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COMPUTING MULTIVARIATE FEKETE AND LEJA POINTS BY NUMERICAL LINEAR ALGEBRA ∗
"... Abstract. We discuss and compare two greedy algorithms, that compute discrete versions of Feketelike points for multivariate compact sets by basic tools of numerical linear algebra. The first gives the socalled “Approximate Fekete Points ” by QR factorization with column pivoting of Vandermondeli ..."
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Cited by 18 (16 self)
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Abstract. We discuss and compare two greedy algorithms, that compute discrete versions of Feketelike points for multivariate compact sets by basic tools of numerical linear algebra. The first gives the socalled “Approximate Fekete Points ” by QR factorization with column pivoting of Vandermondelike matrices. The second computes Discrete Leja Points by LU factorization with row pivoting. Moreover, we study the asymptotic distribution of such points when they are extracted
Geometric Weakly Admissible Meshes, Discrete Least Squares Approximations and Approximate Fekete Points
, 2009
"... Using the concept of Geometric Weakly Admissible Meshes (see §2 below) together with an algorithm based on the classical QR factorization of matrices, we compute efficient points for discrete multivariate least squares approximation and Lagrange interpolation. ..."
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Cited by 14 (11 self)
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Using the concept of Geometric Weakly Admissible Meshes (see §2 below) together with an algorithm based on the classical QR factorization of matrices, we compute efficient points for discrete multivariate least squares approximation and Lagrange interpolation.
Growth of balls of holomorphic sections and energy at equilibrium
, 2008
"... Let L be a big line bundle on a compact complex manifold X. Given a nonpluripolar compact subset K of X and the weight φ of a continuous Hermitian metric e −φ on L, we define the energy at equilibrium of (K, φ) as the AubinMabuchi energy of the extremal psh weight associated to (K, φ). We prove t ..."
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Cited by 9 (1 self)
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Let L be a big line bundle on a compact complex manifold X. Given a nonpluripolar compact subset K of X and the weight φ of a continuous Hermitian metric e −φ on L, we define the energy at equilibrium of (K, φ) as the AubinMabuchi energy of the extremal psh weight associated to (K, φ). We prove the differentiability of the energy at equilibrium with respect to φ, and we show that this energy describes the asymptotic behaviour as k → ∞ of the volume of the supnorm unit ball induced by (K, kφ) on the space of global holomorphic sections H 0 (X, kL). As a consequence of these results, we recover and extend Rumely’s Robintype formula for the transfinite diameter. We also obtain an asymptotic description of the analytic torsion, and extend Yuan’s equidistribution theorem for algebraic points of small height to the case of a big line bundle.
Capacities and weighted volumes for line bundles
, 2008
"... Let (L, h) be an arbitrary Hermitian holomorphic line bundle over a compact Kähler manifold X. We introduce a natural capacity for compact subsets K of X, which describes the volume growth of the corresponding unit L ∞ (K)ball of global sections of L ⊗k as k → ∞. The main theorem expresses this ca ..."
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Cited by 6 (1 self)
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Let (L, h) be an arbitrary Hermitian holomorphic line bundle over a compact Kähler manifold X. We introduce a natural capacity for compact subsets K of X, which describes the volume growth of the corresponding unit L ∞ (K)ball of global sections of L ⊗k as k → ∞. The main theorem expresses this capacity as an energy functional, which is a mixed MongeAmpère formula involving the corresponding equilibrium metric obtained as the nonnegatively curved envelope of h. As a corollary we obtain various expressions for the (weighted) Leja transfinite diameter in C n. We also study variational properties of the energy (proving convexity, differentiability etc...). We obtain as applications a generalization of Yuan’s arithmetic equidistribution theorem to big line bundles, and a description of the asymptotic behaviour of the RaySinger analytic torsion with respect to a smooth metric of arbitrary curvature.
The GelfondSchnirelman Method In Prime Number Theory
 Canad. J. Math
"... The original GelfondSchnirelman method, proposed in 1936, uses polynomials with integer coe#cients and small norms on [0, 1] to give a Chebyshevtype lower bound in prime number theory. We study a generalization of this method for polynomials in many variables. Our main result is a lower bound for t ..."
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Cited by 4 (4 self)
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The original GelfondSchnirelman method, proposed in 1936, uses polynomials with integer coe#cients and small norms on [0, 1] to give a Chebyshevtype lower bound in prime number theory. We study a generalization of this method for polynomials in many variables. Our main result is a lower bound for the integral of Chebyshev's #function, expressed in terms of the weighted capacity. This extends previous work of Nair and Chudnovsky, and connects the subject to the potential theory with external fields generated by polynomialtype weights. We also solve the corresponding potential theoretic problem, by finding the extremal measure and its support. 1. Lower bounds for arithmetic functions Let #(x) be the number of primes not exceeding x. The celebrated Prime Number Theorem (PNT), suggested by Legendre and Gauss, states that ##.
A ROBIN FORMULA FOR THE FEKETELEJA TRANSFINITE DIAMETER
, 2005
"... The purpose of this note is to give a formula for the FeketeLeja transfinite diameter on C N, generalizing the classical Robin formula −V (E) d∞(E) = e for the usual transfinite diameter. We will disengage this from a formula for the sectional capacity proved in arithmetic intersection theory ([5] ..."
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Cited by 2 (0 self)
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The purpose of this note is to give a formula for the FeketeLeja transfinite diameter on C N, generalizing the classical Robin formula −V (E) d∞(E) = e for the usual transfinite diameter. We will disengage this from a formula for the sectional capacity proved in arithmetic intersection theory ([5], Theorem 1.1, p.233). First recall the definition of the FeketeLeja transfinite diameter for a compact set E ⊂ CN (see [1], [16]). Consider the set of monomials zk = z k1 1 · · ·z kN N in the polynomial ring C[z] = C[z1,...,zN]. Let Γ(n) ⊂ C[z] be the space of polynomials of total degree at most n, and let K(n) = {k ∈ ZN: ki ≥ 0, k1 + · · · + kN ≤ n} be the index set for the monomial basis of Γ(n). Put qn = #(K(n)) = () n+N n Fixing n, take qn independent vector variables zi = (zi1,..., ziN) ∈ CN, i = 1,...,qn. Let k1,...,kqn be the indices in K(n). The Vandermonde determinant Qn(z1,...,zqn): = det(z kj i)qn i,j=1 is a homogeneous polynomial in the zij of total degree Tn = N · () n+N. For each n, put N+1 dn(E) = The FeketeLeja transfinite diameter is defined by max z1,...,zqn ∈E Qn(z1,..., zqn)  1/Tn. d∞(E) = lim n→∞ dn(E). The existence of the limit is due to Zaharjuta ([16]). Henceforth, we will assume that d∞(E)> 0. This is equivalent to E being nonpluripolar ([10]). For f ∈ C[z1,...,zN] write ‖f‖E = sup z∈E f(z). The Green’s function G ∗ (z, E) is the upper semicontinuous regularization of the Siciak extremal function 1 G(z, E): = lim max log(f(z)). n→ ∞ f∈Γ(n) n
THE MULTIVARIATE INTEGER CHEBYSHEV PROBLEM
"... Abstract. The multivariate integer Chebyshev problem is to find polynomials with integer coefficients that minimize the supremum norm over a compact set in C d. We study this problem on general sets, but devote special attention to product sets such as cube and polydisk. We also establish a multivar ..."
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Abstract. The multivariate integer Chebyshev problem is to find polynomials with integer coefficients that minimize the supremum norm over a compact set in C d. We study this problem on general sets, but devote special attention to product sets such as cube and polydisk. We also establish a multivariate analog of the HilbertFekete upper bound for the integer Chebyshev constant, which depends on the dimension of space. In the case of single variable polynomials in the complex plane, our estimate coincides with the HilbertFekete result. 1. The integer Chebyshev problem and its multivariate counterpart The supremum norm on a compact set E ⊂ C d, d ∈ N, is defined by �f�E: = sup f(z). z∈E We study polynomials with integer coefficients that minimize the sup norm on a set E, and investigate their asymptotic behavior. The univariate case (d = 1) has a long history, but the problem is virtually untouched for d ≥ 2. Let Pn(C) and Pn(Z) be the classes of algebraic polynomials in one variable, of degree at most n, respectively with complex and with integer coefficients. The problem of minimizing the uniform norm on E by monic polynomials from Pn(C) is the classical Chebyshev problem (see [5], [23], [26], etc.) For E = [−1, 1], the explicit solution of this problem is given by the monic Chebyshev polynomial of degree n: Tn(x): = 2 1−n cos(n arccos x), n ∈ N. By a linear change of variable, we immediately obtain that � �n � � b − a 2x − a − b tn(x):=
and
"... Using recent results of Berman and Boucksom [BB2] we show that for a nonpluripolar compact set K ⊂ C d and an admissible weight function w = e −φ any sequence of socalled optimal measures converges weak * to the equilibrium measure µK,φ of (weighted) Pluripotential Theory for K, φ. 1 ..."
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Using recent results of Berman and Boucksom [BB2] we show that for a nonpluripolar compact set K ⊂ C d and an admissible weight function w = e −φ any sequence of socalled optimal measures converges weak * to the equilibrium measure µK,φ of (weighted) Pluripotential Theory for K, φ. 1
A Robin formula for the FeketeLeja transfinite diameter
, 2006
"... We generalize the classical Robin formula to higher dimensions. ..."