in particular, that a degree 9 rule for µD requires at least 18 nodes [VC, page 404] (cf. [CH] =-=[MP]-=-). An 18-node rule had previously been obtained by Haegemans and Piessens [HP], who conjectured its minimality. These results show that the lower bound in Theorem 1.11 is not sharp for n even. We next... |

28 |
An operator Hellinger integral,
- Smul’jan
- 1959
(Show Context)
Citation Context ...OVIC with A = A∗ ∈ L(H1), C = C∗ ∈ L(H2). Theorem 2.5 concerns the case A = M(n)(β), Ã = M(n+ 1)(β̃), so we need a characterization of the case when Ã ≥ 0 and rank Ã = rankA. Proposition 2.6. (cf. =-=[Smu]-=- [Epp]) Suppose Ã is as in (2.6). Then Ã ≥ 0 if and only if A ≥ 0 and there exists W ∈ L(H1,H2) such that B = AW and C ≥ W ∗AW . In this case, W ∗AW is independent of W satisfying B = AW , and when ... |

27 |
Interpolational Cubature Formulas
- Mysovskikh
- 1981
(Show Context)
Citation Context ... are the estimation of the fewest nodes possible in a cubature rule of prescribed degree, and the construction of rules with the fewest nodes possible (cf., [C1], [Mo1] – [Mo3], [My1] – [My3], [My5], =-=[My6]-=-, [P2], [R], [S], [Str1] – [Str4], [SX], [X1] – [X3]). In [R], Radon introduced the technique of constructing minimal cubature rules whose nodes are common zeros of multivariable orthogonal polynomial... |

23 | Recursively generated weighted shifts and the subnormal completion problem, - Curto, Fialkow - 1994 |

21 |
Kubaturformeln mit minimaler
- Möller
(Show Context)
Citation Context ...er the above estimate for ρC can be extended to general centrally symmetric planar measures (so as to recover Theorem 1.3) is an open question. As we discuss in Section 5, other results of Möller in =-=[Mo2]-=- imply that Theorem 1.11 is not sharp when n is even, since the lower bound for N can be increased by at least 1 in this case. Whether, for n even, we can improve the estimate for ρC(C ] C(n + 1)[µD] ... |

21 |
Lower bounds for the number of nodes in cubature formulae, in: Numerische Integration (Tagung,
- Moller
- 1978
(Show Context)
Citation Context ...themes in cubature literature are the estimation of the fewest nodes possible in a cubature rule of prescribed degree, and the construction of rules with the fewest nodes possible (cf., [C1], [Mo1] – =-=[Mo3]-=-, [My1] – [My3], [My5], [My6], [P2], [R], [S], [Str1] – [Str4], [SX], [X1] – [X3]). In [R], Radon introduced the technique of constructing minimal cubature rules whose nodes are common zeros of multiv... |

21 |
On bivariate gaussian cubature formulae
- Schmid, Xu
- 1994
(Show Context)
Citation Context ...possible in a cubature rule of prescribed degree, and the construction of rules with the fewest nodes possible (cf., [C1], [Mo1] – [Mo3], [My1] – [My3], [My5], [My6], [P2], [R], [S], [Str1] – [Str4], =-=[SX]-=-, [X1] – [X3]). In [R], Radon introduced the technique of constructing minimal cubature rules whose nodes are common zeros of multivariable orthogonal polynomials. In the 1960s and 1970s this approach... |

21 | Formules de cubatures mécaniques à coefficients non négatifs - Tchakaloff - 1957 |

20 |
On Tchakaloff’s Theorem,
- Putinar
- 1995
(Show Context)
Citation Context ...[µ] = {βi}|i|≤m: βi = ∫ ti dν, |i| ≤ m, ν ≥ 0, supp ν ⊂ Rd.(1.2) Following a line of results beginning with Tchakaloff’s Theorem [T], and including generalizations due to Mysovskikh [My1] and Putinar =-=[P1]-=-, in [CF5, Theorem 1.4] it was proved that if µ has convergent moments up to at least order m+ 1, then µ admits an inside cubature rule of degree m, with size ≤ 1 + dim(Pdm|suppµ). (An inside rule is ... |

19 | Flat extensions of positive moment matrices: Relations in analytic or conjugate terms, Operator Th.: Adv - Curto, Fialkow |

19 | Solution of the singular quartic moment problem
- Curto, Fialkow
(Show Context)
Citation Context ...[C2]). By contrast, the measures studied by Schmid and Xu [SX] (op. cit.), have Gaussian rules of all degrees and are supported on a region of the plane with nonempty interior. Recently, we showed in =-=[CF4]-=- and [CF8] that if µ (as in Theorem 1.5) is supported in a parabola or ellipse in the plane, then µ always admits a Gaussian rule of degree 2n with size N = rankM(n)[µ]. In the sequel, we refer to a r... |

17 |
extensions and the truncated complex moment problem
- Fialkow, Positivity
(Show Context)
Citation Context ...3 = −(3/10)Z̄+(6/5)ZZ̄2+(1/32)w̄Z5, ZZ̄4 = −(4/5)w̄Z3+(4/5)Z̄3+w̄Z4Z̄, Z̄5 = (48/5)w̄Z − (192/5)w̄Z2Z̄ + 32w̄Z3Z̄2. Since M(7) is a positive extension of M(5), the same relations hold in ColM(7) (cf. =-=[F1]-=-); thus, suppµw is contained in the common zeros of the polynomials corresponding to these relations. For w = 1, we find that there are precisely 19 distinct common zeros, {zk}18k=0, which must theref... |

17 |
Common zeros of polynomials in several variables and higherdimensional quadrature
- Xu
- 1994
(Show Context)
Citation Context ...le in a cubature rule of prescribed degree, and the construction of rules with the fewest nodes possible (cf., [C1], [Mo1] – [Mo3], [My1] – [My3], [My5], [My6], [P2], [R], [S], [Str1] – [Str4], [SX], =-=[X1]-=- – [X3]). In [R], Radon introduced the technique of constructing minimal cubature rules whose nodes are common zeros of multivariable orthogonal polynomials. In the 1960s and 1970s this approach was r... |

16 |
Hankel and Toeplitz Matrices and Forms: Algebraic Theory, Birkhauser-Verlag,
- Iohvidov
- 1982
(Show Context)
Citation Context ...erving Hankel extensions of Hankel matrices are referred to as singular extensions . In the case of the complex plane C that we consider below, a moment matrix block C(n+ 1) is a Toeplitz matrix, and =-=[I]-=- contains a theory for rank-preserving Toeplitz extensions of Toeplitz matrices, and a formula for the rank of an arbitrary Toeplitz matrix [I, Theorem 15.1]. We prove Theorem 1.5 in Section 3 (Theore... |

15 |
On orthogonal polynomials in several variables, in: Special functions, q-series, and related topics. Editors
- Xu
- 1997
(Show Context)
Citation Context ...ow); subsequently, the theory of lower estimates and minimal rules developed in several directions, e. g., [Mo3], [CS], [X1] – [X3], [S]; many of these developments are discussed in the surveys of Xu =-=[X2]-=-, Cools [C1], and Cools et. al. [CMS]. Our moment matrix approach is based on the observation that for a positive Borel measure µ on Rd with convergent moments βi = ∫ ti dµ, |i| ≤ m, the existence of ... |

13 | Solution of the truncated hyperbolic moment problem - Curto, Fialkow |

11 | Cubature formulae and orthogonal polynomials
- Cools, Mysovskikh, et al.
- 2001
(Show Context)
Citation Context ...er estimates and minimal rules developed in several directions, e. g., [Mo3], [CS], [X1] – [X3], [S]; many of these developments are discussed in the surveys of Xu [X2], Cools [C1], and Cools et. al. =-=[CMS]-=-. Our moment matrix approach is based on the observation that for a positive Borel measure µ on Rd with convergent moments βi = ∫ ti dµ, |i| ≤ m, the existence of a cubature rule for µ of degree m is ... |

9 |
A survey of known and new cubature formulas for the unit disk
- Cools, Kim
- 2000
(Show Context)
Citation Context ...tric, one effective strategy for constructing a cubature rule is to design a highly symmetric (if sometimes non-minimal) distribution of the nodes, reflecting the symmetry in suppµ (cf. [Str4], [HP], =-=[CK]-=-, [C1]). By contrast, our approach does not take advantage of symmetry, and is applied in the same manner whether or not suppµ displays symmetry; in [CF4, Example 4.12] moment matrices were used to gi... |

9 |
Truncated complex moment problems with a ZZ̄ relation
- Fialkow
(Show Context)
Citation Context ...e of a sequence of rank increasing positive extensions M(n+ 1), . . . ,M(n+ k), followed by a flat extension M(n+ k+ 1); although a number of concrete existence theorems are known (cf. [CF2] – [CF7], =-=[F3]-=-), much remains to be learned about moment matrix extensions. 2. Moment matrices Let Cdr [z, z̄] denote the space of polynomials with complex coefficients in the indeterminates z ≡ (z1, . . . , zd) an... |

9 | Szegö: Problems and theorems in analysis, Vol. II, Theory of functions – zeros – polynomials – determinants – number theory – geometry, Rev. and enl. translation of the 4th ed, Springer Study Edition - Pólya, G - 1976 |

9 |
Zur mechanischen kubatur
- Radon
- 1948
(Show Context)
Citation Context ...mation of the fewest nodes possible in a cubature rule of prescribed degree, and the construction of rules with the fewest nodes possible (cf., [C1], [Mo1] – [Mo3], [My1] – [My3], [My5], [My6], [P2], =-=[R]-=-, [S], [Str1] – [Str4], [SX], [X1] – [X3]). In [R], Radon introduced the technique of constructing minimal cubature rules whose nodes are common zeros of multivariable orthogonal polynomials. In the 1... |

9 |
On cubature formulae of degree 4k + 1 attaining Möller’s lower bound for integrals with circular symmetry
- Verlinden, Cools
- 1992
(Show Context)
Citation Context ...roblem. Indeed, there is an extensive literature concerning cases where Möller’s lower bounds can be achieved or cases where the estimates cannot be realized (cf. [Mo1] [Mo3] [CH] [CS] [MP] [S] [SX] =-=[VC]-=- [X3]); by contrast, we have concrete estimates for ρ(C]) in only relatively few cases (discussed below), so at this point it is difficult to ascertain when the lower bound of Theorem 1.7 is attainabl... |

7 | A duality proof of Tchakaloff’s theorem - Curto, Fialkow |

6 |
Minimal cubature formulae of degree 2k − 1 for two classical functionals,
- Cools, Schmid
- 1989
(Show Context)
Citation Context ...racterized the cubature rules that attain the lower bounds of [Mo2] (cf. Section 5 below); subsequently, the theory of lower estimates and minimal rules developed in several directions, e. g., [Mo3], =-=[CS]-=-, [X1] – [X3], [S]; many of these developments are discussed in the surveys of Xu [X2], Cools [C1], and Cools et. al. [CMS]. Our moment matrix approach is based on the observation that for a positive ... |

6 | Solution of the truncated parabolic moment problem, preprint 2002
- Curto, Fialkow
(Show Context)
Citation Context ...contrast, the measures studied by Schmid and Xu [SX] (op. cit.), have Gaussian rules of all degrees and are supported on a region of the plane with nonempty interior. Recently, we showed in [CF4] and =-=[CF8]-=- that if µ (as in Theorem 1.5) is supported in a parabola or ellipse in the plane, then µ always admits a Gaussian rule of degree 2n with size N = rankM(n)[µ]. In the sequel, we refer to a rank-preser... |

6 |
Numerical characteristics of orthogonal polynomials in two variables
- Mysovskikh
- 1970
(Show Context)
Citation Context ...erature are the estimation of the fewest nodes possible in a cubature rule of prescribed degree, and the construction of rules with the fewest nodes possible (cf., [C1], [Mo1] – [Mo3], [My1] – [My3], =-=[My5]-=-, [My6], [P2], [R], [S], [Str1] – [Str4], [SX], [X1] – [X3]). In [R], Radon introduced the technique of constructing minimal cubature rules whose nodes are common zeros of multivariable orthogonal pol... |

4 |
Multivariable quadrature and extensions of moment matrices, unpublished manuscript
- Fialkow
- 1996
(Show Context)
Citation Context ... rule of degree 3 if and only if there exists β 6= b, such that cardZ(pβ) ≥ 4 (this occurs, in particular, if |D| 6= 1 [F2, Proposition 1.6]); for each such β, a minimal rule can be constructed as in =-=[F2]-=-. (A similar result, involving a pair of real orthogonal polynomials instead of pβ , was obtained by Goit [G] (cf. [Str4, page 99]).) 2 Concerning the last case in Example 5.1, it is an open question ... |

4 | Numerical evaluation of multiple integrals, - Hammer, Stroud - 1958 |

4 |
A proof of minimality of the number of nodes of a cubature formula for a hypersphere
- Mysovskikh
- 1966
(Show Context)
Citation Context ...in cubature literature are the estimation of the fewest nodes possible in a cubature rule of prescribed degree, and the construction of rules with the fewest nodes possible (cf., [C1], [Mo1] – [Mo3], =-=[My1]-=- – [My3], [My5], [My6], [P2], [R], [S], [Str1] – [Str4], [SX], [X1] – [X3]). In [R], Radon introduced the technique of constructing minimal cubature rules whose nodes are common zeros of multivariable... |

4 |
A dilation theory approach to cubature formulas.
- Putinar
- 2000
(Show Context)
Citation Context ...e estimation of the fewest nodes possible in a cubature rule of prescribed degree, and the construction of rules with the fewest nodes possible (cf., [C1], [Mo1] – [Mo3], [My1] – [My3], [My5], [My6], =-=[P2]-=-, [R], [S], [Str1] – [Str4], [SX], [X1] – [X3]). In [R], Radon introduced the technique of constructing minimal cubature rules whose nodes are common zeros of multivariable orthogonal polynomials. In ... |

4 | Two-dimensional minimal cubature formulas and matrix equations - Schmid - 1995 |

4 |
Quadrature methods for functions of more than one variable
- Stroud
- 1960
(Show Context)
Citation Context ...the fewest nodes possible in a cubature rule of prescribed degree, and the construction of rules with the fewest nodes possible (cf., [C1], [Mo1] – [Mo3], [My1] – [My3], [My5], [My6], [P2], [R], [S], =-=[Str1]-=- – [Str4], [SX], [X1] – [X3]). In [R], Radon introduced the technique of constructing minimal cubature rules whose nodes are common zeros of multivariable orthogonal polynomials. In the 1960s and 1970... |

4 |
Numerical integration formulas of degree two,
- Stroud
- 1960
(Show Context)
Citation Context ...bature rules. In [R] Radon introduced the method of constructing multivariable cubature rules supported on the common zeros of orthogonal polynomials. Using an approach based on matrix theory, Stroud =-=[Str2]-=-, [Str4, Section 3.9, p. 88] constructed a family of 2d-node cubature rules of degree 3 in Rd for a class including centrally symmetric measures; Mysovskih [My1] subsequently showed that these rules a... |

3 | Can a minimal degree 6 cubature rule for the disk have all points inside?, manuscript in preparation
- Easwaran, Fialkow, et al.
(Show Context)
Citation Context ...ures (cf., [Str4, Section 3.12]). In Theorem 5.5 we use Proposition 5.4 to completely parametrize the (minimal) 7-node rules of degree 5 for µD. In a companion paper by C. V. Easwaran and the authors =-=[EFP]-=- we use moment matrix methods to resolve an open problem of [C2] by showing that among the 10-node (minimal) cubature rules of degree 6 for µD, there is no inside cubature rule (although there are man... |

3 |
Cubature formulae and polynomial ideals. Adv
- Xu
- 1999
(Show Context)
Citation Context ... cubature rule of prescribed degree, and the construction of rules with the fewest nodes possible (cf., [C1], [Mo1] – [Mo3], [My1] – [My3], [My5], [My6], [P2], [R], [S], [Str1] – [Str4], [SX], [X1] – =-=[X3]-=-). In [R], Radon introduced the technique of constructing minimal cubature rules whose nodes are common zeros of multivariable orthogonal polynomials. In the 1960s and 1970s this approach was refined ... |

2 |
Construction of symmetric cubature formulae with the number of knots (almost) equal to Möller’s lower bound. Numerical integration, III(Oberwolfach
- Cools, Haegemans
- 1987
(Show Context)
Citation Context ...nd in [Mo2] is an open problem. Indeed, there is an extensive literature concerning cases where Möller’s lower bounds can be achieved or cases where the estimates cannot be realized (cf. [Mo1] [Mo3] =-=[CH]-=- [CS] [MP] [S] [SX] [VC] [X3]); by contrast, we have concrete estimates for ρ(C]) in only relatively few cases (discussed below), so at this point it is difficult to ascertain when the lower bound of ... |

2 |
Cubature formulas of degree nine for symmetric planar regions
- Haegemans, Piessens
- 1975
(Show Context)
Citation Context ... symmetric, one effective strategy for constructing a cubature rule is to design a highly symmetric (if sometimes non-minimal) distribution of the nodes, reflecting the symmetry in suppµ (cf. [Str4], =-=[HP]-=-, [CK], [C1]). By contrast, our approach does not take advantage of symmetry, and is applied in the same manner whether or not suppµ displays symmetry; in [CF4, Example 4.12] moment matrices were used... |

2 | Integration formulas and orthogonal polynomials - Stroud - 1970 |

1 |
Formeln zur numerischen Integration ber Kreisbereiche
- Albrecht
- 1960
(Show Context)
Citation Context ...ules for µD. Proposition 5.10 gives a new proof that there is no degree 9 rule for µD̄ with as few as 17 points. The first example of a degree 9 rule for µD with as few as 19 nodes is due to Albrecht =-=[A]-=-. In Proposition 5.12 we show how Albrecht’s rule (and a related infinite family of 19-node rules) can be derived by a 2-step moment matrix extension M(5) → M(6) → M(7), where rankM(5) = 18 and rankM(... |

1 |
Third degree integration formulas with four points in two dimensions
- Goit
- 1968
(Show Context)
Citation Context ...|D| 6= 1 [F2, Proposition 1.6]); for each such β, a minimal rule can be constructed as in [F2]. (A similar result, involving a pair of real orthogonal polynomials instead of pβ , was obtained by Goit =-=[G]-=- (cf. [Str4, page 99]).) 2 Concerning the last case in Example 5.1, it is an open question whether there always exists some β 6= b for which cardZ(pβ) ≥ 4. In the case when µ is centrally symmetric, w... |

1 |
Polynomideale und Kubaturforeln
- Möller
- 1973
(Show Context)
Citation Context ...current themes in cubature literature are the estimation of the fewest nodes possible in a cubature rule of prescribed degree, and the construction of rules with the fewest nodes possible (cf., [C1], =-=[Mo1]-=- – [Mo3], [My1] – [My3], [My5], [My6], [P2], [R], [S], [Str1] – [Str4], [SX], [X1] – [X3]). In [R], Radon introduced the technique of constructing minimal cubature rules whose nodes are common zeros o... |

1 | Concerning Radon’s work on cubature formulas - Mysovskikh - 1967 |

1 |
On the construction of cubature formulas with the smallest number of nodes
- Mysovskikh
- 1968
(Show Context)
Citation Context ...ure literature are the estimation of the fewest nodes possible in a cubature rule of prescribed degree, and the construction of rules with the fewest nodes possible (cf., [C1], [Mo1] – [Mo3], [My1] – =-=[My3]-=-, [My5], [My6], [P2], [R], [S], [Str1] – [Str4], [SX], [X1] – [X3]). In [R], Radon introduced the technique of constructing minimal cubature rules whose nodes are common zeros of multivariable orthogo... |

1 |
Numerical Integration over Planar Regions
- Pierce
- 1956
(Show Context)
Citation Context ...e 6 for µD, there is no inside cubature rule (although there are many minimal rules with 9 points inside). A 12-node (inside, minimal) rule for µD of degree 7 is cited in [Str4, S2:7–1, pg. 281] (cf. =-=[P]-=-). In Proposition 5.8 we develop a new family of 12-node degree 7 rules for µD. Proposition 5.10 gives a new proof that there is no degree 9 rule for µD̄ with as few as 17 points. The first example of... |

1 | Formules de cubatures mcaniques coefficients non ngatifs - Tchakaloff - 1957 |

1 | Formeln zur numerischen Integration über Kreisbereiche - Albrecht - 1960 |