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## Fekete Points as Norming Sets

### Len Bos

a

Abstract

Suppose that K⊂ Rdis compact. Fekete points of degree n are those points Fn⊂ K that maximize the determinant of the interpolation matrix for polynomial interpolation of degree n. We discuss some special cases where we can show that Fekete points (of uniformly higher degree) are norming sets for K, i.e., for any c> 1, there exists a constant C > 0 such that kpkK≤ CkpkFcn, for all polynomials of degree at most n. It is conjectured that this is true for “general” K.

d

n

n

n

n

n

n

m

m

dme

n

d−1

d

n

n

1

2

N

1

2

N

n

i

i

1

2

N

n

Nj=1

j

j

1

1

2

1

N

1

1

2

2

2

N

2

1

N

2

N

N

N

i

i

i

1

2

N

1

1

2

1

N

1

1

2

2

2

N

2

1

N

2

N

N

N

i

1

i−1

i+1

N

1

N

i

j

i j

n

n

i=1

i

i

### (x)

aUniversity of Verona, Verona, Italy

(2)

x∈K N

i=1

i

1

2

N

1

N

N

1

2

x∈K

i

x∈K N

i=1

i

n

K

Fn

d

X

z∈X

n

n

K

Xn

n

n

n

n

d

K

### (x, y) := sup

n≥1, p∈Pn(K), kpkK=1

−1

−1

6

4

5

1

K

−1

−1

3

10, Prop. 1

d

n

y∈Xn

K

n

K

Xn

K

K

n

K

−1

−1

−1

−1

K

−1

−1

K

Xn

n

m

(3)

n

K

1.2

1.2

m

12, Lemma 1

k

k

j

12

m+1

k=1

k

k

k

0

k

1

m+1

x∈[−1,1]

k

0k

k

2

00

2

m0−1

m

−1/2

00

2

2

00

2

k

13, Thm. 1.82.1

k

k+1

k

k

k

k+1

k

k

k+1

k+1

k

2

2

m0

m0

2

1

1

2

1

2

j

j

j+1

j

k

m

K

k

(4)

2

d

n

s

m

n

s

dlog(n)e

dlog(n)eK

dlog(n)e

K

m

dlog(n)e

Fm

m

dlog(n)eFm

K

m

1/dlog(n)e

Fm

n

1/dlog(n)e

s/ log(n)

s/ log(n)

s

9

d−1

d

d−1

### , as already shown by Dubiner  (cf. [4,

5]), the Dubiner distance is just geodesic

K

−1

d−1

n

Nn

k=1

2

k

Sd−1

2

n

d−1

A

n

A

Sd−1

A2

−1

n

d−1

d−1

Sd−1

A

d−1

A

A

A2

Sd−1

1

2

Nan

k

Sd−1

Nan

k=1

k

k

Nan

k=1

k

A2

k

k

Nan

k=1

k

A2

k

(5)

k

K

Sd−1

1≤k≤N

an

k

Nan

k=1

A2

k

1≤k≤N

an

k

an

an

Nan

k=1

A2

k

1≤k≤N

an

k

an

Sd−1

A2

A

Sd−1

1≤k≤N

an

k

an

(a−1)n

0

1≤k≤N

an

k

0

an

(a−1)n

8, Cor. 4.6

n

Nn

k=1

2

k

Sd−1

2

n

n

1

2

Nn

6

K

i

j

K

i

j

### ) = sup

n≥1, p∈Pn(K), kpkK=1

−1

i

−1

j

−1

i

i

−1

i

j

−1

−1

d

r

d

d−1

r

d−1

K

d

r

d

d

d

d−1

r

d0−1

d−1

d−1

d−1

r

d−1

d−1

r

d−1

d−1

d

K

(6)

2

2

2

2

2

2

2

2

2

2

2

2

2

2

d

n

c/n

2.3,

c/n

d−1

c0/n

d−1

0

c/n

d−1

c0/n

d−1

c00/n

00

0

1

m

c/n

1

m

c00/n

d−1

ac/n

j

ac/n

k

0

0

d−1

c/n

i

ac/n

j

0

d−1

ac/n

j

d−1

c/n

0

d−1

mj=1

ac/n

j

0

d−1

0

d−1

c/n

0

d−1

c,n

d

d−1

n

d−1

d

Tc,n

2

Sd−1

2

8, Lemma 4.2

rd−1

d

n

d−1

Sd−1r

2

Sd−1

2

(7)

0

Sd−1r

2

Sd−1

2

Sd−1

2

2deg(P)

Sd−1

2

n

k=0

k

k

k

k

Sd−1

2

Sd−1

n

k=0

k

k

2

Sd−1

n

k=0

k

k

k

2

n

k=0

2k

2k

Sd−1

2k

2n

n

k=0

2k

Sd−1

2k

2n

Sd−1

2

8, Cor. 4.3

Tc,n

2

Sd−1

2

d

Tc,n

2

d

r=1+c/n r=1−c/n

Sd−1r

d−1

2

Tc,n

2

d

r=1+c/n r=1−c/n

Sd−1r

d−1

2

r=1+c/n r=1−c/n

2n

Sd−1

2

2n

Sd−1

2

2c

Sd−1

2

### We continue with the conclusion of the proof of Proposition

2.2. Indeed, by subharmonicity, there is a constant C such that for all

d−1

2

d

B(z,1/n)

2

(8)

d

n

Nn

k=1

2

k

n

Nn

k=1

d

B(fk,1/n)

2

d

C1,n

2

n

Nn

k=1

B(fk,1/n)

d

C1,n

2

n

Nn

k=1

B(x,1/n)

k

d

C1,n

2

n

d

n

C1,n

2

d

n

Sd−1

2

2.5

Sd−1

2

n

d−1

d−1

n

d−1

n

n

d−1

A

n

n

A

n

n

A

2

n2

Sd−1

n

n

n2

n

n

p

p

d

### References

 T. Bloom, L. Bos, C. Christensen, and N. Levenberg, Polynomial interpolation of holomorphic functions in C and Cn, Rocky Mountain J.

Math., 22 (1992), no. 2, 441-470.

 T. Bloom, L. Bos, J.-P. Calvi, and N. Levenberg, Polynomial interpolation of holomorphic functions in Cd, Ann. Polon. Math., 106 (2012), 53 – 81.

 L. Bos, A Simple Recipe for Modelling a d-cube by Lissajous curves, Dolomites Res. Notes Approx. DRNA 10 (2017), 1 – 4.

 L. Bos, N. Levenberg and S. Waldron, Metrics associated to multivariate polynomial inequalities, Advances in constructive approximation:

Vanderbilt 2003, 133 – 147, Mod. Methods Math., Nashboro Press, Brentwood, TN, 2004.

 L. Bos, N. Levenberg and S. Waldron, Pseudometrics, distances and multivariate polynomial inequalities, J. Approx. Theory 153 (2008), no. 1, 80 – 96.

 M. Dubiner, The theory of multi-dimensional polynomial approximation, J. Anal. Math. 67 (1995), 39 – 116.

 H. Ehlich and K. Zeller, Schwankung von Polynomen zwischen Gitterpunkten, Math. Zeistschr. 86 (1964), 41 – 44.

 J. Marzo, Marcinkiewicz-Zygmund inequalities and interpolation by spherical harmonics, J. Fun. Anal. 250 (2007), 559 – 587.

 J. Marzo and J. Ortega-Cerdà, Equidistribution of Fekete Points on the Sphere, Constr. Approx. 32 (2010), 513 – 521.

 F. Piazzon and M. Vianello, A note on total degree polynomial optimization by Chebyshev grids, Optim. Lett. 12 (2018), 63 – 71.

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 M. Reimer, Multivariate Polynomial Approximation, Birkhauser, 2003.

 B. S¨undermann, Lebesgue constants in Lagrangian interpolation at the Fekete points, Mitt. Math. Ges. Hamburg 11 (1983), no. 2, 204 – 211.

 G. Szeg¨o, Orthogonal Polynomials, AMS Colloq. Publ. Vol. 23, 1939.

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