Issues in Multivariate Polynomial Interpolation Carl de Boor - - PDF document

Issues in Multivariate Polynomial Interpolation Carl de Boor 27sep11 pages.cs.wisc.edu/∼deboor/multiint/multiint.html 46

Univariate polynomial interpolation Given an infinite sequence z := (z1, z2, . . .) ∈ FN, F = R or C, the corresponding sequence ωk,z(x) :=

k

∏

i=1

(x − zi), k = 0, 1, 2, . . . , is a basis for the space Π(F1) of univariate polynomials. 45

Univariate polynomial interpolation Given an infinite sequence z := (z1, z2, . . .) ∈ FN, F = R or C, the corresponding sequence ωk,z(x) :=

k

∏

i=1

(x − zi), k = 0, 1, 2, . . . , is a basis for the space Π(F1) of univariate polynomials. Hence, every p ∈ Π(F1) has a unique expansion as the corresponding Newton series p =:

∞

∑

j=1

ωj−1,z∆(z≤j)p, with ∆(z≤j) := ∆(z1, . . . , zj) the linear functionals on Π(F1) defined thereby, and this par- ticular notation to be justified in a moment. 44

Univariate polynomial interpolation Given an infinite sequence z := (z1, z2, . . .) ∈ FN, F = R or C, the corresponding sequence ωk,z(x) :=

k

∏

i=1

(x − zi), k = 0, 1, 2, . . . , is a basis for the space Π(F1) of univariate polynomials. Hence, every p ∈ Π(F1) has a unique expansion as the corresponding Newton series p =:

∞

∑

j=1

ωj−1,z∆(z≤j)p, with ∆(z≤j) := ∆(z1, . . . , zj) the linear functionals on Π(F1) defined thereby, and this par- ticular notation to be justified in a moment. By Taylor’s theorem, p(x) =

∞

∑

j=1

(x − a)j−1 Dj−1p(a) (j − 1)! , therefore (with zj = a, all j) ∆(a, . . . , a

j terms

)p = Dj−1p(a) (j − 1)! , while p(z1) = ∆(z1)p p(z2) = ∆(z1)p + (x − z1)∆(z1, z2)p + (x − z1)(x − z2)q(x), hence, if x = z2 ̸= z1, then p(z2) − p(z1) z2 − z1 = ∆(z1.z2)p, a divided difference. 43

Univariate polynomial interpolation Given an infinite sequence z := (z1, z2, . . .) ∈ FN, F = R or C, the corresponding sequence ωk,z(x) :=

k

∏

i=1

(x − zi), k = 0, 1, 2, . . . , is a basis for the space Π(F1) of univariate polynomials. Hence, every p ∈ Π(F1) has a unique expansion as the corresponding Newton series p =:

∞

∑

j=1

ωj−1,z∆(z≤j)p, with ∆(z≤j) := ∆(z1, . . . , zj) the linear functionals on Π(F1) defined thereby, and this par- ticular notation to be justified in a moment. Then, for any n, p −

n

∑

j=1

ωj−1,z∆(z≤j)p

• =: pn

= ∑

j>n

ωj−1,z∆(z≤j)p

• =: ωn,zqn

. 42

Univariate polynomial interpolation Given an infinite sequence z := (z1, z2, . . .) ∈ FN, F = R or C, the corresponding sequence ωk,z(x) :=

k

∏

i=1

(x − zi), k = 0, 1, 2, . . . , is a basis for the space Π(F1) of univariate polynomials. Hence, every p ∈ Π(F1) has a unique expansion as the corresponding Newton series p =:

∞

∑

j=1

ωj−1,z∆(z≤j)p, with ∆(z≤j) := ∆(z1, . . . , zj) the linear functionals on Π(F1) defined thereby, and this par- ticular notation to be justified in a moment. Then, for any n, p −

n

∑

j=1

ωj−1,z∆(z≤j)p

• =: pn

= ∑

j>n

ωj−1,z∆(z≤j)p

• =: ωn,zqn

.

• pn is the remainder of the division of p by ωn,z, hence the unique element of Π<n agreeing

with p at z≤n = (z1, . . . , zn) (counting multiplicities), and depends continuously on z≤n . 41

Univariate polynomial interpolation Given an infinite sequence z := (z1, z2, . . .) ∈ FN, F = R or C, the corresponding sequence ωk,z(x) :=

k

∏

i=1

(x − zi), k = 0, 1, 2, . . . , is a basis for the space Π(F1) of univariate polynomials. Then, for any n, p −

n

∑

j=1

ωj−1,z∆(z≤j)p

• =: pn

= ∑

j>n

ωj−1,z∆(z≤j)p

• =: ωn,zqn

.

• pn is the remainder of the division of p by ωn,z, hence the unique element of Π<n agreeing

with p at z≤n = (z1, . . . , zn) (counting multiplicities), and depends continuously on z≤n .

• Hermite interpolation is the limit of Lagrange interpolation as data sites appro-

priately coalesce. 40

Univariate polynomial interpolation Given an infinite sequence z := (z1, z2, . . .) ∈ FN, F = R or C, the corresponding sequence ωk,z(x) :=

k

∏

i=1

(x − zi), k = 0, 1, 2, . . . , is a basis for the space Π(F1) of univariate polynomials. Then, for any n, p −

n

∑

j=1

ωj−1,z∆(z≤j)p

• =: pn

= ∑

j>n

ωj−1,z∆(z≤j)p

• =: ωn,zqn

.

• pn is the remainder of the division of p by ωn,z, hence the unique element of Π<n agreeing

with p at z≤n = (z1, . . . , zn) (counting multiplicities), and depends continuously on z≤n . Note:

• ∆(z≤n) depends on the multiset {z1, . . . , zn} (and not on the ordering).
• ∑

j>n ωj−1,z ωn,z ∆(z≤j)p = qn = ∆(z≤n, · )p

= ⇒ ∆(z≤n+2) = ∆(zn+1, zn+2)∆(z≤n, · ) Hence, by induction,

• ∆(z≤n, · ) = ∆(zn, · )∆(zn−1, · ) · · · ∆(z2, · )∆(z1, · ), showing why ∆(z≤n+1) is called

an nth divided difference. 39

Univariate polynomial interpolation Given an infinite sequence z := (z1, z2, . . .) ∈ FN, F = R or C, the corresponding sequence ωk,z(x) :=

k

∏

i=1

(x − zi), k = 0, 1, 2, . . . , is a basis for the space Π(F1) of univariate polynomials. Then, for any n, p −

n

∑

j=1

ωj−1,z∆(z≤j)p

• =: pn

= ∑

j>n

ωj−1,z∆(z≤j)p

• =: ωn,zqn

.

• pn is the remainder of the division of p by ωn,z, hence the unique element of Π<n agreeing

with p at z≤n = (z1, . . . , zn) (counting multiplicities), and depends continuously on z≤n . Note:

• ∆(z≤n) depends on the multiset {z1, . . . , zn} (and not on the ordering).
• ∑

j>n ωj−1,z ωn,z ∆(z≤j)p = qn = ∆(z≤n, · )p

= ⇒ ∆(z≤n+2) = ∆(zn+1, zn+2)∆(z≤n, · ) Hence, by induction,

• ∆(z≤n, · ) = ∆(zn, · )∆(zn−1, · ) · · · ∆(z2, · )∆(z1, · ), showing why ∆(z≤n+1) is called

an nth divided difference.

• (error `

a la Genocchi-Hermite): Since ∆(x, y)p = ∫ 1

0 Dp(x + s(y − x)) ds (even if x = y),

get qn(x) = ∫ 1 ∫ s1 · · · ∫ sn−1 (Dnp) (z1 + s1(z2 − z1) + · · · + sn(x − zn)) dsn · · · ds1 =: ∫ M(z1, . . . , zn, x | · ) Dnp 38

Polynomial interpolation 37

Multivariate polynomial interpolation dim Π≤1(R2) = 3 < 4 < 6 = dim Π≤2(R2) 36

Multivariate polynomial interpolation Now, even Π≤2(R2) won’t do; need at least Π≤3(R2). 35

Multivariate polynomial interpolation Basic Problem: Given a finite set T of data sites τ in Fd, how to choose a space F of polynomials so that, for every choice of data values a = (a(τ) : τ ∈ T) ∈ FT, there is exactly one element f ∈ F that matches this information, i.e., satisfies f(τ) = a(τ), τ ∈ T. I.e., want F to be correct for T.

unisolvent, poised, properly posed, regular, . . .

F is R or C. 34

Multivariate polynomial interpolation Basic Problem: Given a finite set T of data sites τ in Fd, how to choose a space F of polynomials so that, for every choice of data values a = (a(τ) : τ ∈ T) ∈ FT, there is exactly one element f ∈ F that matches this information, i.e., satisfies f(τ) = a(τ), τ ∈ T. I.e., want F to be correct for T.

unisolvent, poised, properly posed, regular, . . .

F is R or C. Equivalently, want the linear map δT : g → g|T := (g(τ) : τ ∈ T) ∈ FT (of restriction to T) to be onto and 1-1 when restricted to F. Then PT := ((δT)|F )−1δT : g → ∑

τ∈T

ℓτ g(τ) is the linear projector that associates each g with its unique interpolant PTg at T in F and ∑

τ∈T ℓτ g(τ) is its Lagrange form.

33

Multivariate polynomial interpolation Basic Problem: Given a finite set T of data sites τ in Fd, how to choose a polynomial space F correct for T. 32

Multivariate polynomial interpolation Basic Problem: Given a finite set T of data sites τ in Fd, how to choose a polynomial space F correct for T. Want a map T → ΠT with the following properties:

• ΠT is a pol. space correct for T, with PTg the interpolant from ΠT at T to g ;

31

Multivariate polynomial interpolation Basic Problem: Given a finite set T of data sites τ in Fd, how to choose a polynomial space F correct for T. Want a map T → ΠT with the following properties:

• ΠT is a pol. space correct for T, with PTg the interpolant from ΠT at T to g ;
• T → ΠT is monotone, i.e., T ⊂ Σ =

⇒ ΠT ⊂ ΠΣ ; 30

Multivariate polynomial interpolation Basic Problem: Given a finite set T of data sites τ in Fd, how to choose a polynomial space F correct for T. Want a map T → ΠT with the following properties:

• ΠT is a pol. space correct for T, with PTg the interpolant from ΠT at T to g ;
• T → ΠT is monotone, i.e., T ⊂ Σ =

⇒ ΠT ⊂ ΠΣ ;

• translation-invariance, i.e., p ∈ ΠT =

⇒ p(· + σ) ∈ ΠT, hence D-invariance;

• dilation-invariance, i.e., p ∈ ΠT =

⇒ p(h·) ∈ ΠT, i.e., ΠT is spanned by homog. pols; 29

Multivariate polynomial interpolation Basic Problem: Given a finite set T of data sites τ in Fd, how to choose a polynomial space F correct for T. Want a map T → ΠT with the following properties:

• ΠT is a pol. space correct for T, with PTg the interpolant from ΠT at T to g ;
• T → ΠT is monotone, i.e., T ⊂ Σ =

⇒ ΠT ⊂ ΠΣ ;

• translation-invariance, i.e., p ∈ ΠT =

⇒ p(· + σ) ∈ ΠT, hence D-invariance;

• dilation-invariance, i.e., p ∈ ΠT =

⇒ p(h·) ∈ ΠT, i.e., ΠT is spanned by homog. pols;

• for standard T, ΠT is the standard choice. E.g., if Π≤k is correct for T, then ΠT = Π≤k ;

e.g., ΠΣ×P = ΠΣ ⊗ ΠP ; 28

Multivariate polynomial interpolation Basic Problem: Given a finite set T of data sites τ in Fd, how to choose a polynomial space F correct for T. Want a map T → ΠT with the following properties:

• ΠT is a pol. space correct for T, with PTg the interpolant from ΠT at T to g ;
• T → ΠT is monotone, i.e., T ⊂ Σ =

⇒ ΠT ⊂ ΠΣ ;

• translation-invariance, i.e., p ∈ ΠT =

⇒ p(· + σ) ∈ ΠT, hence D-invariance;

• dilation-invariance, i.e., p ∈ ΠT =

⇒ p(h·) ∈ ΠT, i.e., ΠT is spanned by homog. pols;

• for standard T, ΠT is the standard choice. E.g., if Π≤k is correct for T, then ΠT = Π≤k ;

e.g., ΠΣ×P = ΠΣ ⊗ ΠP ;

• degree-reducing, i.e., deg(PTp) ≤ deg p ; hence, PTg is a minimal degree interpolant;

27

Multivariate polynomial interpolation Basic Problem: Given a finite set T of data sites τ in Fd, how to choose a polynomial space F correct for T. Want a map T → ΠT with the following properties:

• ΠT is a pol. space correct for T, with PTg the interpolant from ΠT at T to g ;
• T → ΠT is monotone, i.e., T ⊂ Σ =

⇒ ΠT ⊂ ΠΣ ;

• translation-invariance, i.e., p ∈ ΠT =

⇒ p(· + σ) ∈ ΠT, hence D-invariance;

• dilation-invariance, i.e., p ∈ ΠT =

⇒ p(h·) ∈ ΠT, i.e., ΠT is spanned by homog. pols;

• for standard T, ΠT is the standard choice. E.g., if Π≤k is correct for T, then ΠT = Π≤k ;

e.g., ΠΣ×P = ΠΣ ⊗ ΠP ;

• degree-reducing, i.e., deg(PTp) ≤ deg p ; hence, PTg is a minimal degree interpolant;
• PTp is “easily” constructible;
• · · ·

26

Multivariate polynomial interpolation Basic Problem: Given a finite set T of data sites τ in Fd, how to choose a polynomial space F correct for T. Notation: Π ∋ p : x → ∑

α

p(α)xα, xα := x(1)α(1) · · · x(d)α(d), x ∈ Fd, α ∈ Zd

+

α! := α(1)! · · · α(d)! 25

Multivariate polynomial interpolation Basic Problem: Given a finite set T of data sites τ in Fd, how to choose a polynomial space F correct for T. Notation: Π ∋ p : x → ∑

α

p(α)xα, xα := x(1)α(1) · · · x(d)α(d), x ∈ Fd, α ∈ Zd

+

One solution: least interpolation (dB+Amos Ron’88): Choose F = ΠT := span{f↓ : f ∈ span{eτ : τ ∈ T}

• =: ExpT

}, eτ : x → eτ ∗x = ∑

k

(τ ∗x)k k! = ∑

α

(τ)α α! xα least term (initial term): f↓(x) := ∑ |α| = min{|β| : f(β) ̸= 0}

• f(α)xα,

|α| := ∥α∥1 E.g., (eτ)↓(x) = 1, (eτ − 1)↓(x) = τ ∗x, . . . 24

Multivariate polynomial interpolation Basic Problem: Given a finite set T of data sites τ in Fd, how to choose a polynomial space F correct for T. One solution: least interpolation (dB+Amos Ron’88): Choose F = ΠT := span{f↓ : f ∈ span{eτ : τ ∈ T}

• =: ExpT

}, eτ : x → eτ ∗x = ∑

k

(τ ∗x)k k! = ∑

α

(τ)α α! xα least term (initial term): f↓(x) := ∑ |α| = min{|β| : f(β) ̸= 0}

• f(α)xα,

|α| := ∥α∥1

• eτ represents, on Π, evaluation at τ wrto the standard skew-bilinear form

⟨f, p⟩ := ∑

α

• f(α)α!

p(α) = ⇒ ⟨eτ, p⟩ = ∑

α

τ α p(α) = p(τ) 23

Multivariate polynomial interpolation Basic Problem: Given a finite set T of data sites τ in Fd, how to choose a polynomial space F correct for T. One solution: least interpolation (dB+Amos Ron’88): Choose F = ΠT := span{f↓ : f ∈ span{eτ : τ ∈ T}

• =: ExpT

}, eτ : x → eτ ∗x = ∑

k

(τ ∗x)k k! = ∑

α

(τ)α α! xα

• eτ represents, on Π, evaluation at τ wrto the standard skew-bilinear form

⟨f, p⟩ := ∑

α

• f(α)α!

p(α) = ⇒ ⟨eτ, p⟩ = ∑

α

τ α p(α) = p(τ)

• Use Gram-Schmidt algorithm to derive, from the basis (eτ : τ ∈ T) for ExpT, a basis

B for ExpT for which ⟨b, c↓⟩ = δbc, b, c ∈ B. Set PTp := ∑

c∈B⟨c, p⟩c↓ ∈ ΠT. Then

• ⟨b, PTp⟩ = ∑

c∈B⟨c, p⟩⟨b, c↓⟩ = ⟨b, p⟩, b ∈ B, therefore

• ExpT = span B ⊥ (p − PTp), hence p = PTp on T.

22

Multivariate polynomial interpolation Basic Problem: Given a finite set T of data sites τ in Fd, how to choose a polynomial space F correct for T. One solution: least interpolation (dB+Amos Ron’88): Choose F = ΠT := span{f↓ : f ∈ span{eτ : τ ∈ T}

• =: ExpT

}, eτ : x → eτ ∗x = ∑

k

(τ ∗x)k k! = ∑

α

(τ)α α! xα

• eτ represents, on Π, evaluation at τ wrto the standard skew-bilinear form

⟨f, p⟩ := ∑

α

• f(α)α!

p(α) = ⇒ ⟨eτ, p⟩ = ∑

α

τ α p(α) = p(τ)

• Use Gram-Schmidt algorithm to derive, from the basis (eτ : τ ∈ T) for ExpT, a basis

B for ExpT for which ⟨b, c↓⟩ = δbc, b, c ∈ B. Set PTp := ∑

c∈B⟨c, p⟩c↓ ∈ ΠT. Then

• ⟨b, PTp⟩ = ∑

c∈B⟨c, p⟩⟨b, c↓⟩ = ⟨b, p⟩, b ∈ B, therefore

• ExpT = span B ⊥ (p − PTp), hence p = PTp on T.

Note: deg c↓ > deg p = ⇒ ⟨c, p⟩ = ∑

α

c(α)α! p(α) = 0. Hence,

• deg PTp ≤ deg p.

21

Problem: T → ΠT is as continuous as possible but not more so. 20

Next Problem: Suppose that T → Σ (with #T held constant), and, for each T, PT is a corresponding polynomial interpolation projector (on Π). If Q := limT→Σ PT, exists (say, boundedly pointwise on Π), what can we say about Q? 19

Next Problem: Suppose that T → Σ (with #T held constant), and, for each T, PT is a corresponding polynomial interpolation projector (on Π). If Q := limT→Σ PT, exists (say, boundedly pointwise on Π), what can we say about Q? Q is necessarily an ideal projector, in the sense that ker Q is a polynomial ideal (since, for each T, ker PT = {p ∈ Π : p|T = 0} is a polynomial ideal ). 18

Next Problem: Suppose that T → Σ (with #T held constant), and, for each T, PT is a corresponding polynomial interpolation projector (on Π). If Q := limT→Σ PT, exists (say, boundedly pointwise on Π), what can we say about Q? Q is necessarily an ideal projector, in the sense that ker Q is a polynomial ideal (since, for each T, ker PT = {p ∈ Π : p|T = 0} is a polynomial ideal ). This implies that, for each σ ∈ Σ, there exists a D-invariant polynomial space Rσ ̸= ∅ so that Qp matches at σ all derivatives of p of the form r(D)p := ∑

α

• r(α)Dαp,

Dα := Dα(1)

1

· · · Dα(d)

d

, r ∈ Rσ. E.g., if Σ = limh→0 hT for some fixed T, then Σ = {0} and R0 = ΠT. 17

Next Problem: Suppose that T → Σ (with #T held constant), and, for each T, PT is a corresponding polynomial interpolation projector (on Π). If Q := limT→Σ PT, exists (say, boundedly pointwise on Π), what can we say about Q? Q is necessarily an ideal projector, in the sense that ker Q is a polynomial ideal (since, for each T, ker PT = {p ∈ Π : p|T = 0} is a polynomial ideal ). This implies that, for each σ ∈ Σ, there exists a D-invariant polynomial space Rσ ̸= ∅ so that Qp matches at σ all derivatives of p of the form r(D)p := ∑

α

• r(α)Dαp,

Dα := Dα(1)

1

· · · Dα(d)

d

, r ∈ Rσ. Next Problem: Is ideal interpolation Hermite interpolation, i.e., is every ideal projector the limit of Lagrange projectors? 16

Next Problem: Suppose that T → Σ (with #T held constant), and, for each T, PT is a corresponding polynomial interpolation projector (on Π). If Q := limT→Σ PT, exists (say, boundedly pointwise on Π), what can we say about Q? Q is necessarily an ideal projector, in the sense that ker Q is a polynomial ideal (since, for each T, ker PT = {p ∈ Π : p|T = 0} is a polynomial ideal ). This implies that, for each σ ∈ Σ, there exists a D-invariant polynomial space Rσ ̸= ∅ so that Qp matches at σ all derivatives of p of the form r(D)p := ∑

α

• r(α)Dαp,

Dα := Dα(1)

1

· · · Dα(d)

d

, r ∈ Rσ. Next Problem: Is ideal interpolation Hermite interpolation, i.e., is every ideal projector the limit of Lagrange projectors? Boris Shekhtman, via Geir Ellingsrud: yes for d = 2, no for d > 2. closely related question: can any sequence (M1, . . . , Md) of commuting matrices be ap- proximated by a commuting sequence of diagonalizable ones? yes for d = 2, no for d > 2. Next Problem: Characterize limits of Lagrange projectors. 15

Next Problem: Given a correct pair (T, F), how to represent the error p − PTp = ???, p ∈ Π. 14

Next Problem: Given a correct pair (T, F), how to represent the error p − PTp = ???, p ∈ Π. In the univariate case, with n := #T, and F = Π<n, p(x) − PTp(x) = ∏

τ∈T

(x − τ) ∫ M(T, x | ·) Dnp . 13

Next Problem: Given a correct pair (T, F), how to represent the error p − PTp = ???, p ∈ Π. In the univariate case, with n := #T, and F = Π<n, p(x) − PTp(x) = ∏

τ∈T

(x − τ) ∫ M(T, x | ·) Dnp . In general, ker PT = {p − PTp : p ∈ Π} = {q ∈ Π : q|T = 0} =: ideal(T)

(the radical ideal with variety T), hence (for a “good” F) hope for

p(x) − PTp(x) = ∑

b∈BT

b(x) ∫ Mb(x, ·) hb(D)p with

• BT

a ‘nice’ (finite) generating set for ideal(T);

• hb a homogeneous polynomial associated with b;

12

Next Problem: Given a correct pair (T, F), how to represent the error p − PTp = ???, p ∈ Π. In general, ker PT = {p − PTp : p ∈ Π} = {q ∈ Π : q|T = 0} =: ideal(T)

(the radical ideal with variety T), hence (for a “good” F) hope for

p(x) − PTp(x) = ∑

b∈BT

b(x) ∫ Mb(x, ·) hb(D)p “good” F? Would need that F = ∩b∈BT ker hb(D). Fortunately, in least interpolation, F = ΠT = ∩b∈BT ker b↑(D) . q↑(x) := ∑

|α|=deg q

q(α)xα 11

Next Problem: Given a correct pair (T, F), how to represent the error p − PTp = ???, p ∈ Π. In general, ker PT = {p − PTp : p ∈ Π} = {q ∈ Π : q|T = 0} =: ideal(T)

(the radical ideal with variety T), hence (for a “good” F) hope for

p(x) − PTp(x) = ∑

b∈BT

b(x) ∫ Mb(x, ·) hb(D)p For example, there is such a formula when F = R and T is a natural lattice (Chung&Yao), i.e., the collection of all intersections of d hyperplanes from a collection of d + k hyperplanes in general position. In that case, F = ΠT = Π≤k and #BT = #{α ∈ Zd

+ : |α| = k + 1}.

Moreover, hb(D)c = 0, b ̸= c ∈ BT. Sauer&Xu’95 have a formula involving many more terms but valid for any T correct for Π≤k for some k. Unfortunately, M¨

• ßner&Reif’07 show by an example that, in bicubic Taylor interpolation,

i.e., with F = ker D4

1 ∩ ker D4 2,

it is impossible to bound the error g − Pg in terms of D4

1g and D4 2g.

10

Multivariate polynomial interpolation mainstream Basic Problem: Given a polynomial space F (usually, F = Π≤k for some k), how to choose a T correct for F. 9

mainstream Basic Problem: Given a polynomial space F (usually, F = Π≤k for some k), how to choose a T correct for F. (Berzolari’14) Radon’48 recipe: If P is correct for Π<k, and Σ ⊂ H is correct for Π≤k(H) for some hyperplane H ⊂ Fd\P, then P ∪ Σ is correct for Π≤k. k = 4, d = 2 H is a maximal hyperplane for T := P ∪ Σ. 8

mainstream Basic Problem: Given a polynomial space F (usually, F = Π≤k for some k), how to choose a T correct for F. (Berzolari’14) Radon’48 recipe: If P is correct for Π<k, and Σ ⊂ H is correct for Π≤k(H) for some hyperplane H ⊂ Fd\P, then P ∪ Σ is correct for Π≤k. Chung-Yao’77 definition: T ⊂ Fd is called a GCk-set (short for a set satisfying the Geometric Conditions of degree k), if #T ≥ dim Π≤k and, for each τ ∈ T, T\{τ} lies in a union of ≤ k hyperplanes that does not contain τ. simplicial lattice natural lattice 7

mainstream Basic Problem: Given a polynomial space F (usually, F = Π≤k for some k), how to choose a T correct for F. (Berzolari’14) Radon’48 recipe: If P is correct for Π<k, and Σ ⊂ H is correct for Π≤k(H) for some hyperplane H ⊂ Fd\P, then P ∪ Σ is correct for Π≤k. Chung-Yao’77 definition: T ⊂ Fd is called a GCk-set (short for a set satisfying the Geometric Conditions of degree k), if #T ≥ dim Π≤k and, for each τ ∈ T, T\{τ} lies in a union of ≤ k hyperplanes that does not contain τ. simplicial lattice natural lattice Gasca-Maeztu Conjecture (1982). Any bivariate GCk-set is obtainable by Radon recipe. (Proven for k < 5) d > 2 ??? 2011 Ph.D. thesis of Armen Apozyan has GC2-set in R6 without any maximal hyperplane. 6

mainstream Basic Problem: Given a polynomial space F (usually, F = Π≤k for some k), how to choose a T correct for F. 5

mainstream Basic Problem: Given a polynomial space F (usually, F = Π≤k for some k), how to choose a T correct for F. Good interpolation sites for Π≤k: ∥p − PTp∥ ≤ (1 + ∥PT∥) dist (p, Π≤k) Look for a d-variate version of the (univariate) Chebyshev points for K = [−1 . . 1], {cos((2j + 1)π 2(k + 1) ) : j = 0, . . . , k}

−1 1

i.e., for sites T whose corresponding projector PT onto Π≤k ⊂ C(K) has norm O((ln k)d). 4

mainstream Basic Problem: Given a polynomial space F (usually, F = Π≤k for some k), how to choose a T correct for F. Good interpolation sites for Π≤k: ∥p − PTp∥ ≤ (1 + ∥PT∥) dist (p, Π≤k) Look for a d-variate version of the (univariate) Chebyshev points for K = [−1 . . 1], {cos((2j + 1)π 2(k + 1) ) : j = 0, . . . , k}

−1 1

i.e., for sites T whose corresponding projector PT onto Π≤k ⊂ C(K) has norm O((ln k)d). Ideally, get optimal points T which minimize ∥PT∥ := ∥ ∑

τ∈T |ℓτ|∥∞,K over correct T ⊂ K.

3

mainstream Basic Problem: Given a polynomial space F (usually, F = Π≤k for some k), how to choose a T correct for F. Good interpolation sites for Π≤k: ∥p − PTp∥ ≤ (1 + ∥PT∥) dist (p, Π≤k) Look for a d-variate version of the (univariate) Chebyshev points for K = [−1 . . 1], {cos((2j + 1)π 2(k + 1) ) : j = 0, . . . , k}

−1 1

i.e., for sites T whose corresponding projector PT onto Π≤k ⊂ C(K) has norm O((ln k)d). Ideally, get optimal points T which minimize ∥PT∥ := ∥ ∑

τ∈T |ℓτ|∥∞,K over correct T ⊂ K.

Perhaps, must be satisfied with Fekete points. These maximize |V (T)|, with V (T) := det(τ α : τ ∈ T, |α| ≤ k),

• ver T ⊂ K (with #T = #{α : |α| ≤ k). For such T,

|ℓτ(x)| =

• V (T(τ ← x))

V (T)

• ≤ 1,

hence ∥PT∥ ≤ #T = dim Π≤k. Fekete points are only known explicitly for an interval but can be computed numerically, in principle. Len Bos et al 2

mainstream Basic Problem: Given a polynomial space F (usually, F = Π≤k for some k), how to choose a T correct for F. Good interpolation sites for Π≤k: ∥p − PTp∥ ≤ (1 + ∥PT∥) dist (p, Π≤k) Look for a d-variate version of the (univariate) Chebyshev points for K = [−1 . . 1], {cos((2j + 1)π 2(k + 1) ) : j = 0, . . . , k}

−1 1

i.e., for sites T whose corresponding projector PT onto Π≤k ⊂ C(K) has norm O((ln k)d). Difficult, strongly dependent on the domain K. One very recent and very striking example are the Padua points of Marco Vianello, further worked on by Marco Caliari and Stefano De Marchi, for which their joint analysis with Len Bos gave O((ln k)2) in C([−1 . . 1]2) (which is optimal). ↓ matlab 1

−1 −0.5 0.5 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1

the Padua points for degree < 11

↓

the Padua points for degree < 11

{ ( cos( jπ k + 1), cos(jπ k ) ) : j = 0, . . . , k(k + 1)} dim Π≤k(R2) = (k + 1)(k + 2)/2 What about K ̸∼ [−1 . . 1]2, e.g., K = {x : ∥x∥2 ≤ 1}? d > 2? FINIS

• 1

• Lemma. The linear map Q on Π is an ideal projector iff

Q(fg) = Q(fQg), f, g ∈ Π. Proof: The identity can be rewritten as Q(f · (id − Q)g) = 0, f, g ∈ Π, showing, with f = 1, that Q = Q2, hence Q is a projector, in particular ker Q = ran(id − Q), while Q(Π · (id − Q)Π) = {0}, i.e., Π · ker Q ⊂ ker Q.

• 2

standard pairing: A0 × Π : (g, p) → ⟨g, p⟩ := ∑

α

g(α)α! p(α) = p(D)g(0) [BR91-92]. ΠT = ∩p|T=0 ker p↑(D) =: K. Proof: p|T = 0 = ⇒ Exp(T) ⊥ p ⟨g, p⟩ = 0 = ⇒ ⟨g↓, p↑⟩ = 0 = ⇒ ΠT ⊥ p↑ ⟨g↓, p↑⟩ = ⟨p↓, g↑⟩ = ⇒ ∀{g ∈ ΠT} p↑(D)g(0) = 0 Dα(ΠT) ⊂ ΠT = ⇒ ∀{g ∈ ΠT}∀{α} p↑(D)Dαg(0) = 0 = ⇒ ∀{g ∈ ΠT} p↑(D)g = 0

• 3

standard pairing: A0 × Π : (g, p) → ⟨g, p⟩ := ∑

α

g(α)α! p(α) = p(D)g(0) [BR91-92]. ΠT = ∩p|T=0 ker p↑(D) =: K. Proof: p|T = 0 = ⇒ Exp(T) ⊥ p = ⇒ ΠT ⊥ p↑ = ⇒ ∀{g ∈ ΠT} p↑(D)g(0) = 0 = ⇒ ∀{g ∈ ΠT}∀{α} p↑(D)Dαg(0) = 0 = ⇒ ∀{g ∈ ΠT} p↑(D)g = 0 Hence ΠT ⊆ K. dim ΠT < ∞ = ⇒ k := max{deg f : f ∈ ΠT} < ∞ = ⇒ ∀{|α| = k + 1} (()α − PT()α)↑ = ()α = ⇒ K ⊂ ∩|α|=k+1 ker Dα = Π≤k ⊂ Π Hence, g ∈ K implies that Π ∋ p := g −PTg ∈ K ⊂ ker p↑(D), i.e., p↑(D)p = 0, hence p = 0, so g ∈ ΠT.

• 4