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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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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