PPLURIPOTENTIAL NUMERICS
FEDERICO PIAZZONA bstract . We introduce numerical methods for the approximation of the main(global) quantities in Pluripotential Theory as the extremal plurisubharmonicfunction V ∗ E of a compact L -regular set E ⊂ C n , its transfinite diameter δ ( E ) , andthe pluripotential equilibrium measure µ E : = (cid:16) dd c V ∗ E (cid:17) n . The methods rely on the computation of a polynomial mesh for E and nu-merical orthonormalization of a suitable basis of polynomials. We prove theconvergence of the approximation of δ ( E ) and the uniform convergence of ourapproximation to V ∗ E on all C n ; the convergence of the proposed approximationto µ E follows. Our algorithms are based on the properties of polynomial meshesand Bernstein Markov measures.Numerical tests are presented for some simple cases with E ⊂ R to illustratethe performances of the proposed methods. C ontents
1. Introduction 22. Preliminaries 52.1. Pluripotential theory: some definitions 52.2. Admissible meshes and Bernstein Markov measures 83. Approximating the extremal function 113.1. Theoretical results 113.2. The SZEF and SZEF-BW algorithms 143.3. Numerical Tests of SZEF and SZEF-BW 174. Approximating the transfinite diameter 224.1. Theoretical result 224.2. Implementation of the TD-GD algorithm 264.3. Numerical test of the TD-GD algorithm 275. Approximating the equilibrium measure 30Acknowledgements 33References 33
Date : October 7, 2018.1991
Mathematics Subject Classification.
MSC 65E05 and MSC 41A10 and MSC 32U35 andMSC 42C05.
Key words and phrases.
Pluripotential theory, Orthogonal Polynomials, admissible meshes. a r X i v : . [ m a t h . NA ] A p r FEDERICO PIAZZON
1. I ntroduction
Let E ⊂ C be a compact infinite set. Polynomial interpolation of holomorphicfunctions on E and its asymptotic are intimately related with logarithmic potentialtheory, i.e., the study of subharmonic function of logarithmic growth on C . Thisis a well established classical topic whose study goes back to Bernstein, Fekete,Leja, Szeg¨o, Walsh and many others; we refer the reader to [37], [47] and [41] foran extensive treatment of the subject.To study polynomial interpolation on a given compact set E one introduces theVandermonde determinant (usually with respect to the monomial basis) and, forany degree k ∈ N , tries to maximize its modulus among the tuples of k + k for E . A primaryinterest on Fekete arrays is that one immediately has the bound Λ k ( z , . . . , z k ) : = sup z ∈ E sup f ∈ C ( E ) , f (cid:44) | I k [ f ]( z ) |(cid:107) f (cid:107) E ≤ ( k + Λ k (i.e., the norm of the polynomial interpolation oper-ator I k ) for any Fekete array of order k for E . On the other hand Fekete arrays provide the link of polynomial interpolationwith potential theory. Indeed, the logarithmic energy cap( E ) of a unit charge dis-tribution on E at equilibriumcap( E ) : = exp (cid:32) − min µ ∈M ( E ) (cid:90) E (cid:90) E log 1 | z − ζ | d µ ( z ) d µ ( ζ ) (cid:33) turns out to coincide with certain asymptotic of the modulus of the Vandermondedeterminant computed at any sequence of Fekete points and with respect to themonomial basis. Since the considered asymptotic is a geometric mean of mutualdistances of Fekete points, it is termed transfinite diameter of E and denoted by δ ( E ) , δ ( E ) : = lim k | Vdm( z , . . . , z k ) | P k , for ( z , . . . , z k ) Fekete array . The Fundamental Theorem of Logarithmic Potential Theory asserts that δ ( E ) = cap( E ) = τ ( E ), where τ ( E ) is the Chebyshev constant of E and is defined bymeans of asymptotic of certain normalized monic polynomials. Moreover, pro-vided δ ( E ) (cid:44)
0, for any sequence of arrays having the same asymptotic of theVandermonde determinants as Fekete points the sequence of uniform probabilitymeasures supported on such arrays converges weak star to the unique minimizer ofthe logarithmic energy minimization problem, that is the equilibrium measure µ E of E .The other fundamental object in this theory is the Green function with pole atinfinity g E ( · , ∞ ) for the domain C \ ˆ E , where ˆ E is the polynomial hull of E . (1) g E ( z , ∞ ) : = lim sup ζ → z (cid:0) sup { u ( ζ ) : u ∈ L ( C ) , u | E ≤ } (cid:1) . Here L ( C ) is the Lelong class of subharmonic function of logarithmic growth, i.e., u ( z ) − log | z | is bounded near infinity. LURIPOTENTIAL NUMERICS 3
It turns out that, provided δ ( E ) (cid:44)
0, one has(2) g E ( z , ∞ ) = lim sup ζ → z (cid:32) sup (cid:40) p log | p ( ζ ) | , p ∈ P , (cid:107) p (cid:107) E ≤ (cid:41)(cid:33) . It follows that the Green function of E is intimately related to polynomial in-equalities and polynomial interpolation: for instance, one has the Bernstein WalshInequality(3) | p ( z ) | ≤ (cid:107) p (cid:107) E exp(deg p g E ( z , ∞ )) , ∀ p ∈ P and the Bernstein Walsh Theorem [47], that relates the rate of decrease of the errorof best polynomial uniform approximation on E of a function f ∈ hol (int E ) ∩ C ( E ) to the possibility of extending f holomorphically to a domain of the form { g E ( z , ∞ ) < c } . Lastly, it is worth to recall that, using the fact that log | · | is the fundamentalsolution of the Laplace operator in C , it is possible to show that ∆ g E ( z , ∞ ) = µ E in the sense of distributions.When we move from the complex plane to the case of E ⊂ C n , n >
1, thesituation becomes much more complicated. Indeed, one can still define Feketepoints and look for asymptotic of Vandermonde determinants with respect to thegraded lexicographically ordered monomial basis computed at these points, butthis is no more related to a linear convex functional on the space of probabilitymeasures (the logarithmic energy) neither to a linear partial di ff erential operator(the Laplacian) as in the case of C . During the last two decades (see for instance [25], [27]), a non linear potentialtheory in C n has been developed: Pluripotential Theory is the study of plurisub-harmonic functions , i.e., functions that are upper semicontinuous and subharmonicalong each complex line. Plurisubharmonic functions in this setting enjoy the roleof subharmonic functions in C , while maximal plurisubharmonic functions replaceharmonic ones; the geometric-analytical relation among the classes of functionsbeing the same. It turned out that this theory is related to several branches ofComplex Analysis and Approximation Theory, exactly as happens for LogarithmicPotential Theory in C .It was first conjectured by Leja that Vandermonde determinants computed atFekete points should still have a limit and that the associated probability measuressequences should converge to some unique limit measure, even in the case n > . The existence of the asymptotic of Vandermonde determinants and its equivalencewith a multidimensional analogue of the Chebyshev constant was proved by Zahar-juta [48], [49]. Finally the relation of Fekete points asymptotic with the equilibriummeasure and the transfinite diameter in Pluripotential Theory (even in a much moregeneral setting) has been explained by Berman Boucksom and Nymstr¨om very re-cently in a series of papers; [6], [5]. Indeed, the situation in the several complexvariables setting is very close to the one of logarithmic potential theory, provided
FEDERICO PIAZZON a suitable translation of the definitions, though the proof and the theory itself ismuch much more complicated.Since Logarithmic Potential Theory has plenty of applications in Analysis, Ap-proximation Theory and Physics, many numerical methods for computing approx-imations to Greens function, transfinite diameters and equilibrium measure hasbeen developed following di ff erent approaches as Riemann Hilbert problem [31],numerical conformal mapping [24], linear or quadratic optimization [39, 38] anditerated function systems [28].On the other hand, to the best authors’ knowledge, there are no algorithms forapproximating the corresponding quantities in Pluripotential Theory; the aim ofthis paper is to start such a study. This is motivated by the growing interest thatPluripotential Theory is achieving in applications during the last years. We men-tion, among the others, the quest for nearly optimal sampling in least squares re-gression [30, 42, 22], random polynomials [14, 50] and estimation of approxima-tion numbers of a given function [46].Our approach, first presented in the doctoral dissertation [32, Part II Ch. 6], isbased on certain sequences, first introduced by Calvi and Levenberg [21], of finitesubsets of a given compact set termed admissible polynomial meshes having slowincreasing cardinality and for which a sampling inequality for polynomials holdstrue. The core idea of the present paper is inspired by the analogy of sequences ofuniform probability measures supported on an admissible mesh with the class ofBernstein Markov measures (see for instance [32], [15] and [8]).Indeed, we use L methods with respect to these sequences of measures: we canprove rigorously that our L maximixation procedure leads to the same asymptoticas the L ∞ maximization that appears in the definitions of the transfinite diameter(or other objects in Pluripotential Theory), this is due to the sampling property ofadmissible meshes. On the other hand the slow increasing cardinality of the admis-sible meshes guarantees that the complexity of the computations is not growing toofast.We warn the reader that, though all proposed examples and tests are for realsets E ⊂ R n ⊂ C n , our results hold in the general case E ⊂ C n . This choice hasbeen made essentially for two reasons: first, the main examples for which we haveanalytical expression to compare our computation with are real, second, the caseof E ⊂ R is both computationally less expensive and easier from the point of viewof representing the obtained results.The methods we are introducing in the present work are suitable to be extendedin at least three directions. First, one may consider weighted pluripotential theory (see for instance [12]) instead of the classical one: the theoretical results we provehere can be recovered in such a more general setting by some modifications. Itis worth to mention that a relevant part of the proofs of our results rest upon thisweighted theory even if is not presented in such a framework, since we extensivelyuse the results of the seminal paper [6]. However, in order to produce the samealgorithms in the weighted framework, one needs to work with weighted polyno-mials, i.e., functions of the type p ( z ) w deg p ( z ) for a given weight function w andtypically E needs no more to be compact in this theory, these changes cause some LURIPOTENTIAL NUMERICS 5 theoretical di ffi culties in constructing suitable admissible meshes and may carrynon trivial numerical issues as well.Second, we recall that pluripotential theory has been developed on certain ”lowerdimensional sets” of C n as sub-manifolds and a ffi ne algebraic varieties. If E is acompact subset of an algebraic subset A of C n , then one can extend the pluripo-tential theory of the set A reg of regular points of A to the whole variety and usetraces of global polynomials in C n to recover the extremal plurisubharmonic func-tion V ∗ E ( z , A ); see [40]. This last direction is probably even more attractive, due tothe recent development of the theory itself especially when E lies in the set of realpoints of A , see for instance [29] and [7].Lastly, we mention an application of our methods that is ready at hand. Very re-cently polynomial spaces with non-standard degree ordering (e.g., not total degreenor tensor degree) start to attract a certain attention in the framework of randomsampling [22], Approximation Theory [46], and Pluripotential Theory [18]. Forinstance, one can consider spaces of polynomials of the form P kq : = span { z α , α i ∈ N n , q ( α ) < k } , where q is any norm or even, more generally, P kP : = span { z α , α i ∈ N n , α ∈ kP } for any P ⊂ R n + closed and star-shaped with respect to 0 such that ∪ k ∈ N kP = R n + . Since this spaces are being used only very recently, many theoreti-cal questions, from the pluripotential theory point of view, arise. Our methods canbe used to investigate conjectures in this framework by a very minor modificationof our algorithms.The paper is structured as follows. In
Section 2 we introduce admissible meshes and all the definitions and the tools we need from
Pluripotential Theory .Then we present our algorithms of approximation: for each of them we provethe convergence and we illustrate their implementation and their performances bysome numerical tests; we stress that, despite the fact that we will consider onlycases of E ⊂ R for relevance and simplicity, our techniques are fine for general E ⊂ C n . We consider the extremal function V ∗ E (Pluripotential Theory counterpartof the Green function, see (3) below) in Section 3 , the transfinite diameter δ ( E ) in Section 4 and the pluripotential equilibrium measure µ E : = (cid:16) dd c V ∗ E (cid:17) n in Section5 . All the experiment are performed with the MATLAB software
PPN package ,seehttp: // / ∼ fpiazzon / software.2. P reliminaries Pluripotential theory: some definitions.
Let Ω ⊂ C be any domain and u : Ω → R ∪ {−∞} , u is said to be subharmonic if u is upper semicontinuous and u ( z ) ≤ π r (cid:82) | z − ζ | = r u ( ζ ) ds ( ζ ) for any r > z ∈ Ω such that B ( z , r ) ⊆ Ω . A function u : Ω → R ∪ {−∞} , where Ω ⊂ C n , is said to be plurisubharmonic if u is upper semicontinuous and is subharmonic along each complex line (i.e., eachcomplex one dimensional a ffi ne variety); the class of such functions is usually de-noted by PSH ( Ω ) . It is worth to stress that the class of plurisubharmonic functionsis strictly smaller than the class of subharmonic function on Ω as a domain in R n . FEDERICO PIAZZON
The
Lelong class of plurisubharmonic function with logarithmic growth at in-finity is denoted by L ( C n ) and u ∈ L ( C n ) i ff u ∈ PSH ( C n ) is a locally boundedfunction such that u ( z ) − log | z | is bounded near infinity.Let E ⊂ C n be a compact set. The extremal function V ∗ E (also termed pluricom-plex Green function) of E is defined mimicking one of the possible definitions in C of the Green function with pole at infinity; see (1). V E ( ζ ) : = sup { u ( ζ ) ∈ L ( C n ) , u | E ≤ } , (4) V ∗ E ( z ) : = lim sup ζ → z V E ( ζ ) . (5)It turns out that the extremal function enjoys the same relation with polynomials ofthe Green function; precisely Siciak introduced [44]˜ V E ( ζ ) : = sup (cid:40) p log | p ( ζ ) | , p ∈ P , (cid:107) p (cid:107) E ≤ (cid:41) , (6) ˜ V ∗ E ( z ) : = lim sup ζ → z V E ( ζ ) . (7)and shown (the general statement has been proved by Zaharjuta) that ˜ V ∗ E ≡ V ∗ E and˜ V E = q.e. V E . Here q.e. stands for quasi everywhere and means for each z ∈ C n \ P where P is a pluripolar set , i.e., a subset of the {−∞} level set of a plurisubharmonicfunction not identically −∞ . In the case that V ∗ E is also continuous, the set E istermed L -regular.A remarkable consequence of this equivalence is that the Bernstein Walsh In-equality (3) and Theorem (see [44]) hold even in the several complex variablesetting simply replacing g E ( z , ∞ ) by V ∗ E ( z ); see for instance [27].In Pluripotential Theory the role of the Laplace operator is played by the com-plex Monge Ampere operator (dd c ) n . Here dd c = i ¯ ∂∂ , where ∂ u ( z ) : = (cid:80) nj = ∂ u ( z ) ∂ z j dz j ,¯ ∂ u ( z ) : = (cid:80) nj = ∂ u ( z ) ∂ ¯ z j d ¯ z j and (dd c u ) n = dd c u ∧ dd c u · · · ∧ dd c u . For u ∈ C ( C n , R )one has (dd c u ) n = c n det[ ∂ u /∂ z i ∂ ¯ z j ] d Vol C n . The Monge Ampere operator extendsto locally bounded plurisubharmonic functions as shown by Bedford and Taylor[4], [3], (dd c u ) n being a positive Borel measure.The equation (dd c u ) n = Ω (in the sense of currents) characterizethe maximal plurisubharmonic functions ; recall that u is maximal if for any openset Ω (cid:48) ⊂⊂ Ω and any v ∈ PSH ( Ω ) such that v | ∂ Ω (cid:48) ≤ u | ∂ Ω (cid:48) we have v ( z ) ≤ u ( z ) forany z ∈ Ω (cid:48) . Given a compact set E ⊂ C n , two situations may occur: either V ∗ E ≡ + ∞ , or V ∗ E is a locally bounded plurisubharmonic function. The first case is when E ispluripolar, it is too small for pluripotential theory . In the latter case (cid:16) dd c V ∗ E (cid:17) n = C n \ E (i.e., V ∗ E is maximal on such a set), in other words the positive measure (cid:16) dd c V ∗ E (cid:17) n is supported on E . Such a measure is usually denoted by µ E and termedthe (pluripotential) equilibrium measure of E by analogy with the one dimensionalcase. LURIPOTENTIAL NUMERICS 7
Let us introduce the graded lexicographical strict order ≺ on N n . For any α, β ∈ N n we have α ≺ β if | α | < | β | or | α | = | β | α (cid:44) βα ¯ j < β ¯ j , (8) where ¯ j : = min { j ∈ { , , . . . , n } : α j (cid:44) β j } . (9)This is clearly a total (strict) well-order on N n and thus it induces a bijective map α : N −→ N n . On the other hand the map e : N n → P ( C n )(10) α (cid:55)→ e α ( z ) : = z α · z α · · · · · z α n n (11)is a isomorphism having the property that deg e α ( z ) = | α | . Thus, if we denote by e i ( z ) the i -th monomial function e α ( i ) ( z ), we have P k ( C n ) = span { e i ( z ) , ≤ i ≤ N k } = : span M kn , where N k : = dim P k ( C n ) = (cid:32) n + kn (cid:33) . From now on we will refer to M kn as the graded lexicographically ordered mono-mial basis of degree k . For any array of points { z , . . . z N k } ⊂ E N k we introduce the Vandermonde deter-minant of order k as Vdm k ( z , . . . z N k ) : = det[ e i ( z j )] i , j = ,..., N k . If a set { z , . . . , z N k } of points of E satisfies (cid:12)(cid:12)(cid:12) Vdm k ( z , . . . z N k ) (cid:12)(cid:12)(cid:12) = max ζ ,... ζ Nk ∈ E (cid:12)(cid:12)(cid:12) Vdm k ( ζ , . . . ζ N k ) (cid:12)(cid:12)(cid:12) it is said to be a array of Fekete points for E . Clearly Fekete points do not need tobe unique. Zaharjuta proved in his seminal work [48] that the sequence δ k ( E ) : = max ζ ,... ζ Nk ∈ E (cid:12)(cid:12)(cid:12) Vdm k ( ζ , . . . ζ N k ) (cid:12)(cid:12)(cid:12) n + nkNk does have limit and defined, by analogy with the case n =
1, the transfinite diameter of E as(12) δ ( E ) : = lim k δ k ( E ) . It turns out that the condition δ ( E ) = C n as itcharacterize polar subset of C . Let { z ( k ) } k ∈ N : = { ( z ( k )1 , . . . , z ( k ) N k ) } k ∈ N be a sequence of Fekete points for the com-pact set E , we consider the canonically associated sequence of uniform probabilitymeasures ν k : = (cid:80) N k j = N k δ z ( k ) j . FEDERICO PIAZZON
Berman and Boucksom [5] showed that the sequence ν k converges weak star tothe pluripotential equilibrium measure µ E as it happens in the case n = . We willuse both this result (in Section 5) and a remarkable intermediate step of its proof(in Section 3 and Section 4) termed
Bergman Asymptotic , see (21) below.2.2.
Admissible meshes and Bernstein Markov measures.
We recall that a com-pact set E ⊂ R n (or C n ) is said to be polynomial determining if any polynomialvanishing on E is necessarily the null polynomial.Let us consider a polynomial determining compact set E ⊂ R n (or C n ) and let A k be a subset of E . If there exists a positive constant C k such that for any polynomial p ∈ P k ( C n ) the following inequality holds(13) (cid:107) p (cid:107) E ≤ C k (cid:107) p (cid:107) A k , then A k is said to be a norming set for P k ( C n ) . Let { A k } be a sequence of norming sets for P k ( C n ) with constants { C k } , supposethat both C k and Card( A k ) grow at most polynomially with k (i.e., max { C k , Card( A k ) } = O ( k s ) for a suitable s ∈ N ), then { A k } is said to be a weakly admissible mesh (WAM) for E ; see [21]. Observe that necessarily(14) Card A k ≥ N k : = dim P k ( C n ) = (cid:32) k + nk (cid:33) = O ( k n )since a (W)AM A k is P k ( C n )-determining by definition.If C k ≤ C ∀ k , then { A k } N is said an admissible mesh (AM) for E ; in the sequel,with a little abuse of notation, we term (weakly) admissible mesh not only thewhole sequence but also its k -th element A k . When Card( A k ) = O ( k n ), followingKro´o [26], we refer to { A k } as an optimal admissible mesh , since this grow rate forthe cardinality is the minimal one in view of equation (14).Weakly admissible meshes enjoy some nice properties that can be also usedtogether with classical polynomial inequalities to construct such sets. For instance,WAMs are stable under a ffi ne mappings, unions and cartesian products and wellbehave under polynomial mappings. Moreover any sequence of interpolation nodeswhose Lebesgue constant is growing sub-exponentially is a WAM.It is worth to recall other nice properties of (weakly) admissible meshes. Namely,they enjoy a stability property under smooth mapping and small perturbations bothof E and A k itself; [34]. For a survey on WAMs we refer the reader to [16].Weakly admissible meshes are related to Fekete points. For instance assume aFekete triangular array { z ( k ) } = { ( z ( k )1 , . . . , z ( k ) N k ) } for E is known, then setting A k : = z ( k log k ) for all k ∈ N we obtain an admissible mesh for E ; [11].Conversely, if we start with an admissible mesh { A k } for E it has been provedin [19] that it is possible to extract (by numerical linear algebra) a set z ( k ) : = The results of [5] and [6] hold indeed in the much more general setting of high powers of a linebundle on a complex manifold. The original definition in [21] is actually slightly weaker (sub-exponential growth instead ofpolynomial growth is allowed), here we prefer to use the present one which is now the most commonin the literature.
LURIPOTENTIAL NUMERICS 9 { z ( k )1 , . . . , z ( k ) N k } ⊂ A k from each A k such that the sequence { z ( k ) } is an asymptoticallyFekete sequence of arrays, i.e.,(15) (cid:12)(cid:12)(cid:12)(cid:12) Vdm k ( z ( k )1 , . . . , z ( k ) N k ) (cid:12)(cid:12)(cid:12)(cid:12) n + nkNk → δ ( E )as happens for Fekete points. By the deep result of Berman and Boucksom [6] (andsome further refining, see [10]) it follows that the sequence of uniform probabilitymeasures { ν k } canonically associated to z ( k ) converges weak star to the pluripoten-tial equilibrium measure.In [6] authors pointed out, among other deep facts, the relevance of a class ofmeasures for which a strong comparability of uniform and L norms of polyno-mials holds, they termed such measures Bernstein Markov measures. Precisely,a Borel finite measure µ with support S µ ⊆ E is said to be a Bernstein Markovmeasure for E if we have(16) lim sup k sup p ∈ P k \{ } (cid:107) p (cid:107) E (cid:107) p (cid:107) L µ / k ≤ . Let us denote by { q j ( z , µ ) } , j = , . . . , N k the orthonormal basis (obtained byGram-Schmidt orthonormalizaion starting by M kn ) of the space P k endowed bythe scalar product of L µ . The reproducing kernel of such a space is K µ k ( z , ¯ z ) : = (cid:80) N k j = ¯ q j ( z , µ ) q j ( z , µ ), we consider the related Bergman functionB µ k ( z ) : = K µ k ( z , z ) = N k (cid:88) j = | q j ( z , µ ) | . As a side product of the proof of the asymptotic of Fekete points Berman Bouck-som and Nystr¨om deduces the so called
Bergman Asymptotic (17) B µ k N k µ (cid:42) ∗ µ E , for any positive Borel measure µ with support on E and satisfying the BernsteinMarkov property.Note that, by Parseval Identity, the property (16) above can be rewritten as(18) lim sup k (cid:107) B µ k (cid:107) / (2 k ) E ≤ . Bernstein Markov measures are very close to the
Reg class defined (in the com-plex plane) by Stahl and Totik and studied in C n by Bloom [8]. Recently BernsteinMarkov measures have been studied by di ff erent authors, we refer the reader to[15] for a survey on their properties and applications.Our methods in the next sections rely on the fact admissible meshes are gooddiscrete models of Bernstein Markov measures ; let us illustrate this. From now on • we assume E ⊂ C n to be a compact L -regular set and hence polynomialdetermining, • we denote by µ k the uniform probability measure supported on A k = { z ( k )1 , . . . , z ( k ) M k } ,i.e., µ k : = A k Card A k (cid:88) j = δ z ( k ) j , • we denote by B k ( z ) the function B µ k k ( z ) and by K k ( z , ζ ) the function K µ k k ( z , ζ ) . Assume that an admissible mesh { A k } of constant C for the compact polynomialdetermining set E ⊂ C n is given. Now pick ˆ z ∈ E such that B k (ˆ z ) = max E B k , wenote that B k (ˆ z ) = N k (cid:88) j = c j q j (ˆ z , µ ) : = p (ˆ z ) , c j : = ¯ q j (ˆ z , µ ) . By Parseval inequality we have (cid:107) p (cid:107) E ≤ N k (cid:88) j = | c j | / max z ∈ E N k (cid:88) j = | q j ( z , µ ) | / = B k (ˆ z ) = (cid:107) B k (cid:107) E . Therefore we an write (cid:107) B k (cid:107) E = p (ˆ z ) ≤ (cid:107) p (cid:107) E ≤ C (cid:107) p (cid:107) A k ≤ C (cid:112) B k (ˆ z ) (cid:107) B k (cid:107) / E , thus(19) (cid:107) B k (cid:107) E ≤ C (cid:107) B k (cid:107) A k . On the other hand for any polynomial p ∈ P k we have (cid:107) p (cid:107) E ≤ C (cid:107) p (cid:107) A k ≤ C (cid:112) Card A k (cid:107) p (cid:107) L µ k . Recall that it follows by the definition of (weakly) admissible meshes that ( C Card A k ) / (2 k ) → . Thus, the sequence of probability measures associated to the mesh has the prop-erty (20) lim sup k (cid:107) B k (cid:107) / (2 k ) E ≤ lim sup k ( C (cid:112) Card A k ) / k = , which closely resembles (18).Conversely, assume { µ k } to be a sequence of probabilities on E with Card supp µ k = O ( k s ) for some s , then we have (cid:107) p (cid:107) E ≤ (cid:112) (cid:107) B k (cid:107) E (cid:107) p (cid:107) supp µ k , ∀ p ∈ P k . Therefore, if (cid:107) B k (cid:107) E = O ( k t ) for some t , the sequence of sets { supp µ k } is a weaklyadmissible mesh for E . LURIPOTENTIAL NUMERICS 11
3. A pproximating the extremal function
Theoretical results.
In this section we introduce certain sequences of func-tions, namely u k , v k , ˜ u k and ˜ v k , that can be constructed starting by a weakly admis-sible mesh, all of them having the property of local uniform convergence to V ∗ E , provided E is L -regular. Theorem 3.1.
Let E ⊂ C n be a compact L -regular set and { A k } a weakly admissi-ble mesh for E, then, uniformly in C n , we have lim k v k : = lim k k log B k = V ∗ E , (21) lim k u k : = lim k k log (cid:90) E | K k ( · , ζ ) | d µ k ( ζ ) = V ∗ E . (22) Proof.
We first prove (21), for we introduce F ( k ) E : = { p ∈ P k : (cid:107) p (cid:107) E ≤ } log Φ ( k ) E ( z ) : = sup (cid:40) k log | p ( z ) | , p ∈ F ( k ) E (cid:41) . The sequence of function Φ ( k ) E has been defined by Siciak and has been shown toconverge to exp ˜ V ∗ E (see equation (7)) for E L -regular, moreover we have V ∗ E ≡ ˜ V ∗ E ;[44], see also [43].Let us denote by F ( k )2 the family { p ∈ P k : (cid:107) p (cid:107) L µ k ≤ } we notice that, due tothe Parseval Identity, we have B k ( z ) = sup p ∈F ( k )2 | p ( z ) | . Let us pick p ∈ F ( k )2 , we have (cid:107) p (cid:107) E ≤ √(cid:107) B k (cid:107) E (cid:107) p (cid:107) L µ k for the reason above, thus q : = p (cid:107) B k (cid:107) − / E ∈ F ( k ) E . Hencelog Φ ( k ) E ( z ) ≥ k log | q ( z ) | = k log | p ( z ) | − k log (cid:107) B k (cid:107) E , ∀ p ∈ F ( k )2 . It follows that log Φ ( k ) E ( z ) + k log (cid:107) B k (cid:107) E ≥ v k ( z ) . On the other hand, since µ k is a probability measure, we have (cid:107) p (cid:107) E ≥ (cid:107) p (cid:107) L µ k for any polynomial. Hence if p ∈ F ( k ) E it follows that p ∈ F ( k )2 . Thus v k ( z ) ≥ log Φ ( k ) E ( z ) . Therefore we havelog Φ ( k ) E ( z ) + k log (cid:107) B k (cid:107) E ≥ v k ( z ) ≥ log Φ ( k ) E ( z ) . Note that we have lim sup k (cid:107) B k (cid:107) / kE ≤ { A k } is weakly admissible (seeequation (20)), hence we can conclude that locally uniformly we have V ∗ E ( z ) ≤ lim inf k (cid:32) log Φ ( k ) E ( z ) − k log (cid:107) B k (cid:107) E (cid:33) ≤ lim inf v k ( z ) ≤ lim sup v k ( z ) ≤ lim sup k log Φ ( k ) E ( z ) = V ∗ E ( z ) . This concludes the proof of (21), let us prove (22).It follows by Cauchy-Schwarz and Holder Inequalities and by (cid:82) B k d µ k = N k that (cid:90) | K k ( z , ζ ) | d µ k ( ζ ) ≤ (cid:90) N k (cid:88) j = | q j ( z , µ k ) | / N k (cid:88) j = | q j ( ζ, µ k ) | / d µ k ( ζ ) ≤ (cid:13)(cid:13)(cid:13)(cid:13) (cid:112) B k ( ζ ) (cid:13)(cid:13)(cid:13)(cid:13) L µ k · (cid:112) B k ( z ) ≤ N / k (cid:112) B k ( z ) . Thus it follows that(23) u k ( z ) ≤ k log[ N k ] + v k ( z ) uniformly in C n . On the other hand, for any p ∈ P k we have | p ( z ) | = (cid:12)(cid:12)(cid:12)(cid:12) (cid:104) K k ( z , ζ ); p ( ζ ) (cid:105) L µ k (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) K k ( z , ζ ) p ( ζ ) d µ k ( ζ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:107) p (cid:107) L ∞ µ k (cid:90) | K k ( z , ζ ) | d µ k ( ζ ) ≤ (cid:107) p (cid:107) E (cid:90) | K k ( z , ζ ) | d µ k ( ζ ) , hence, using the definition of Siciak function, (cid:90) | K µ k k ( z , ζ ) | d µ k ( ζ ) ≥ sup p ∈ P k \{ } | p ( z ) |(cid:107) p (cid:107) E = ( Φ ( k ) E ) k . Finally, using (23), we havelog Φ ( k ) E ( z ) ≤ u k ( z ) ≤ v k ( z ) + log N / (2 k ) k , uniformly in C n , this concludes the proof of (22) since N / (2 k ) k → v k andlog Φ ( k ) E converge to V ∗ E uniformly in C n . (cid:3) It is worth to notice that both u k and v k are defined in terms of orthonormalpolynomials with respect to µ k , hence they can be computed with a finite number LURIPOTENTIAL NUMERICS 13 of operations at any point z ∈ C n , indeed we have v k ( z ) = k log (cid:90) | K k ( z , ζ ) | d µ k ( ζ ) = k log A k Card A k (cid:88) h = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N k (cid:88) j = q j ( z , µ k ) ¯ q j ( ζ h , µ k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (24)Also we note that Theorem 3.1 can be understood as a generalization of the originalSiciak statement [44, Th. 4.12]. Indeed, if we take A k = { z ( k )1 , . . . , z ( k ) N k } a set ofFekete points of order k for E we get q j ( z , µ k ) = √ N k (cid:96) j , k ( z ), where (cid:96) j , k ( z ) is the j -th Lagrange polynomial, hence we have1 N k N k (cid:88) h = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N k (cid:88) j = q j ( z , µ k ) ¯ q j ( ζ h , µ k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = √ N k N k (cid:88) h = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N k (cid:88) j = q j ( z , µ k ) δ | j − h | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = √ N k N k (cid:88) h = (cid:12)(cid:12)(cid:12) q h ( z , µ k ) (cid:12)(cid:12)(cid:12) = N k (cid:88) h = | (cid:96) h , k ( z ) | = : Λ A k ( z ) . Here Λ A k ( z ) is the Lebesgue function of the interpolation points A k . Therefore, for A k being a Fekete array of order k , we have u k ( z ) = log (cid:0) Λ A k ( z ) (cid:1) / k , this is preciselyexp (cid:16) k Φ (2) k ( z ) (cid:17) in the Siciak notation.In Section 5 we will deal with measures of the form ν k : = B µ k N k µ for a BernsteinMarkov measure µ for E , or, more generally ν k : = B µ kk N k µ k , where the sequence { µ k } has the property lim sup k (cid:107) B µ k k (cid:107) / kE =
1; we refer to such a sequence { µ k } asa Bernstein Markov sequence of measures.
Due to a modification of the BermanBoucksom and Nymstrom result, ν k converges weak star to µ E (see Proposition 5.1below). Note that ν k is still a probability measure since B k ( z ) ≥ z ∈ C n and (cid:90) E d ν k = N k (cid:90) E B k d µ k = N k N k (cid:88) j = (cid:107) q j ( z ) (cid:107) L µ k = . Here we point out another (easier, but very useful to our aims) property of the”weighted” sequence ν k : actually they are a Bernstein Markov sequence of mea-sure, more precisely the following theorem holds. Theorem 3.2.
Let E ⊂ C n be a compact L -regular set and { A k } a weakly admissi-ble mesh for E . Let us set ˜ µ k : = B µ kk N µ k , where µ k is the uniform probability measureon A k . Then the following holds.i) For any k ∈ N and any z ∈ C n (25) B ˜ µ k k ( z ) ≤ N min E B k B µ k k ( z ) ≤ N B µ k k ( z ) . Thus lim sup k (cid:107) B ˜ µ k k (cid:107) / kE = . ii) We have lim k ˜ v k : = lim k k log B ˜ µ k k = V ∗ E , (26) lim k ˜ u k : = lim k k log (cid:90) E | K ˜ µ k k ( · , ζ ) | d ˜ µ k ( ζ ) = V ∗ E , (27) uniformly in C n . From now on we use the notations˜ B k : = B ˜ µ k k ( z ) , ˜ µ k : = B k N µ k , where B k is as above B µ k k and µ k will be clarified by the context. Proof.
We prove (25), then lim sup k (cid:107) ˜ B k (cid:107) / kE = k (cid:107) B k (cid:107) / kE = k N / kk = . The proof of (26) and (27) is identical to the ones of Theorem3.1 so we do not repeat them.We simply notice that, for any sequence of polynomials { p k } with deg p k ≤ k ,we have (cid:107) p k (cid:107) L µ k = (cid:90) E | p k ( z ) | d µ k = (cid:90) E NB k ( z ) | p k ( z ) | B k ( z ) N d µ k ≤ max E NB k (cid:90) E | p k ( z ) | B k ( z ) N d µ k = N min E B k (cid:107) p k (cid:107) L µ k . Now, for any z ∈ E , we pick a sequence { p k } such that it maximizes (for any k ) theratio ( | q ( z ) | (cid:107) q (cid:107) − L µ k ) / k among q ∈ P k and we get˜ B k ( z ) / = | p k ( z ) |(cid:107) p k (cid:107) L µ k ≤ N min E B k | p k ( z ) |(cid:107) p k (cid:107) L µ k ≤ N min E B k B k ( z ) / . Here the last inequality follows by the definition of B k . Note in particular that B k ( z ) = + | q ( z ) | + . . . , thus N min E B k ≤ N = O ( k n ) and (cid:107) ˜ B k (cid:107) / kE ≤ N / k (cid:107) B k (cid:107) / kE ∼(cid:107) B k (cid:107) / kE as k → ∞ . (cid:3) Remark 3.3.
We stress that the upper bound (25) is in many cases quite rough,though su ffi cient to prove the convergence result (26) . Indeed, since (cid:82) B k d µ k = for any k, it follows that N min E B k is always larger than , but we warn the reader that (cid:107) ˜ B k (cid:107) E does not need to be larger than (cid:107) B k (cid:107) E in general . Hence the measure ˜ µ k maybe more suitable than µ k for our approximation purposes. The SZEF and SZEF-BW algorithms.
The function V ∗ E , at least for a regu-lar set E , can be characterized as the unique continuous solution of the followingproblem (dd c u ) n = , in C n \ Eu ≡ , on Eu ∈ L ( C n ) . LURIPOTENTIAL NUMERICS 15
It is rather clear that writing a pseudo-spectral or a finite di ff erences scheme forsuch a problem is a highly non trivial task, as one needs to deal with a unboundedcomputational domain C n \ E , with a positivity constraint on (dd c u ) n , and with aprescribed growth rate at infinity (both encoded by u ∈ L ( C n )).Here we present the SZEF and
SZEF-BW algorithms (which stands for forSiciak Zaharjuta Extremal Function and Siciak Zaharjuta Extremal Function byBergman weight) to compute the values of the functions u k and v k (see Theorem3.1) and the functions ˜ u k and ˜ v k (see Theorem 3.2) respectively at a given set ofpoints. In our methods both the growth rate and the plurisubharmonicity are en-coded in the particular structure of the approximated solutions u k or v k , while theunboundedness of C n \ E does not carry any issue since all the sampling points usedto build the solutions lie on E . Indeed, once the approximated solution is computedon a set of points and the necessary matrices are stored, it is possible to compute u k or v k on another set of points by few very fast matrix operations; this will be moreclear in a while.To implement our algorithms we make the following assumptions. • Let E ⊂ C n be a compact regular set, for simplicity let us assume E tobe a real body (i.e., the closure of a bounded domain), but notice that thisassumption is not restrictive neither from the theoretical nor from the com-putational point of view. • We further assume that we are able to compute a weakly admissible mesh { A k } = { z , . . . , z M k } for E with constants C k , k ∈ N . Note that an algo-rithmic construction of an admissible mesh is available in the literature forseveral classes of sets [36, 35, 33], since the study of (weakly) admissiblemeshes is attracting certain interest during last years. • We assume n = • Let us fix a computational grid { ζ (1)1 , . . . , ζ ( L )1 } × { ζ (1)2 , . . . , ζ ( L )2 } = : Ω ⊂ C with finite cardinality L : = L · L and let us denote by Ω E the (possiblyempty) set Ω ∩ E , while by Ω the set Ω \ Ω E . We will reconstruct thevalues u k ( ζ ) , v k ( ζ ) ∀ ζ ∈ Ω , however we will test the performance of ouralgorithm (see Subsection 3.3 below) only in term of error on Ω . Thischoice is motivated as follows. First, we have to mention the fact thatthe point-wise error of u k and v k exhibits two rather di ff erent behavioursdepending whether the considered point ζ lies on E or not: the convergencefor ζ ∈ E is much slower. In contrast, for any regular set E , the function V ∗ E identically vanishes on E , hence there is no point in trying to approximateit on E . Note that the function V ∗ E ( ζ ) (cid:44) ζ ∈ C n \ E , thus we canmeasure the error of u k of v k on Ω ⊂ C \ E both in the absolute and inthe relative sense. • Let L : C → C be the invertible a ffi ne map, mapping A k in the square[ − , and defined by P i ( z ) : = b i − a i (cid:16) z i − a i + b i (cid:17) , a i : = min z ∈ A k z i , b i : = max z ∈ A k z i . We denote by T j ( z ) the classical j -th Chebyshev polynomialand we set φ j ( z ) : = T α ( j ) ( P ( z )) T α ( j ) ( P ( z )) , ∀ j ∈ N , j > , where α : N → N is the one defined in (11). The set T kP : = { φ j ( z ) , ≤ j ≤ ( k + k + / } is a (adapted Chebyshev) basis of P k ( C ) . We will use thebasis T kP for computing and orthogonalizing the Vandermonde matrix ofdegree k computed at A k . This choice has been already fruitfully used, forinstance in [19, 16], when stable computations with Vandermonde matricesare needed and is on the background of the widely used matlab packageChebFun2 [45].3.2.1. SZEF Algorithm Implementation.
The first step of SZEF is the computation of the Vandermonde matrices with re-spect to the basis T kP VT : = ( φ j ( z i )) i = ,..., M k : = Card A k , j = ,..., ( k + k + / (28) WT : = ( φ j ( ζ i )) i = ,..., L , j = ,..., ( k + k + / . (29)Here VT is computed simply by the formula T h ( x ) = cos( h arccos( x )) while for WT we prefer to use the recursion algorithm to improve the stability of the computationsince the points P ( ζ i ) , ζ i ∈ Ω , may in general lie outside of [ − , or even becomplex.The second step of the algorithm is the most delicate: we perform the orthonor-malization of VT and store the triangular matrix defining the change of basis. Moreprecisely the orthonormalization procedure is performed by applying the QR algo-rithm twice (following the so called twice is enough principle). First we apply theQR algorithm to VT and we store the obtained R , then we apply the QR algorithmto VT \ R , we obtain Q and R that we store. Here \ is the matlab backslash operator implementing the backward substitution; this is much more stable thanthe direct inversion of the matrix R , which may be ill conditioned. Note that Q i , j = p j ( z i ) , where, since Q is an orthogonal matrix, M k (cid:90) p j ( z ) ¯ p h ( z ) d µ k ( z ) = M (cid:88) i = p j ( z i ) ¯ p h ( z i ) = ( QQ T ) j , h = δ j , h . Therefore √ M k Q i , j = q j ( z i ) . Step three. We compute the orthonormal polynomials evaluated at the points { ζ , . . . ζ L } = Ω . Again we prefer the backslash operator rather than the matrixinversion to cope with the possible ill-conditioning of R and R : we compute W : = WT \ R \ R . Note that W i , j = p j ( ζ i ) and thus q j ( ζ i ) = W i , j √ M k . We also compute the matrix K = Q · W T , here we have K i , h = N k (cid:88) j = p j ( z i ) p j ( ζ h ) = M k N k (cid:88) j = q j ( z i ) q j ( ζ h ) . LURIPOTENTIAL NUMERICS 17
Step 4.
Finally we have(30) ( v k ( ζ ) , . . . , v k ( ζ L )) i = ,..., L = V : = k log N k (cid:88) j = M k W i , j i = ,..., L and(31) ( u k ( ζ ) , . . . , u k ( ζ L )) i = ,..., L = U : = k log M k (cid:88) i = | K i , h | h = ,..., L SZEF-BW Algorithm Implementation.
First the step 1 and step 2 of SZEF algorithm are performed.
Step 3.
The Bergman weight σ = ( B k / N ( z ) , . . . , B k ( z M ) / N ) is computed by σ i = M k N k N k (cid:88) j = Q i , j . Then the weighted Vandermonde matrix V ( w ) i , j : = √ σ i Q i , j is computed by a matrixproduct. Step 4.
We compute the orthonormal polynomials by another orthonormaliza-tion by the QR algorithm: V ( w ) = Q ( w ) · R ( w ) . We get˜ q j ( z i ) = (cid:112) M k (cid:113) σ − i V ( w ) i , j = : ˜ Q i , j . We also compute the evaluation of ˜ q j s at Ω by˜ q j ( ζ i ) = WT \ R \ R \ R ( w ) = : ˜ W i , j . Step 5.
We perform the step 4 of the SZEF algorithm, where Q and W arereplaced by ˜ Q and ˜ W . Numerical Tests of SZEF and SZEF-BW.
The extremal function V ∗ E can becomputed analytically for very few instances as real convex bodies and sub-levelsets of complex norms. For the unit real ball B Lundin (see for instance [25])proved that(32) V B ( z ) ≡ V ∗ B ( z ) =
12 log h ( (cid:107) z (cid:107) + |(cid:104) z , ¯ z (cid:105) − | ) , where h ( ζ ) : = ζ + (cid:112) − ζ denotes the inverse of the Joukowski function and mapsconformally the set C \ [ − ,
1] onto C \ { z ∈ C : | z | ≤ } . Let E be any real convex set, one can define the convex dual set E ∗ as E ∗ : = { x ∈ R n : (cid:104) x , y (cid:105) ≤ , ∀ y ∈ E } . Baran [1, 2] proved, that if E is a convex compact set symmetric with respect to 0and containing a neighbourhood of 0, the following formula holds.(33) V ∗ E ( z ) = sup { log | h ( (cid:104) z , w (cid:105) ) | , w ∈ extrE ∗ } . Here extrF is the set of points of F that are not interior points of any segmentlaying in F . In order to be able to compare our approximate solution with the true extremalfunction, we build our test cases as particular instances of the Baran’s Formula: • test case 1, E : = [ − , , • test case 2, E m the real m -agon centred at 0 , • test case 3, E ∞ : = B , the real unit disk.We measure the error with respect to the true solution V E in terms of e ( f , V ∗ E , Ω ) : = Ω Ω (cid:88) i = | f ( ζ i ) − V E ( ζ i ) | , which should be intended as a quadrature approximation for the L norm of theerror.Also we consider, as an approximation of the relative L error, the quantity e rel ( f , V ∗ E , Ω ) : = (cid:80) Card Ω i = | f ( ζ i ) − V E ( ζ i ) | (cid:80) Card Ω i = V E ( ζ i ) , and the following approximation of the error in the uniform norm e ∞ ( f , V ∗ E , Ω ) : = max ζ i ∈ Ω | f ( ζ i ) − V E ( ζ i ) | . In all the tests we performed the rate of convergence is experimentally sub-linear, i.e., (cid:107) s k + (cid:107) / (cid:107) s k (cid:107) → k → ∞ , where s k : = f k + − f k for f k being one of u k , ˜ u k , v k , ˜ v k and (cid:107) · (cid:107) being one of the pseudo-norms we used above for defining e and e ∞ . This slow convergence may be overcame by extrapolation at infinity withthe vector rho algorithm (see [20]). We present below experiments regarding boththe original and the accelerated algorithm.3.3.1.
Test case 1.
In this case equation (33) reads as V ∗ [ − , ( z ) : = max ζ ∈{− , } log | h ( ζ ) | . To build an admissible mesh for [ − , we can use the Cartesian product of anadmissible mesh X k for one dimensional polynomials up to degree k ; see [16]. Wecan pick X k : = − cos( θ j ) , θ j : = j π/ (cid:100) mk (cid:101) , with j = , , . . . , (cid:100) mk (cid:101) .First we compare the performance of our four approximations in terms of L error behaviour as k grows large. Also, in order to better understand the rate ofconvergence, we compute the ratios s k : = e ( f k + , f k + , Ω ) e ( f k + , f k , Ω ) , for f k in { u k , v k ˜ u k , ˜ v k } and Ω = [ − , . We report the results in Figure 1 and 2respectively. The profile of convergence is slow but monotone, indeed the asymp-totic constants s k become rather close to 1 . This suggest a sub-linear convergencerate. It is worth to say that in all the tests we made the point-wise error is muchsmaller far from E than for points that lie near to E : in the experiment reportedin Figure 1 and Figure 2 Ω is an equispaced real grid in [0 , , but the results are LURIPOTENTIAL NUMERICS 19 F igure
1. Comparison of the e ( · , Ω ) error of u k , ˜ u k , v k and ˜ v k for E = [ − , and Ω the 10000 points equispaced coordinate gridin [ − , -3 -2 -1 BergmanKernelw-Bergmanw-Kernel F igure
2. Comparison of the s k behaviour for f k being u k (Bergman), ˜ u k (w-Bergman), v k (Kernel) or ˜ v k (w-Kernel) for E = [ − , and Ω the 10000 points equispaced coordinate gridin [ − , . BergmanKernelw-Bergmanw-Kernel the same for larger bounds and becomes much better if Ω ∩ E = ∅ or when Ω is apurely imaginary set, i.e., Ω ∩ R = ∅ . Fortunately, even if the convergence rate shown by the experiment of Figure 1,the vector rho algorithm , see for instance [20], works e ff ectively on our approxima-tion sequences and actually allows to produce much better approximations than theoriginal sequences { u k } , { ˜ u k } , { v k } , { ˜ v k } . We report in Figure 3 a comparison amongthe errors, on [ − , and [0 , respectively, of the original sequence { u k } andthe accelerated sequence produced by the vector rho algorithm, together with alinear (with respect to log-log scale) fitting of the original errors. In contrast withthe sub-linear convergence enlightened above, the accelerated sequence exhibits(in the considered interval for k ) a super-quadratic behaviour.3.3.2. Test case 2.
To construct a weakly admissible mesh on the real regular m -agon E m : = conv (cid:40)(cid:32) (cid:33) , (cid:32) cos π m sin π m (cid:33) , . . . . . . , (cid:32) cos m − π m sin m − π m (cid:33)(cid:41) , we use the algorithm proposed in [17]. First the convex polygon is subdividedin non overlapping triangles, then an admissible mesh { ˆ A k , l } k for each triangle l , l = , , . . . , m is created by the Du ff y transformation; finally, the union { ˜ A k } : = {∪ ml = ˆ A k , l } of all meshes (with no point repeated) is an admissible mesh for thepolygon by definition.The resulting mesh ˜ A k has good approximation properties: for instance it canbe constructed in such a way to have a very small constant, however ˜ A k is nottailored to our problem. Points of ˜ A k cluster, as one could expect, at any cornerof the regular polygon, but also they cluster near the edges of each triangle inthe subdivision. This ”spurious” clustering is not coming from the geometry of theproblem, instead it is an e ff ect of the method we used to solve it. More importantly,this issue tends to deteriorate the convergence of SZEF algorithm. Let us brieflygive an insight on why this phenomena occurs in the following remark. Remark 3.4.
In Section 5 we will prove (see Proposition 5.1) that the sequence ofmeasures { ˆ µ k } : = B µ k k N k µ k (cid:42) ∗ µ E , where (cid:42) ∗ denotes the convergence in the weak ∗ topology on Borel measures. As-sume for simplicity that µ E is absolutely continuous with respect to the Lebesguemeasure m restricted to E and consider an admissible mesh that tends to clus-ter points on a ball B ( z , r ) ⊂ E for which µ E dm ( z ) , z ∈ B ( z , r ) , is very small.The asymptotic above implies in particular that B µ k k ( z ) will be very small for z ∈ supp µ k ∩ B ( z , r ) . On the other hand we have N − k (cid:82) B µ k d µ = for any measure and thus there willbe some points ˆ z ∈ supp µ k \ B ( z , r ) for which B µ k k (ˆ z ) is very large. Recall that the uniform convergence of Theorem 3.1 relies on the fact that (cid:107) B µ k k (cid:107) / kE → , thus we should aim to get an admissible mesh { A k } having small constant, whosecardinality is slowly increasing, and whose Bergman function is ”not too large”, LURIPOTENTIAL NUMERICS 21 F igure
3. Log-log plot of e ( ˜ u k , V ∗ E , Ω ), e rel ( ˜ u k , Ω ) and e ∞ ( ˜ u k , V ∗ E , Ω ) and the same quantities for the accelerated se-quence of approximations for E = [ − , and Ω the 10000points equispaced coordinate grid in [ − , (above) and in[0 , (below). -4 -3 -2 -1 absolute L errorlin fittingrelative L-1 errorlin fittingabsolute max errorlin fittingAcc. absolute L-1 errorAcc. relative L errorAcc. absolute max error -5 -4 -3 -2 -1 absolute L errorlin fittingrelative L-1 errorlin fittingabsolute max errorlin fittingAcc. absolute L-1 errorAcc. relative L errorAcc. absolute max error e.g., is ”not too oscillating”. From the observation above a good heuristic toachieve such an aim is to mimic the density of the equilibrium measure.In order to overcome this issue we can apply two strategies.The first one is to get rid of these extra points. To this aim we can use the AFPalgorithm [19] to extract a set of discrete Fekete points A k of order 2 k from ˜ A k . Let us motivate this choice. First A k has been shown (numerically) to be aweakly admissible mesh in many test cases, see [19] and reference therein. Also,even more importantly, the resulting Bergman function is (experimentally) muchless oscillating than the one that can be computed starting by ˜ A k and this turnsin a smaller uniform norm on E m of such a function. Lastly, heuristically speak-ing in view of Remark 3.4, we would like to use a mesh which is mimicking thedistribution of the equilibrium measure. We invite the reader to compare this lastdiscussion with [10] where the definition of optimal measures is introduced.We can also perform a di ff erent choice which rests upon Theorem 3.2. Indeedthe SZEF-BW algorithm uses a Bergman weighting of µ k to prevent the phenomenawe discussed in Remark 3.4. We tested our algorithm SZEF-BW for di ff erentvalues of m and several choices of Ω , here we consider the case m = . Again, the convergence is quite slow but (numerically) monotone and, apartfrom the region of Ω close to ∂ R n E , it is not a ff ected by the particular choice of Ω ; that is, we can appreciate numerically the global uniform convergence provedin Theorem 3.2. Moreover, extrapolation at infinity is successful even in this testcase. We report profiles of convergence relative to two possible choices of Ω inFigure 4.3.3.3. Test case 3.
Lastly we consider E = E ∞ : = { ( z , z ) : (cid:61) ( z i ) = , (cid:60) ( z ) + (cid:60) ( z ) ≤ } ; in such a case the true solution is computed by the Lundin Formula(32). We can easily construct an admissible mesh of degree k for the real unitdisk following [16], for, it is su ffi cient to take a set of Chebyshev-Lobatto points { η , η , . . . , η s } , s > k and set A k : = (cid:26) η h (cid:18) cos (cid:18) π j s (cid:19) , sin (cid:18) π j s (cid:19)(cid:19) , j = , , . . . , s − , h = , , . . . , s (cid:27) . We report the behaviour of the error and the convergence profile in Figure 5, againthe sub-linear rate of convergence is rather evident.Also, we compare the profile of the error function e k ( z ) : = | ˜ v k ( z ) − V ∗ E ∞ ( z ) | on dif-ferent two real dimensional squares in Figure 6. The global uniform convergenceof ˜ v k to V ∗ E ∞ theoretically proven in Theorem 3.2 reflects on our experiments: e k issmall (approximately 10 − for k ≥
30) and very flat away from E ∞ while it attainsits maximum on E ∞ with a fast oscillation near ∂ R E ∞ . Again, the extrapolation atinfinity improves the quality of our approximation.4. A pproximating the transfinite diameter
In this section we present a method for approximating the transfinite diameterof a real compact set.4.1.
Theoretical result.
Given any basis B k = { b , . . . , b N k } of the space P k anda measure µ inducing a norm on P k , we denote by G k ( µ, B k ) its Gram matrix ,namely we set G k ( µ, B k ) : = [ (cid:104) b i ; b j (cid:105) L µ ] i , j = ,..., N k . LURIPOTENTIAL NUMERICS 23 F igure
4. Log-log plot of e ( ˜ u k , V ∗ E , Ω ), e rel ( ˜ u k , Ω ) and e ∞ ( ˜ u k , V ∗ E , Ω ) and the same quantities for the accelerated se-quence of approximations Ω the 10000 points equispaced coor-dinate grid in [ − , (above), and in [0 , (below). -4 -3 -2 -1 absolute L errorlin fittingrelative L-1 errorlin fittingabsolute max errorlin fittingAcc. absolute L-1 errorAcc. relative L errorAcc. absolute max error -4 -3 -2 -1 absolute L errorlin fittingrelative L-1 errorlin fittingabsolute max errorlin fittingAcc. absolute L-1 errorAcc. relative L errorAcc. absolute max error The hermitian matrix G k ( µ, B k ) has square root, indeed introducing the generalizedVandermonde matrix V k ( µ, B k ) : = [ (cid:104) q i ( · , µ ); b j (cid:105) L µ ] i , j = ,..., N k , we have V k ( µ, B k ) H V k ( µ, B k ) = G k ( µ, B k ) . Note that, for µ being the uniform prob-ability measure supported on an array of unisolvent points of degree k , V k ( µ, B k ) isthe standard Vandermonde matrix for the basis B k and divided by √ N k . F igure
5. Comparison of e ( ˜ u k , Ω ), e rel ( ˜ u k , Ω ) and e ∞ ( ˜ u k , Ω ), theirlinear (in the log-log scale) fitting and the same quantities com-puted on the accelerated sequences of approximations for k = , . . . , E ∞ , the real unit disk, and Ω the 10000 points eq-uispaced coordinate grid in [ − , (above) and in [0 , (be-low). -3 -2 -1 absolute L errorlin fittingrelative L-1 errorlin fittingabsolute max errorlin fittingAcc. absolute L-1 errorAcc. relative L errorAcc. absolute max error -4 -3 -2 -1 absolute L errorlin fittingrelative L-1 errorlin fittingabsolute max errorlin fittingAcc. absolute L-1 errorAcc. relative L errorAcc. absolute max error We recall, see [13], the following relation between Gram determinants and L norms of Vandermonde determinants. (34) det G k ( µ, M k ) = N k ! (cid:90) · · · (cid:90) | det Vdm k ( ζ , . . . , ζ N k ) | d µ ( ζ ) . . . d µ ( ζ N k ) , LURIPOTENTIAL NUMERICS 25 F igure
6. Profile of the error | ˜ v k − V ∗ B ∞ | in logarithmic scale for k =
40, and Ω the 40000 points equispaced coordinate grid in[0 , (high-left), [100 , (above-right), (0 . i + [0 , (belowleft) and ( i + [0 , (below right)Where M k denotes the (graded lexicographically ordered) monomial basis.Here is the main result this section. Theorem 4.1.
Let E ⊂ C n be a compact L -regular set and { A k } a (weakly) admis-sible mesh for E then, denoting by µ k the uniform probability measure on A k , wehave (35) lim k (det G k ( µ k , M k )) n + nkNk = δ ( E ) . Proof.
We use equation (34). Since µ k is a probability measure, it follows thatdet G k ( µ k , M k ) / ≤ max ζ ,...,ζ Nk ∈ E | det Vdm k ( ζ , . . . , ζ N k ) | hence lim sup k (det G k ( µ, M k )) n + nkNk ≤ lim sup k (cid:32) max ζ ,...,ζ Nk ∈ E | det Vdm k ( ζ , . . . , ζ N k ) | (cid:33) n + nkNk ≤ lim k (cid:32) max ζ ,...,ζ Nk ∈ E | det Vdm k ( ζ , . . . , ζ N k ) | (cid:33) n + nkNk = δ ( E ) . (36)On the other hand, by the sampling property of admissible meshes it follows that,for any polynomial p of degree at most k we have (cid:107) p (cid:107) E ≤ C (cid:107) p (cid:107) A k ≤ C √ Card A k (cid:107) p (cid:107) L µ k . Thus, since det Vdm k ( ζ , . . . , ζ N k ) is a polynomial in each variable ζ i ∈ C n of de-gree not larger than k , we get(det G k ( µ k , M k )) n + nkNk = (cid:107) . . . (cid:107) Vdm k ( ζ , . . . , ζ N k ) (cid:107) L µ k . . . (cid:107) n + nkNk L µ k ≥ (cid:32) C √ Card A k (cid:107) . . . (cid:107) max z ∈ E Vdm k ( z , ζ . . . , ζ N k ) (cid:107) L µ k . . . (cid:107) L µ k (cid:33) n + nkNk ≥ . . . ≥ (cid:32) C √ Card A k (cid:33) n + nk (cid:32) max z ,..., z Nk ∈ E | det Vdm k ( z , . . . , z N k ) | (cid:33) n + nkNk Since (Card A k ) / k →
1, being { A k } weakly admissible, it follows thatlim inf k (det G k ( µ k , M k )) n + nkNk ≥ δ ( E ) . (cid:3) In principle Theorem 4.1 provides an approximation procedure for δ ( E ), given { A k } , however the straightforward computation of the left hand side of (35) leadsto stability issues. In the next subsection we present our implementation of analgorithm based on Theorem 4.1 and we discuss a possible way to overcome suchdi ffi culties.4.2. Implementation of the TD-GD algorithm.
We recall that we denote by T j ( z ) the classical j -th Chebyshev polynomial and we set T k : = { φ , . . . , φ N k } , where φ j ( z ) : = T α ( j ) ( π z ) T α ( j ) ( π z ) , ∀ j ∈ N , j > , where α : N → N is the one defined in (11) and π h is the h -coordinate projection.We denote by V k = V k ( A k , T k ) the Vandermonde matrix of degree k with respectthe mesh A k : = { ( x , y ) , . . . , ( x M k , y M k ) } and the basis T k , that is V k : = (cid:104) φ j ( z i ) (cid:105) i = ,..., M k , j = ,..., N k , similarly we define W k : = V k ( A k , M k ) where the chosen reference basis is thelexicographically ordered monomial one.Now we notice that, setting M k : = Card A k , (cid:104) m α , m β (cid:105) L µ k = M − k M k (cid:88) h = ( W k ) α, h ( W k ) h ,β , thus we have det G k ( µ k ) = det W t k W k M k . The direct application of this procedure leads to a unstable computation that actu-ally does not converge.On the other hand, the computation of the Gram determinant in the Chebyshevbasis, det G k ( µ k , T k ) : = det V t k V k M k , LURIPOTENTIAL NUMERICS 27 is more stable and we have(det G k ( µ k )) n + nkNk = (cid:32) det W t k W k M k (cid:33) n + nkNk = (cid:32) det P t k V t k V k P k M k (cid:33) n + nkNk = (det( P k )) n + nkNk det G k ( µ, T k ) n + nkNk . Here the matrix P k is the matrix of the change of basis. Again the numericalcomputation of det P k becomes severely ill-conditioned as k grows large.Instead, our approach is based on noticing that P k does not depend on the par-ticular choice of E , thus we can compute the term (det( P k )) n + nkNk once we know(det G k ( ˆ µ k )) n + nkNk and (det ˜ G k ( ˆ µ k )) n + nkNk for a particular ˆ µ k which is a Bernstein Markovmeasure for ˆ E ⊆ [ − , as(37) (det( P k )) n + nkNk = (cid:32) det G k ( ˆ µ k )det ˜ G k ( ˆ µ k ) (cid:33) n + nkNk . Also we can introduce a further approximation, since det G k ( ˆ µ k ) n + nkNk → δ ( ˆ E ) , wereplace in the above formula det G k ( ˆ µ k ) n + nkNk by δ ( ˆ E ) . Finally, we pick ˆ E : = [ − , and ˆ µ k uniform probability measure on an admissible mesh for the square, forinstance the Chebyshev Lobatto grid with (2 k + points, thus our approximationformula becomes(38) δ ( E ) ≈ (cid:32) det ˜ G k ( µ k ) 1det ˜ G k ( ˆ µ k ) (cid:33) n + nkNk , where we used δ ([ − , ) = /
2; [9].Finally, to compute the determinants of the Gram matrices on the right handside of equation (38) we use the SVD factorization of the square root of the Grammatrices, note that for instancedet G k ( ˆ µ k ) = det (cid:32) M k V Hk V k (cid:33) = (det S k ) = N k (cid:89) j = σ j , where V k and M k has been defined above and S k = diag ( σ , . . . , σ N k ) is the diago-nal matrix with the singular values the matrix V k / √ M k . Numerical test of the TD-GD algorithm.
In order to illustrate how our al-gorithm works in practice, we perform two numerical tests for real compact setswhose transfinite diameters have been computed analytically in [9]. Namely, weconsider the case of the unit disk B = { x ∈ R : | x | ≤ } and the unit simplex S : = { x ∈ R + : x + x ≤ } For such sets Bos and Levenberg computed formulasthat in the specific case of dimension n = δ ( B ) = √ e , δ ( S ) = e . F igure
7. The admissible meshes ˆ A (left) and A (right) of de-gree 25 used below for the approximation of the transfinite diam-eter of the unit disk. −1 −0.5 0 0.5 1−1−0.8−0.6−0.4−0.200.20.40.60.81 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1−0.8−0.6−0.4−0.200.20.40.60.81 TD-GD test case 1: the unit ball in R . To compute the approximation of( G k ( ˆ µ k )) n + nkNk we pick ˆ µ k to be the probability measure on the point setˆ A k : = { (cos( i π/ (2 k )) , cos( j π/ (2 k ))) , i , j ∈ { , , . . . , k }} . This is a well known admissible mesh of constant 2 of the square [ − , forthe space of k tensor degree polynomials that in particular include the space P k ;[23, 16].For µ k we use an admissible mesh A k of degree k built as in [16] using the radialsymmetry of the unit disk. Here A k : = { cos( i π/ (2 k ))(cos( j π/ (2 k )) , sin( j π/ (2 k ))) , i , j ∈ { , , . . . , k }} . The admissible meshes A k and ˆ A k are displayed in Figure 7. We compute the righthand side of equation (38) for the sequence of values k = , , . . . ,
28 and we re-port both the absolute and the relative errors in Figure 8 (continuous line withoutand with diamonds respectively). On one hand we notice that the convergencerate is very slow, but on the other hand the sequence of approximations is mono-tone and the error structure is good for the application of the extrapolation. Indeedwe report the absolute and relative error (dashed line without and with diamondsrespectively) of the sequence obtained by the diagonal of the rho table (rho algo-rithm) in the same figure. Notice that the absolute error of the accelerated sequenceat degree 28 is 4 . · − , that is, six digits of δ ( B ) are computed exactly. TD-GD test case 2: the unit simplex in R . The first part of the algorithmfor the computation of δ ( S ), i.e., the computation of the factor in (38) comingfrom (37), is identical to the one we performed for δ ( B ) . Then we pick an admissible mesh on S following [16]. Our mesh A k at the k -thstage is the image under the Du ff y transformation of a Chebyshev grid on [ − , formed by (4 k + points, see Figure 9. We recall that the Du ff y transformation(with a suitable choice of parameters) maps the unit square onto the simplex andany degree k polynomial on the simplex is pulled back by the Du ff y transformation LURIPOTENTIAL NUMERICS 29 F igure
8. Behaviour of the absolute and relative error of the ap-proximation of the transfinite diameter of the unit disk by the for-mula (38) (continuous lines) and by the diagonal of the obtainedrho table (dashed lines). −6 −5 −4 −3 −2 −1 abs errrel errabs err accrel err acc F igure
9. The admissible meshes of degree 15 used below for theapproximation of the transfinite diameter of the unit simplex. −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8−0.4−0.3−0.2−0.100.10.20.30.4 onto the square to a polynomial of degree not larger than 2 k . It follows that { A k } isan admissible mesh of constant 2 for the simplex; [16]. Once the mesh has been defined the numerical computations to get the righthand side of (38) are performed as above. The obtained results, both in terms ofabsolute and relative errors, are displayed in Figure 10.Again, the defined algorithm is very slowly converging, nevertheless using ex-trapolation at infinity by the rho algorithm we get a sequence rather fast converging.Indeed, more than six exact digits of δ ( S ) can be computed in less than 10 secondseven on a rather outdated laptop , e.g., Intel CORE i3-3110M CPU, 4 Gb RAM.F igure
10. Behaviour of the absolute and relative error of the ap-proximation of the transfinite diameter of the unit simplex by theformula (38) (continuous lines) and by the third column of the ob-tained rho table (dashed lines). −7 −6 −5 −4 −3 −2 −1 abs errrel errabs err accrel err acc
5. A pproximating the equilibrium measure
Fekete points are (at least theoretically) the first tool to investigate how the equi-librium measure looks like for a given regular compact set E ⊂ C n . Indeed, themain result of [6] asserts that the sequence of uniform probability measures sup-ported at the k -th stage on a Fekete array of order k is converging weak ∗ to theequilibrium measure of the considered set. Unfortunately Fekete arrays are knownanalytically for very few instances and they are characterized in general as solu-tions of an extremely hard optimization problem, hence, though its strong theoret-ical motivation, this method is not of practical interest.However, the results in [6] are in fact more general (as shown also in [10]):one can take asymptotically Fekete arrays (see equation (15)) and obtain the same LURIPOTENTIAL NUMERICS 31 result. This is indeed the approach of [16, Th. 1], where the asymptotically Feketearrays are produced by a discretizing the optimization problem using an admissiblemesh as optimization domain. A 50-th stage of an asymptotic Fekete array for aregular hexagon is reported in Figure 11.F igure
11. A degree 50 asymptotic Fekete points set for a regularhexagon computed by the AFP algorithm. -1 -0.5 0 0.5 1-0.8-0.6-0.4-0.200.20.40.60.8
Another strategy to get a sequence of (weighted) point masses approaching theequilibrium measure in the weak ∗ sense is based on the Bergman Asymptotic (21)and the use of admissible meshes. We summarize this in the following proposition,which is a consequence of the results of [6] and [10]. Proposition 5.1.
Let E ⊂ C n a regular compact set. Let { A k } be a weakly admis-sible mesh for E and let µ k be the uniform probability measure supported on A k . Denote by ˜ µ k the measure N − k B µ k k . Then we ˜ µ k converges weak ∗ to µ E . Sketch of the proof.
First we note that the sequence of measures { µ k } leads to thetransfinite diameter , i.e. one has the asymptotic result (35) as proven above. Thisis the starting point for applying the machinery of [10].Indeed for such a sequence of measures one still has Lemma 2.7 and Lemma2.8 of [10], thus using the derivative of the Aubin-Mabuchi energy functional andLemma 3.1 of [6], one getslim k n + nN k (cid:90) E u ( z ) B µ k k d µ k ( z ) = n + n (cid:90) E u ( z ) d µ E , ∀ u ∈ C ( E ) . (cid:3) Approximations of the equilibrium measure built by means of point masses mayhave certain interest when one aims to perform approximated computations withequilibrium measure, for instance computing orthogonal series, since the approx-imation is given in terms of a quadrature rule. On the other hand, such a kind of approximation can not be easily represented to get an insight on how the equilib-rium measure looks like for a given E ; this property becomes relevant if one aimsto test or argue conjectures.In the rest of the section we will introduce an approximation scheme for µ E based on absolutely continuous measures with respect to the standard Lebesguemeasure.Our method is based on the following lemma. Lemma 5.1.
Let µ be a positive Borel measure. Let us set, for any k ∈ N , D = ( D k ( z , µ )) h , i : = ( ∂ i q h ( z , µ )) ∈ C N k × n b = ( b k ( z , µ )) i : = ( q ( z , µ ) , . . . q N k ( z , µ )) T ∈ C N k , then we have (39) det (cid:32) ∂ ¯ ∂ k log B µ k ( z ) (cid:33) = det( D H D ) − b H D adj( D H D ) D H b (2 k | b | ) n . Here adj denotes the adjugate of a matrix.Proof.
First notice that B µ k ( z ) = (cid:80) N k h = q h ( z , µ ) q h ( z , µ ) is a smooth function nevervanishing in C n , hence we can use classical di ff erentiation with no problems.We have ∂ ¯ ∂ k log B µ k ( z ) = k ∂ ¯ ∂ B µ k B µ k − ¯ ∂ B µ k B µ k ( ¯ B µ k ) T B µ k k = kB µ k ∂ ¯ ∂ B µ k + i ¯ ∂ B µ k (cid:113) B µ k i ( ¯ ∂ B µ k ) T (cid:113) B µ k . Also, using the linearity of di ff erentiation and the tensor structure of B µ k = b H b weget ∂ B µ k = D T ¯ b , ¯ ∂ B µ k = D H b , ∂ ¯ ∂ B µ k = D H D . So we can write ∂ ¯ ∂ k log B µ k ( z ) = k | b | (cid:32) D H D + iD H b | b | i ¯ bD | b | (cid:33) . Lastly we use the Matrix Determinant Lemma, i.e., det( A + uv T ) = det A + v T adj( A ) u , and the fact that det( λ A ) = λ n det A to get equation (39). (cid:3) We already shown, see Theorem 3.1, that, for the sequence { µ k } of uniformprobability measures supported on a weakly admissible mesh for E , one has theasymptotic lim k k log B µ k k ( z ) = V ∗ E ( z )locally uniformly. We recall also that the Monge Ampere operator is continuousunder the local uniform limit (see for instance [25]), thuslim k (cid:32) dd c k log B µ k k (cid:33) n = lim k (2 i ) n det (cid:32) ∂ ¯ ∂ k log B µ k k (cid:33) Vol C n = µ E , where the limit is to be intended in the sense of the weak ∗ topology of Borel mea-sures. Therefore we have the following. LURIPOTENTIAL NUMERICS 33
Theorem 5.1.
Let E ⊂ C n a regular compact set. Let { A k } be a weakly admissiblemesh for E and denote by µ k the uniform probability measure supported on A k . Letus denote by η k the sequence of functions η k : = det (cid:32) ∂ ¯ ∂ k log B µ k ( z ) (cid:33) . The sequence (2 i ) n η k d Vol C n converges weak ∗ to µ E . In particular, when D has fullrank, we have (40) η k : = (cid:81) nl = σ l (2 k | b | ) n b H | b | (cid:16) I N k − DS − D H (cid:17) b | b | , where S = diag ( σ , . . . , σ n ) is the diagonal matrix R H R and D = QR is the stan-dard QR factorization of D . Proof.
The only thing that remains to prove is equation (40). It is su ffi cient tosimply notice that if A is any invertible matrix, then adj( A ) = (det( A )) A − . In ourspecific case, in which S = R H R for a triangular matrix R , det S = (cid:81) nl = σ l factorsout and (40) follows (cid:3) Remark 5.2.
Note that the measures η k are not a priori supported on E, howeverit follows trivially by the above theorem that also the sequence of measures havingdensity η k χ E (i.e., the restriction to E of η k ) has the same weak ∗ limit. A cknowledgements The findings of this work are essentially a part of the doctoral dissertation [32].Consequently, much of what we present here have been deeply influenced by thediscussions with the Advisor Prof. N. Levenberg (Indiana University).All the software used in the numerical tests we performed above has been devel-oped in collaboration with Prof. M. Vianello (University of Padova). The authordeeply thanks him both for the scientific collaboration and the support.R eferences [1] M. Baran. Siciak’s extremal function of convex sets in C N . Ann. Polon. Math. , 48(3):275–280,1988.[2] M. Baran.
Siciak’s extremal function and complex equilibrium measures for compact sets of R n .PhD thesis, Jagellonian University (Krakow). PhD Dissertation, 1989.[3] E. Bedford and B. A. Taylor. The Dirichlet proiblem for a complex Monge Ampere equation. Inventiones Mathematicae , 50:129–134, 1976.[4] E. Bedford and B. A. Taylor. A new capacity for plurisubharmonic functions.
Acta Mathemat-ica , 149(1):1–40, 1982.[5] R. Berman and S. Boucksom. Growth of balls of holomorphic sections and energy at equilib-rium.
Invent. Math. , 181(2):337–394, 2010.[6] R. Berman, S. Boucksom, and D. Witt Nystr¨om. Fekete points and convergence towards equi-librium measures on complex manifolds.
Acta Math. , 207(1):1–27, 2011.[7] R. Berman and J. Ortega-Cerd´a. Sampling of real multivariate polynomials and pluripotentialtheory.
Arxiv preprint , arxiv.org / abs / C n . Indiana Univ. Math. J. , 46(2):427–452, 1997.[9] T. Bloom, L. Bos, and N. Levenberg. The transfinite diameter of the real ball and simplex.
Ann.Polon. Math. , 106:83–96, 2012. [10] T. Bloom, L. Bos, N. Levenberg, and S. Waldron. On the convergence of optimal measures.
Constr. Approx. , 32(1):159–179, 2010.[11] T. Bloom, L. P. Bos, J. Calvi, and N. Levenberg. Approximation in C n . Ann. Polon. Math. ,(106):53–81, 2012.[12] T. Bloom and N. Levenberg. Weighted pluripotential theory in C N . Amer. J. Math. , 125(1):57–103, 2003.[13] T. Bloom and N. Levenberg. Transfinite diameter notions in C N and integrals of Vandermondedeterminants. Ark. Mat. , 48(1):17–40, 2010.[14] T. Bloom and N. Levenberg. Random polynomials and pluripotential-theoretic extremal func-tions.
Potential Anal. , 42(2):311–334, 2015.[15] T. Bloom, N. Levenberg, F. Piazzon, and F. Wielonsky. Bernstein-Markov: a survey.
DRNADolomites Research Notes on Approximation , 8:75–91, 2015.[16] L. Bos, J.-P. Calvi, N. Levenberg, A. Sommariva, and M. Vianello. Geometric weakly admissi-ble meshes, discrete least squares approximations and approximate Fekete points.
Math. Comp. ,80(275):1623–1638, 2011.[17] L. Bos and M. Vianello. Low cardinality admissible meshes on quadrangles, triangles and disks.
Math. Inequal. Appl. , 15(1):229–235, 2012.[18] L. P. Bos and N. Levenberg. Bernstein-Walsh theory associated to convex bodies and applica-tions to multivariate approximation theory. arxiv , 1701.05613, 2017.[19] L. P. Bos, S. D. Marchi, A. Sommariva, and M. Vianello. Weakly admissible meshes and dis-crete extremal sets.
Numer. Math. Theory Methods Appl. , 41(1):1–12, 2011.[20] C. Brezinski.
Acc´el´eration de la convergence en analyse num´erique . Lecture Notes in Mathe-matics, Vol. 584. Springer-Verlag, Berlin-New York, 1977.[21] J.-P. Calvi and N. Levenberg. Uniform approximation by discrete least squares polynomials.
J.Approx. Theory , 152(1):82–100, 2008.[22] A. Cohen and G. Migliorati. Multivariate approximation in downward closed polynomialspaces. arxiv , 1612.06690, 2016.[23] H. Ehlich and K. Zeller. Schwankung von Polynomen zwischen Gitterpunkten.
Math. Z. , 86:41–44, 1964.[24] M. Embree and L. N. Trefethen. Green’s functions for multiply connected domains via confor-mal mapping.
SIAM Rev. , 41(4):745–761, 1999.[25] M. Klimek.
Pluripotential Theory . Oxford Univ. Press, 1991.[26] A. Kro´o. On optimal polynomial meshes.
J. Approx. Theory , 163(9):1107–1124, 2011.[27] N. Levenberg. Ten lectures on weighted pluripotential theory.
Dolomites Notes on Approxima-tion , 5:1–59, 2012.[28] G. Mantica. Computing the equilibrium measure of a system of intervals converging to a cantorset.
Dolomites Res. Notes Approx. DRNA , 6:51–61, 2013.[29] S. Ma (cid:44) u. Chebyshev constants and transfinite diameter on algebraic curves in C . Indiana Univ.Math. J. , 60(5):1767–1796, 2011.[30] A. Narayan, J. D. Jakeman, and T. Zhou. A Christo ff el function weighted least squares algo-rithm for collocation approximations. Mathematics of Computation , to appear, 2016.[31] S. Olver. Computation of equilibrium measures.
J. Approx. Theory , 163(9):1185–1207, 2011.[32] F. Piazzon.
Bernstein Markov Properties . PhD thesis, University of Padova Department ofMathemathics. Advisor: N. Levenberg, 2016.[33] F. Piazzon. Optimal polynomial admissible meshes on some classes of compact subsets of R d . J. Approx. Theory , 207:241–264, 2016.[34] F. Piazzon and M. Vianello. Small perturbations of polynomial meshes.
Appl. Anal. ,92(5):1063–1073, 2013.[35] F. Piazzon and M. Vianello. Constructing optimal polynomial meshes on planar starlike do-mains.
Dolomites Res. Notes Approx. DRNA , 7:22–25, 2014.[36] F. Piazzon and M. Vianello. Suboptimal polynomial meshes on planar Lipschitz domains.
Nu-mer. Funct. Anal. Optim. , 35(11):1467–1475, 2014.
LURIPOTENTIAL NUMERICS 35 [37] T. Ransford.
Potential theory in the complex plane , volume 28 of
London Mathematical SocietyStudent Texts . Cambridge University Press, Cambridge, 1995.[38] T. Ransford and J. Rostand. Computation of capacity.
Math. Comp. , 76(259):1499–1520, 2007.[39] J. Rostand. Computing logarithmic capacity with linear programming.
Experiment. Math. ,6(3):221–238, 1997.[40] A. Sadullaev. An estimates for polynomials on analytic sets.
Math. URSS Izvestiya , 20(3):493–502, 1982.[41] E. B. Sa ff and V. Totik. Logarithmic potentials with external fields . Springer-Verlag Berlin,1997.[42] Y. Shin and D. Xiu. On a near optimal sampling strategy for least squares polynomial regres-sion.
J. Comput. Phys. , 326:931–946, 2016.[43] J. Siciak. On some extremal functions and their application to the theory of analytic functionsof several complex variables.
Trans. of AMS , 105(2):322–357, 1962.[44] J. Siciak. Extremal plurisubharmonic functions in C n . Ann Polon. Math. , 319:175–211, 1981.[45] A. Townsend and L. N. Trefethen. An extension of Chebfun to two dimensions.
SIAM J. Sci.Comput. , 35(6):C495–C518, 2013.[46] L. N. Trefethen. Multivariate polynomial approximation in the hypercube. arxiv , 1608.02216,2017.[47] J. L. Walsh.
Interpolation and approximation by rational function on complex domains . AMS,1929.[48] V. P. Zaharjuta. Extremal plurisubharmonic functions, Hilbert scales, and the isomorphism ofspaces of analytic functions of several variables. i, (russian).
Teor. Funkci˘i Funkcional. Anal. iPriloˇzen. , 127(19):133–157, 1974.[49] V. P. Zaharjuta. Extremal plurisubharmonic functions, hilbert scales, and the isomorphism ofspaces of analytic functions of several variables. ii, (russian).
Teor. Funkci˘i Funkcional. Anal. iPriloˇzen. , 127(21):65–83, 1974.[50] O. Zeitouni and S. Zelditch. Large deviations of empirical measures of zeros of random poly-nomials.
Int. Math. Res. Not. IMRN , (20):3935–3992, 2010.D epartment of M athematics Tullio Levi-Civita , U niversit ´ a di P adova , I taly . E-mail address : [email protected] URL ::