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Near optimal interpolation and quadrature in two variables: the Padua points

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Near optimal interpolation and quadrature in two variables: the Padua points

M. C ALIARI a , S. D E M ARCHI a , A. S OMMARIVA b AND M. V IANELLO b

a

Department of Computer Science, University of Verona, ITALY

b

Department of Pure and Applied Mathematics, University of Padua, ITALY

Abstract

The Padua points, recently studied during an international collaboration at the University of Padua, are the first known example of near-optimal point set for bivariate polynomial in- terpolation of total degree. The associate algebraic cubature formulas are both, in some sense, near-optimal.

Which are the Padua points?

Chebyshev-Lobatto points in [−1, 1]

C

n+1

:= {− cos(jπ/n), j = 0, . . . , n}

card(C

n+1

) = n + 1 = dim(P

1n

)

Padua points in [−1, 1] × [−1, 1]

Pad

n

:= (C

n+1odd

× C

n+2even

) ∪ (C

n+1even

× C

n+2odd

) ⊂ C

n+1

× C

n+2

card(Pad

n

) = (n + 1)(n + 2)

2 = dim(P

2n

)

Alternative representation as self-intersections and boundary contacts of the generating curve

g (t) := (− cos((n + 1)t), − cos(nt)) , t ∈ [0, π]

F

IGURE

1. The Padua points with their generating curve for n = 12 (left, 91 points) and n = 13 (right, 105 points), also as union of two Chebyshev-Lobatto subgrids (open bullets and filled bullets).

-1 -0,5 0 0,5 1

-1 -0,5 0 0,5 1

-1 -0,5 0 0,5 1

-1 -0,5 0 0,5 1

-1 -0,5 0 0,5 1

-1 -0,5 0 0,5 1

-1 -0,5 0 0,5 1

-1 -0,5 0 0,5 1

There are 4 families of such points, corresponding to succes- sive rotations of 90 degrees.

Interpolation at the Padua points

trigonometric quadrature on the generating curve

algebraic cubature at the Padua points for the product Chebyshev measure with weights

w

ξ

:= 1 n(n + 1) ·

 

 

1/2 if ξ is a vertex point 1 if ξ is an edge point 2 if ξ is an interior point near exactness in P

22n

Lagrange interpolation formula L

Padn

f (x) = X

ξ∈Padn

f (ξ) L

ξ

(x) , L

ξ

(η) = δ

ξη

L

ξ

(x) = w

ξ

(K

n

(ξ, x) − T

n

1

)T

n

(x

1

))

T

n

(·) = cos (n arccos (·)) and K

n

(x, y) reproducing kernel of the product Chebyshev orthonormal basis

The Lebesgue constant

Unisolvence gives the projection operator

L

Padn

: C([−1, 1]

2

, k · k

) → P

2n

([−1, 1]

2

, k · k

)

Theorem

kL

Padn

k = max

x∈[−1,1]2

X

ξ∈Padn

|L

ξ

(x)| = O(log

2

(n))

i.e., the Lebesgue constant has optimal order of growth (cf.

[7]).

Riferimenti

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