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Divided differences Tomas Sauer Lehrstuhl fr Mathematik, Schwerpunkt Digitale Bildverarbeitung FORWISS Universitt Passau MAIA 2013, Erice, September 26, 2013 In part joint work with J. Carnicer (Zaragoza) Tomas Sauer (Universitt Passau)

  1. Different Points of View Point set constructions Given a space P ⊂ Π find sites Ξ such that there always exists a unique p ∈ F with p ( Ξ ) = f ( Ξ ) . Answers: Chung–Yao, GPL, . . . Point set constructions Find smallest subspace P ⊂ Π such that for any Ξ with # Ξ = N there exists p ∈ P with p ( Ξ ) = f ( Ξ ) . � n + 2 � Answers: Π N − 1 , Π 2 n − 1 , = N , for s = 2. 2 Space constructions Given Ξ find P such that there always exists p ∈ P with p ( Ξ ) = f ( Ξ ) . Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 4 / 24

  2. Different Points of View Point set constructions Given a space P ⊂ Π find sites Ξ such that there always exists a unique p ∈ F with p ( Ξ ) = f ( Ξ ) . Answers: Chung–Yao, GPL, . . . Point set constructions Find smallest subspace P ⊂ Π such that for any Ξ with # Ξ = N there exists p ∈ P with p ( Ξ ) = f ( Ξ ) . � n + 2 � Answers: Π N − 1 , Π 2 n − 1 , = N , for s = 2. 2 Space constructions Given Ξ find P such that there always exists p ∈ P with p ( Ξ ) = f ( Ξ ) . Answers: Buchberger–Möller, deBoor–Ron, ideal remainders Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 4 / 24

  3. Notation Polynomials Π polynomials = finite sums 1 � p α x α . p ( x ) = | α | ≤ n deg p := n . 2 Monomials or terms of degree k , . . . , n : row vectors 3 x k : n := (( · ) α : k ≤ | α | ≤ n ) , x n = x n : n Coefficients as column vectors p k = ( p α : | α | = k ) . 4 n � x k p k . Polynomials: p ( x ) = 5 k = 0 Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 5 / 24

  4. Notation Polynomials Π polynomials = finite sums 1 � p α x α . p ( x ) = | α | ≤ n deg p := n . 2 Monomials or terms of degree k , . . . , n : row vectors 3 x k : n := (( · ) α : k ≤ | α | ≤ n ) , x n = x n : n Coefficients as column vectors p k = ( p α : | α | = k ) . 4 n � x k p k . Polynomials: p ( x ) = 5 k = 0 Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 5 / 24

  5. Notation Polynomials Π polynomials = finite sums 1 � p α x α . p ( x ) = | α | ≤ n Degree deg p := n . 2 Monomials or terms of degree k , . . . , n : row vectors 3 x k : n := (( · ) α : k ≤ | α | ≤ n ) , x n = x n : n Coefficients as column vectors p k = ( p α : | α | = k ) . 4 n � x k p k . Polynomials: p ( x ) = 5 k = 0 Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 5 / 24

  6. Notation Polynomials Π polynomials = finite sums 1 � p α x α . p ( x ) = | α | ≤ n Total degree deg p := n . 2 Monomials or terms of degree k , . . . , n : row vectors 3 x k : n := (( · ) α : k ≤ | α | ≤ n ) , x n = x n : n Coefficients as column vectors p k = ( p α : | α | = k ) . 4 n � x k p k . Polynomials: p ( x ) = 5 k = 0 Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 5 / 24

  7. Notation Polynomials Π polynomials = finite sums 1 � p α x α . p ( x ) = | α | ≤ n Total degree deg p := n . 2 Monomials or terms of degree k , . . . , n : row vectors 3 x k : n := (( · ) α : k ≤ | α | ≤ n ) , x n = x n : n Coefficients as column vectors p k = ( p α : | α | = k ) . 4 n � x k p k . Polynomials: p ( x ) = 5 k = 0 Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 5 / 24

  8. Notation Polynomials Π polynomials = finite sums 1 � p α x α . p ( x ) = | α | ≤ n Total degree deg p := n . 2 Monomials or terms of degree k , . . . , n : row vectors 3 x k : n := (( · ) α : k ≤ | α | ≤ n ) , x n = x n : n Coefficients as column vectors p k = ( p α : | α | = k ) . 4 n � x k p k . Polynomials: p ( x ) = 5 k = 0 Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 5 / 24

  9. Notation Polynomials Π polynomials = finite sums 1 � p α x α . p ( x ) = | α | ≤ n Total degree deg p := n . 2 Monomials or terms of degree k , . . . , n : row vectors 3 x k : n := (( · ) α : k ≤ | α | ≤ n ) , x n = x n : n Coefficients as column vectors p k = ( p α : | α | = k ) . 4 n � x k p k . Polynomials: p ( x ) = 5 k = 0 Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 5 / 24

  10. Interpolation Correctness A subspace P ⊂ Π is called correct for Ξ ⊂ R s if for any f : Ξ → R there exists unique L Ξ f ∈ P such that L Ξ f ( Ξ ) = f ( Ξ ) . Correctness for Π n Vandermonde matrix � � ξ ∈ Ξ x 0 : n ( Ξ ) = ξ α : | α | ≤ n Then: L Ξ f = x 0 : n p , Fundamental “Theorem” Correctness = nonsingularity of Vandermonde matrix. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 6 / 24

  11. Interpolation Correctness A subspace P ⊂ Π is called correct for Ξ ⊂ R s if for any f : Ξ → R there exists unique L Ξ f ∈ P such that L Ξ f ( Ξ ) = f ( Ξ ) . Correctness for Π n Vandermonde matrix � � ξ ∈ Ξ x 0 : n ( Ξ ) = ξ α : | α | ≤ n Then: L Ξ f = x 0 : n p , x 0 : n ( Ξ ) p = f ( Ξ ) . Fundamental “Theorem” Correctness = nonsingularity of Vandermonde matrix. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 6 / 24

  12. Interpolation Correctness A subspace P ⊂ Π is called correct for Ξ ⊂ R s if for any f : Ξ → R there exists unique L Ξ f ∈ P such that L Ξ f ( Ξ ) = f ( Ξ ) . Correctness for Π n Vandermonde matrix � � ξ ∈ Ξ x 0 : n ( Ξ ) = ξ α : | α | ≤ n Then: p = x 0 : n ( Ξ ) − 1 f ( Ξ ) . L Ξ f = x 0 : n p , Fundamental “Theorem” Correctness = nonsingularity of Vandermonde matrix. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 6 / 24

  13. Interpolation Correctness A subspace P ⊂ Π is called correct for Ξ ⊂ R s if for any f : Ξ → R there exists unique L Ξ f ∈ P such that L Ξ f ( Ξ ) = f ( Ξ ) . Correctness for Π n Vandermonde matrix � � ξ ∈ Ξ x 0 : n ( Ξ ) = ξ α : | α | ≤ n Then: p = x 0 : n ( Ξ ) − 1 f ( Ξ ) . L Ξ f = x 0 : n p , Fundamental “Theorem” Correctness = nonsingularity of Vandermonde matrix. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 6 / 24

  14. Interpolation Spaces Degree of a subspace deg P = max { deg p : p ∈ P } . Degree reducing interpolation f ∈ Π deg L Ξ f ≤ deg f . ⇒ Minimal degree interpolation space Q interpolation space deg Q ≥ deg P . ⇒ Facts Degree reducing is always minimal degree. 1 None of them is unique for general Ξ . 2 Different elimination strategies for x 0 : n ( Ξ ) . 3 Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 7 / 24

  15. Interpolation Spaces Degree of a subspace deg P = max { deg p : p ∈ P } . Degree reducing interpolation f ∈ Π deg L Ξ f ≤ deg f . ⇒ Minimal degree interpolation space Q interpolation space deg Q ≥ deg P . ⇒ Facts Degree reducing is always minimal degree. 1 None of them is unique for general Ξ . 2 Different elimination strategies for x 0 : n ( Ξ ) . 3 Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 7 / 24

  16. Interpolation Spaces Degree of a subspace deg P = max { deg p : p ∈ P } . Degree reducing interpolation f ∈ Π deg L Ξ f ≤ deg f . ⇒ Minimal degree interpolation space Q interpolation space deg Q ≥ deg P . ⇒ Facts Degree reducing is always minimal degree. 1 None of them is unique for general Ξ . 2 Different elimination strategies for x 0 : n ( Ξ ) . 3 Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 7 / 24

  17. Interpolation Spaces Degree of a subspace deg P = max { deg p : p ∈ P } . Degree reducing interpolation f ∈ Π deg L Ξ f ≤ deg f . ⇒ Minimal degree interpolation space Q interpolation space deg Q ≥ deg P . ⇒ Facts Degree reducing is always minimal degree. 1 None of them is unique for general Ξ . 2 Different elimination strategies for x 0 : n ( Ξ ) . 3 Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 7 / 24

  18. Interpolation Spaces Degree of a subspace deg P = max { deg p : p ∈ P } . Degree reducing interpolation f ∈ Π deg L Ξ f ≤ deg f . ⇒ Minimal degree interpolation space Q interpolation space deg Q ≥ deg P . ⇒ Facts Degree reducing is always minimal degree. 1 None of them is unique for general Ξ . 2 Different elimination strategies for x 0 : n ( Ξ ) . 3 Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 7 / 24

  19. Interpolation Spaces Degree of a subspace deg P = max { deg p : p ∈ P } . Degree reducing interpolation f ∈ Π deg L Ξ f ≤ deg f . ⇒ Minimal degree interpolation space Q interpolation space deg Q ≥ deg P . ⇒ Facts Degree reducing is always minimal degree. 1 None of them is unique for general Ξ . 2 Different elimination strategies for x 0 : n ( Ξ ) . 3 Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 7 / 24

  20. Interpolation Spaces Degree of a subspace deg P = max { deg p : p ∈ P } . Degree reducing interpolation f ∈ Π deg L Ξ f ≤ deg f . ⇒ Minimal degree interpolation space Q interpolation space deg Q ≥ deg P . ⇒ Facts Degree reducing is always minimal degree. 1 None of them is unique for general Ξ . 2 Different elimination strategies for x 0 : n ( Ξ ) . 3 Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 7 / 24

  21. Interpolation Spaces Degree of a subspace deg P = max { deg p : p ∈ P } . Degree reducing interpolation f ∈ Π deg L Ξ f ≤ deg f . ⇒ Minimal degree interpolation space Q interpolation space deg Q ≥ deg P . ⇒ Facts Degree reducing is always minimal degree. 1 None of them is unique for general Ξ . 2 Different elimination strategies for x 0 : n ( Ξ ) . 3 Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 7 / 24

  22. The Two Faces of Polynomials The linear part Computation of interpolant: solve Vandermonde system . 1 But . . . Polynomials can be multiplied . 1 Polynomial algebra . . . 2 T. Pratchett, Jingo There’s al–gebra. That’s like sums with letters. For . . . for people whose brains aren’t clever enough for numbers, see? Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 8 / 24

  23. The Two Faces of Polynomials The linear part Computation of interpolant: solve Vandermonde system . 1 But . . . Polynomials can be multiplied . 1 Polynomial algebra . . . 2 T. Pratchett, Jingo There’s al–gebra. That’s like sums with letters. For . . . for people whose brains aren’t clever enough for numbers, see? Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 8 / 24

  24. The Two Faces of Polynomials The linear part Computation of interpolant: solve Vandermonde system . 1 But . . . Polynomials can be multiplied . 1 Polynomial algebra . . . 2 T. Pratchett, Jingo There’s al–gebra. That’s like sums with letters. For . . . for people whose brains aren’t clever enough for numbers, see? Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 8 / 24

  25. The Two Faces of Polynomials The linear part Computation of interpolant: solve Vandermonde system . 1 But . . . Polynomials can be multiplied . 1 Polynomial algebra . . . 2 T. Pratchett, Jingo There’s al–gebra. That’s like sums with letters. For . . . for people whose brains aren’t clever enough for numbers, see? Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 8 / 24

  26. The Two Faces of Polynomials The linear part Computation of interpolant: solve Vandermonde system . 1 But . . . Polynomials can be multiplied . 1 Polynomial algebra . . . 2 T. Pratchett, Jingo There’s al–gebra. That’s like sums with letters. For . . . for people whose brains aren’t clever enough for numbers, see? Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 8 / 24

  27. The Two Faces of Polynomials The linear part Computation of interpolant: solve Vandermonde system . 1 But . . . Polynomials can be multiplied . 1 Polynomial algebra . . . 2 T. Pratchett, Jingo There’s al–gebra. That’s like sums with letters. For . . . for people whose brains aren’t clever enough for numbers, see? Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 8 / 24

  28. Ideals Ideals Ideal I : 1 I ( Ξ ) := { f ∈ Π : f ( Ξ ) = 0 } . 2 Ideal generated by F ⊂ Π : 3 F basis of I = � F � . 4 Hilbert’s Basissatz Any polynomial ideal has a finite basis. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 9 / 24

  29. Ideals Ideals Ideal I : 1 I ( Ξ ) := { f ∈ Π : f ( Ξ ) = 0 } . 2 Ideal generated by F ⊂ Π : 3 F basis of I = � F � . 4 Hilbert’s Basissatz Any polynomial ideal has a finite basis. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 9 / 24

  30. Ideals Ideals Ideal I : I + I = I 1 I ( Ξ ) := { f ∈ Π : f ( Ξ ) = 0 } . 2 Ideal generated by F ⊂ Π : 3 F basis of I = � F � . 4 Hilbert’s Basissatz Any polynomial ideal has a finite basis. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 9 / 24

  31. Ideals Ideals Ideal I : I + I = I and I · Π = I . 1 I ( Ξ ) := { f ∈ Π : f ( Ξ ) = 0 } . 2 Ideal generated by F ⊂ Π : 3 F basis of I = � F � . 4 Hilbert’s Basissatz Any polynomial ideal has a finite basis. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 9 / 24

  32. Ideals Ideals Ideal I : I + I = I and I · Π = I . 1 I ( Ξ ) := { f ∈ Π : f ( Ξ ) = 0 } . 2 Ideal generated by F ⊂ Π : 3 F basis of I = � F � . 4 Hilbert’s Basissatz Any polynomial ideal has a finite basis. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 9 / 24

  33. Ideals Ideals Ideal I : I + I = I and I · Π = I . 1 I ( Ξ ) := { f ∈ Π : f ( Ξ ) = 0 } . 2 Ideal generated by F ⊂ Π : 3 F basis of I = � F � . 4 Hilbert’s Basissatz Any polynomial ideal has a finite basis. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 9 / 24

  34. Ideals Ideals Ideal I : I + I = I and I · Π = I . 1 I ( Ξ ) := { f ∈ Π : f ( Ξ ) = 0 } . 2 Ideal generated by F ⊂ Π : 3 f F basis of I = � F � . 4 Hilbert’s Basissatz Any polynomial ideal has a finite basis. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 9 / 24

  35. Ideals Ideals Ideal I : I + I = I and I · Π = I . 1 I ( Ξ ) := { f ∈ Π : f ( Ξ ) = 0 } . 2 Ideal generated by F ⊂ Π : 3 f g f : g f ∈ Π F basis of I = � F � . 4 Hilbert’s Basissatz Any polynomial ideal has a finite basis. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 9 / 24

  36. Ideals Ideals Ideal I : I + I = I and I · Π = I . 1 I ( Ξ ) := { f ∈ Π : f ( Ξ ) = 0 } . 2 Ideal generated by F ⊂ Π : 3 � f g f : g f ∈ Π f ∈ F F basis of I = � F � . 4 Hilbert’s Basissatz Any polynomial ideal has a finite basis. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 9 / 24

  37. Ideals Ideals Ideal I : I + I = I and I · Π = I . 1 I ( Ξ ) := { f ∈ Π : f ( Ξ ) = 0 } . 2 Ideal generated by F ⊂ Π : 3     � � F � =: f g f : g f ∈ Π  .  f ∈ F F basis of I = � F � . 4 Hilbert’s Basissatz Any polynomial ideal has a finite basis. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 9 / 24

  38. Ideals Ideals Ideal I : I + I = I and I · Π = I . 1 I ( Ξ ) := { f ∈ Π : f ( Ξ ) = 0 } . 2 Ideal generated by F ⊂ Π : 3     � � F � =: f g f : g f ∈ Π  .  f ∈ F F basis of I = � F � . 4 Hilbert’s Basissatz Any polynomial ideal has a finite basis. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 9 / 24

  39. Ideals Ideals Ideal I : I + I = I and I · Π = I . 1 I ( Ξ ) := { f ∈ Π : f ( Ξ ) = 0 } . 2 Ideal generated by F ⊂ Π : 3     � � F � =: f g f : g f ∈ Π  .  f ∈ F F basis of I = � F � . 4 Hilbert’s Basissatz Any polynomial ideal has a finite basis. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 9 / 24

  40. From Interpolation to Bases Construction of basis P degree reducing interpolation space for Ξ . 1 P basis for P . 2 � � Set P ∗ = ( · ) j p : p ∈ P , j = 1, . . . , s . 3 Set 4 Then: I ( Ξ ) = � F Ξ � . 5 Even better Basis is H–basis : Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 10 / 24

  41. From Interpolation to Bases Construction of basis P degree reducing interpolation space for Ξ . 1 P basis for P . 2 � � Set P ∗ = ( · ) j p : p ∈ P , j = 1, . . . , s . 3 Set 4 Then: I ( Ξ ) = � F Ξ � . 5 Even better Basis is H–basis : Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 10 / 24

  42. From Interpolation to Bases Construction of basis P degree reducing interpolation space for Ξ . 1 P basis for P . 2 � � Set P ∗ = ( · ) j p : p ∈ P , j = 1, . . . , s . 3 Set 4 Then: I ( Ξ ) = � F Ξ � . 5 Even better Basis is H–basis : Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 10 / 24

  43. From Interpolation to Bases Construction of basis P degree reducing interpolation space for Ξ . 1 P basis for P . 2 � � Set P ∗ = ( · ) j p : p ∈ P , j = 1, . . . , s . 3 Set 4 Then: I ( Ξ ) = � F Ξ � . 5 Even better Basis is H–basis : Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 10 / 24

  44. From Interpolation to Bases Construction of basis P degree reducing interpolation space for Ξ . 1 P basis for P . 2 � � Set P ∗ = ( · ) j p : p ∈ P , j = 1, . . . , s . (Pre-)border basis. 3 Set 4 Then: I ( Ξ ) = � F Ξ � . 5 Even better Basis is H–basis : Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 10 / 24

  45. From Interpolation to Bases Construction of basis P degree reducing interpolation space for Ξ . 1 P basis for P . 2 � � Set P ∗ = ( · ) j p : p ∈ P , j = 1, . . . , s . (Pre-)border basis. 3 Set 4 Then: I ( Ξ ) = � F Ξ � . 5 Even better Basis is H–basis : Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 10 / 24

  46. From Interpolation to Bases Construction of basis P degree reducing interpolation space for Ξ . 1 P basis for P . 2 � � Set P ∗ = ( · ) j p : p ∈ P , j = 1, . . . , s . (Pre-)border basis. 3 Set 4 p ∈ P ∗ Then: I ( Ξ ) = � F Ξ � . 5 Even better Basis is H–basis : Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 10 / 24

  47. From Interpolation to Bases Construction of basis P degree reducing interpolation space for Ξ . 1 P basis for P . 2 � � Set P ∗ = ( · ) j p : p ∈ P , j = 1, . . . , s . (Pre-)border basis. 3 Set 4 p − L Ξ p : p ∈ P ∗ Then: I ( Ξ ) = � F Ξ � . 5 Even better Basis is H–basis : Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 10 / 24

  48. From Interpolation to Bases Construction of basis P degree reducing interpolation space for Ξ . 1 P basis for P . 2 � � Set P ∗ = ( · ) j p : p ∈ P , j = 1, . . . , s . (Pre-)border basis. 3 Set 4 F Ξ := { p − L Ξ p : p ∈ P ∗ } . Then: I ( Ξ ) = � F Ξ � . 5 Even better Basis is H–basis : Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 10 / 24

  49. From Interpolation to Bases Construction of basis P degree reducing interpolation space for Ξ . 1 P basis for P . 2 � � Set P ∗ = ( · ) j p : p ∈ P , j = 1, . . . , s . (Pre-)border basis. 3 Set 4 F Ξ := { p − L Ξ p : p ∈ P ∗ } . Then: I ( Ξ ) = � F Ξ � . 5 Even better Basis is H–basis : Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 10 / 24

  50. From Interpolation to Bases Construction of basis P degree reducing interpolation space for Ξ . 1 P basis for P . 2 � � Set P ∗ = ( · ) j p : p ∈ P , j = 1, . . . , s . (Pre-)border basis. 3 Set 4 F Ξ := { p − L Ξ p : p ∈ P ∗ } . Then: I ( Ξ ) = � F Ξ � . 5 Even better Basis is H–basis : Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 10 / 24

  51. From Interpolation to Bases Construction of basis P degree reducing interpolation space for Ξ . 1 P basis for P . 2 � � Set P ∗ = ( · ) j p : p ∈ P , j = 1, . . . , s . (Pre-)border basis. 3 Set 4 F Ξ := { p − L Ξ p : p ∈ P ∗ } . Then: I ( Ξ ) = � F Ξ � . 5 Even better Basis is H–basis : I ( Ξ ) ∋ g Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 10 / 24

  52. From Interpolation to Bases Construction of basis P degree reducing interpolation space for Ξ . 1 P basis for P . 2 � � Set P ∗ = ( · ) j p : p ∈ P , j = 1, . . . , s . (Pre-)border basis. 3 Set 4 F Ξ := { p − L Ξ p : p ∈ P ∗ } . Then: I ( Ξ ) = � F Ξ � . 5 Even better Basis is H–basis : � I ( Ξ ) ∋ g = g f f f ∈ F Ξ Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 10 / 24

  53. From Interpolation to Bases Construction of basis P degree reducing interpolation space for Ξ . 1 P basis for P . 2 � � Set P ∗ = ( · ) j p : p ∈ P , j = 1, . . . , s . (Pre-)border basis. 3 Set 4 F Ξ := { p − L Ξ p : p ∈ P ∗ } . Then: I ( Ξ ) = � F Ξ � . 5 Even better Basis is H–basis : � I ( Ξ ) ∋ g = g f f , deg g f f ≤ deg g . f ∈ F Ξ Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 10 / 24

  54. Implicit Interpolation The case s = 1 � ( · − ξ ) . ω = 1 ξ ∈ Ξ Division with remainder: 2 L Ξ f := r interpolation polynomial. 3 Algebraic interpretation Principal ideal: I ( Ξ ) = � ω � . 1 Ideal + Quotient Space. 2 Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 11 / 24

  55. Implicit Interpolation The case s = 1 � ( · − ξ ) . ω = 1 ξ ∈ Ξ Division with remainder: 2 L Ξ f := r interpolation polynomial. 3 Algebraic interpretation Principal ideal: I ( Ξ ) = � ω � . 1 Ideal + Quotient Space. 2 Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 11 / 24

  56. Implicit Interpolation The case s = 1 � ( · − ξ ) . ω = 1 ξ ∈ Ξ Division with remainder: 2 L Ξ f := r interpolation polynomial. 3 Algebraic interpretation Principal ideal: I ( Ξ ) = � ω � . 1 Ideal + Quotient Space. 2 Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 11 / 24

  57. Implicit Interpolation The case s = 1 � ( · − ξ ) . ω = 1 ξ ∈ Ξ Division with remainder: 2 f = ω p + r , L Ξ f := r interpolation polynomial. 3 Algebraic interpretation Principal ideal: I ( Ξ ) = � ω � . 1 Ideal + Quotient Space. 2 Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 11 / 24

  58. Implicit Interpolation The case s = 1 � ( · − ξ ) . ω = 1 ξ ∈ Ξ Division with remainder: 2 deg r ≤ deg ω − 1 f = ω p + r , L Ξ f := r interpolation polynomial. 3 Algebraic interpretation Principal ideal: I ( Ξ ) = � ω � . 1 Ideal + Quotient Space. 2 Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 11 / 24

  59. Implicit Interpolation The case s = 1 � ( · − ξ ) . ω = 1 ξ ∈ Ξ Division with remainder: 2 deg r ≤ deg ω − 1 = # Ξ − 1. f = ω p + r , L Ξ f := r interpolation polynomial. 3 Algebraic interpretation Principal ideal: I ( Ξ ) = � ω � . 1 Ideal + Quotient Space. 2 Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 11 / 24

  60. Implicit Interpolation The case s = 1 � ( · − ξ ) . ω = 1 ξ ∈ Ξ Division with remainder: 2 deg r ≤ deg ω − 1 = # Ξ − 1. f = ω p + r , L Ξ f := r interpolation polynomial. 3 Algebraic interpretation Principal ideal: I ( Ξ ) = � ω � . 1 Ideal + Quotient Space. 2 Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 11 / 24

  61. Implicit Interpolation The case s = 1 � ( · − ξ ) . ω = 1 ξ ∈ Ξ Division with remainder: 2 deg r ≤ deg ω − 1 = # Ξ − 1. f = ω p + r , L Ξ f := r interpolation polynomial. 3 Algebraic interpretation Principal ideal: I ( Ξ ) = � ω � . 1 Ideal + Quotient Space. 2 Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 11 / 24

  62. Implicit Interpolation The case s = 1 � ( · − ξ ) . ω = 1 ξ ∈ Ξ Division with remainder: 2 deg r ≤ deg ω − 1 = # Ξ − 1. f = ω p + r , L Ξ f := r interpolation polynomial. 3 Algebraic interpretation Principal ideal: I ( Ξ ) = � ω � . 1 Ideal + Quotient Space. 2 Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 11 / 24

  63. Implicit Interpolation The case s = 1 � ( · − ξ ) . ω = 1 ξ ∈ Ξ Division with remainder: 2 deg r ≤ deg ω − 1 = # Ξ − 1. f = ω p + r , L Ξ f := r interpolation polynomial. 3 Algebraic interpretation Principal ideal: I ( Ξ ) = � ω � . 1 Ideal + Quotient Space. 2 Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 11 / 24

  64. Implicit Interpolation II Division with remainder Division by set F : 1 � g = g f f + ν F ( g ) . f ∈ F Ideal + normal form. 2 Normal form is 3 interpolant. 1 unique if F is H–basis. 2 Constructive chain Remark All degree reducing interpolants can be constructed this way. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 12 / 24

  65. Implicit Interpolation II Division with remainder Division by set F : 1 � g = g f f + ν F ( g ) . f ∈ F Ideal + normal form. 2 Normal form is 3 interpolant. 1 unique if F is H–basis. 2 Constructive chain Remark All degree reducing interpolants can be constructed this way. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 12 / 24

  66. Implicit Interpolation II Division with remainder Division by set F : 1 � g = g f f + ν F ( g ) . f ∈ F Ideal + normal form. 2 Normal form is 3 interpolant. 1 unique if F is H–basis. 2 Constructive chain Remark All degree reducing interpolants can be constructed this way. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 12 / 24

  67. Implicit Interpolation II Division with remainder Division by set F : 1 � g = g f f + ν F ( g ) . f ∈ F Ideal + normal form. 2 Normal form is 3 interpolant. 1 unique if F is H–basis. 2 Constructive chain Remark All degree reducing interpolants can be constructed this way. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 12 / 24

  68. Implicit Interpolation II Division with remainder Division by set F : 1 � g = g f f + ν F ( g ) . f ∈ F Ideal + normal form. 2 Normal form is 3 interpolant. 1 unique if F is H–basis. 2 Constructive chain Remark All degree reducing interpolants can be constructed this way. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 12 / 24

  69. Implicit Interpolation II Division with remainder Division by set F : 1 � g = g f f + ν F ( g ) . f ∈ F Ideal + normal form. 2 Normal form is 3 interpolant. 1 unique if F is H–basis. 2 Constructive chain Remark All degree reducing interpolants can be constructed this way. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 12 / 24

  70. Implicit Interpolation II Division with remainder Division by set F : 1 � g = g f f + ν F ( g ) . f ∈ F Ideal + normal form. 2 Normal form is 3 interpolant. 1 unique if F is H–basis. 2 Constructive chain Remark All degree reducing interpolants can be constructed this way. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 12 / 24

  71. Implicit Interpolation II Division with remainder Division by set F : 1 � g = g f f + ν F ( g ) . f ∈ F Ideal + normal form. 2 Normal form is 3 interpolant. 1 unique if F is H–basis. 2 Constructive chain Ξ Remark All degree reducing interpolants can be constructed this way. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 12 / 24

  72. Implicit Interpolation II Division with remainder Division by set F : 1 � g = g f f + ν F ( g ) . f ∈ F Ideal + normal form. 2 Normal form is 3 interpolant. 1 unique if F is H–basis. 2 Constructive chain Ξ � → I ( Ξ ) Remark All degree reducing interpolants can be constructed this way. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 12 / 24

  73. Implicit Interpolation II Division with remainder Division by set F : 1 � g = g f f + ν F ( g ) . f ∈ F Ideal + normal form. 2 Normal form is 3 interpolant. 1 unique if F is H–basis. 2 Constructive chain Ξ � → I ( Ξ ) � → H–basis F Remark All degree reducing interpolants can be constructed this way. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 12 / 24

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