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Author's personal copy650J.S. Brauchart et al.that characterizes μQ,s in terms of its potential. The proof of this proposition follows closelythe proof of [28, Theorem I.1.3]. It could also be derived as a particular case from the moregeneral results in [32] (see especially Theorems 1 and 2, and Proposition 1 of that paper).Proposition 3 Let 0 s d.1 For the minimal energy problem on Sd with external fieldQ the following properties hold:(a)(b)(c)VQ,s is finite.There exists a unique s-extremal measure μQ,s M(Sd ) associated with Q. Moreover, the support SQ,s : supp(μQ,s ) of this measure is contained in the compact setEM : {x Sd : Q(x) M} for some M 0.The measure μQ,s satisfies the variational inequalitiesμQ,sUs(x) Q(x) FQ,sμUs Q,s (x) Q(x) FQ,swhere(4)everywhere on SQ,s ,(5) FQ,s : VQ,s (d)q.e. on Sd ,Q(x) dμQ,s (x).(6)Inequalities (4) and (5) completely characterize the s-extremal measure μQ in thesense that if ν M(Sd ) is a measure with finite s-energy such that for some constantC we haveUsν (x) Q(x) CUsν (x) Q(x) Cq.e. on Sd ,(7)everywhere on supp(ν),(8)then ν μQ,s and C FQ,s .Observe that, if the external field Q is continuous on Sd , then the inequality in (7) holdseverywhere on Sd .Remark 1 Proposition 3 remains true if Sd is replaced with any compact subset K Sdwith caps (K) 0. Notationally, the dependence on K will be indicated by a subscript K(e.g., μK,Q,s , FK,Q,s , etc.).In the case when d 1 s d, [10, Theorem 1.3] analyzes further the characterizationproperty from Proposition 3(d) by studying the supremum and the essential infimum of theweighted potential Usν (x) Q(x). Our first theorem extends this analysis to the larger ranged 2 s d. To state the theorem we introduce the notation “ inf ”x E to denote theessential infimum of f with respect to a set E Sd ; that is,“ inf ” f (x) : sup{c : f (x) c q.e. on E};x Ein other words, the infimum is taken quasi-everywhere.Theorem 4 Let d 2 s d, Q be an external field on Sd , and FQ,s be defined as in (6).For any measure λ M(Sd ) we have “ inf ” Usλ (x) Q(x) FQ,s(9)x SQ,s1Asimilar result holds for the logarithmic case.

Author's personal copyRiesz External Field Problems on the Hypersphereandsupx supp(λ) λ Us (x) Q(x) FQ,s .651(10)If equality holds in both inequalities, then λ μQ,s .For the restricted range d 1 s d, the proof of this theorem as given in [10]utilizes the principle of domination for Riesz potentials, which generally is stated for theparameter range d 1 s d 1 and measures supported on any subsets of Rd 1 . (Arestricted version of the principle of domination for d 2 s d was established in[10, Lemma 5.1].) Via a different approach that utilizes the following restricted maximumprinciple on the sphere, we are able to prove the result for s in the extended range d 2 s d; see Section 6.Theorem 5 (Sphere Maximum Principle) Let d 2 s d. Suppose μ is a positiveμmeasure with supp(μ) Sd such that for some M 0, the relation Us (x) M holdsμddμ-almost everywhere on S . Then Us (x) M holds everywhere on S .An essential part of the analysis of external field problems is the determination the sextremal (equilibrium) measure on Sd associated with the external field Q and, in particular,its support. In principle, if the latter is known, the measure μQ,s can be recovered by solving an integral equation for the weighted s-potential of μQ,s arising from the variationalinequalities (4) and (5). A substantially easier problem is to find a (signed) measure that hasconstant weighted s-potential everywhere on Sd . The solution of this problem turns out tobe useful in solving the harder problem. This motivates the study of the signed equilibriummeasure associated with an external field which is defined as follows.Definition 6 Given a compact subset K Rp (p 3) and an external field Q, we call asigned measure ηK,Q ηK,Q,s supported on K and of total charge ηK,Q (K) 1 a signeds-equilibrium on K associated with Q if its weighted Riesz s-potential is constant on K;that is,ηUs K,Q (x) Q(x) GK,Q,sfor all x K.(11)We note that if a signed equilibrium exists, then it is unique (see [2, Lemma 23]).A remarkable connection exists to the Riesz analog of the Mhaskar-Saff F -functionalfrom classical logarithmic potential theory in the plane (see [23] and [28, Chapter IV, p.194]).Definition 7 The Fs -functional of a compact subset E Sd of positive s-capacity isdefined as Fs (E) : Ws (E) Q(x) dμE (x),(12)where Ws (E) is the s-energy of E and μE is the s-equilibrium measure (without externalfield) on E.Let d 2 s d with s 0. If the signed equilibrium on a compact set K Sdassociated with Q exists, then integration of (11) with respect to μK shows thatFs (K) GK,Q,s .The essential property of the Fs -functional is the following (cf. [2, Theorem 9]).(13)

Author's personal copy652J.S. Brauchart et al.Proposition 8 Let d 2 s d with s 0 and Q be an external field on a compactsubset K Sd with caps (K) 0. Then the Fs -functional is minimized for the support ofthe s-extremal measure μK,Q,s on K associated with Q; that is, for every compact subsetE K with caps (E) 0,Fs (E) Fs (supp(μK,Q,s )) FK,Q,s .SK,Q,s of μK,Q,s is defined byGiven a compact subset K Sd , the extended support μ (14)SK,Q,s : x K : Us K,Q,s (x) Q(x) FK,Q,s .The following theorem, which is the Riesz analog of [9, Theorem 2.6] and [18, Lemma 3],establishes a relation between the extended support SQ,s of μQ,s (by (5) this set contains the support of μQ,s ) and the support of the positive part ηQof the Jordan decomposition dηQ ηQ of the signed equilibrium ηQ ηSd ,Q,s on S associated with Q.Theorem 9 Let d 2 s d and suppose that Q is an external field such that a signeds-equilibrium ηρ η ρ ,Q,s on a spherical cap ρ {x Sd : x p ρ} exists. ThenμQ,s ρ ηρ SQ,sSQ,s ρ supp(η ).andSQ,s ρ supp(ηρ ).Furthermore, if FQ,s G ρ ,Q,s , then Remark 2 The theorem remains true if the s-extremal measure on a compact subsetK Sd with caps (K) 0 and signed s-equilibria ηE on compact subsets E Sd with supp ηE K are considered.The characterization results for the shape of the support of the s-extremal measure on Sdassociated with a rotational symmetric external field given in [2, Theorem 10] immediatelycarry over to the external fields with extended range.Proposition 10 Let d 2 s d with s 0 and the external field Q : Sd ( , ]be rotationally invariant about the polar axis; that is, Q(z) f (ξ ), where ξ is the altitude1 ξ 2 z, ξ , z Sd 1 . Suppose that f is a convex function on [ 1, 1]. Then theof z support of the s-extremal measure μQ on Sd is a spherical zone; namely, there are numbers 1 t1 t2 1 such that (15)1 u2 x, u : t1 u t2 , x Sd 1 .supp(μQ ) t1 ,t2 : Moreover, if additionally f is increasing, then t1 1 and the support of μQ is a sphericalcap centered at the South Pole.Next we focus on the discretized version of the Riesz external field problem given inDefinition 1. Recall that the normalized counting measure associated with an n-point setXn {x1 , x2 , . . . , xn } is defined as1 δxj ,nnμXn : j 1where δx is the Dirac-delta measure with unit mass at x. The continuous and discrete externalfield minimization problems are related in the following way.

Author's personal copyRiesz External Field Problems on the Hypersphere653Proposition 11 Let 0 s d or s log. ThenEsQ (n) VQ,s IQ,s (μQ,s ).n n2dFurthermore, if {Xn,Q,s } n 2 is any sequence of n-point (Q, s)-Fekete sets on S (see Definition 2), then the sequence of the normalized counting measures μXn,Q,s associated withXn,Q,s converges in the weak-star sense to the s-extremal measure μQ,s .limThe proof follows from a standard argument and utilizes the uniqueness result stated inProposition 3(b).We are interested in determining sets that contain all the (Q, s)-Fekete sets. For this purpose it is useful to investigate the weighted s-potential of the normalized counting measureμXn which is defined asμXnhXn (x) : Us(x) Q(x) n1 1nx xjj 1s Q(x),x Sd .(16)As an application of Theorem 4 we deduce the following result.Theorem 12 Let d 2 s d. Let Xn Sd be a set of n distinct points, and supposethat, for some constant M, the associated weighted potential satisfies the inequalityhXn (x) Mq.e. on SQ,s supp(μQ,s ).(17)Then (cf. (6))μXnUsμQ,s(x) M Us(x) FQ,severywhere on Sd .(18)Furthermore,hXn (x) Mq.e. on Sd .(19)We point out that this is an extension of [10, Theorem 1.7], which, as with Theorem 4above, was originally established for d 1 s d. As in [10, Corollary 1.9], Theorems 4and 12 yield the following.Corollary 13 For d 2 s d, every (Q, s)-Fekete set is contained in the extendedsupport SQ,s .We note that for most of the above theorems, s d 2 marks the lower end of the statedrange of the Riesz parameter s. It turns out that the case s d 2 is distinctive because newphenomena arise in the solution of the signed equilibrium problem, see Section 4. Moreover,for s in the interval (0, d 2), the Riesz-s kernel becomes strictly superharmonic whenconsidered in the stereographic projection space of Sd ; consequently maximum principlesand domination principles do not apply.3 Application to Point SeparationGood separation of points is generally associated with the stability of an approximation orinterpolation method (e.g., by splines or radial basis functions (RBF)); cf., e.g., [12, 22, 29].In this section we shall apply results from Section 2 (especially Theorem 9 and Corollary 13)to obtain explicit point separation estimates for sequences of n-point (Q, s)-Fekete sets

Author's personal copy654J.S. Brauchart et al.(cf. Definition 2) associated with a large class of external fields Q and establish that suchsequences are “well-separated” in the following sense. Letδ(Xn ) : min{ xj xk : xj , xk Xn , j k}denote the minimum distance among the points in Xn . Then a sequence {Xn }n 2 , Xn Sdfor all n, is called well-separated if δ(Xn ) is of order n 1/d as n . (It suffices to showthe existence of a constant C such thatδ(Xn ) C n 1/dfor sufficiently large n,(20)n 1/d ;cf. [4].)since δ(Xn ) cannot exceed the best-packing distance which is of orderIn the potential-theoretical and field-free setting (Q 0) it has been known sinceDahlberg [5] that Fekete point sets (harmonic case s d 1) on a sufficiently smoothclosed bounded d-dimensional surface in Rd 1 that separates Rd 1 into two parts will forma well-separated sequence (but no explicit constant for the lower bound of δ(Xn ) has beengiven). Götz [13] studied the discrete external field problem on surfaces in Rd where theenergy functional is defined in terms of the Green function for a domain X Rd . His separation result generalizes Dahlberg’s result. Well-separation of minimal logarithmic energyconfigurations on S2 in the field-free setting was first established by Rakhmanov et al. [26,27], and with an improved constant by Dragnev [8]. For minimal Riesz s-energy configurations on Sd in the field-free case, well-separation was established by Kuijlaars et al. [20] fors (d 1, d) and by Dragnev and Saff [10] for s (d 2, d). Damelin and Maymeskul[6] give a separation result of order n 1/(s 2) , 0 s d 2, which is of sharp order inthe boundary case s d 2. It is expected but still unproven that minimal logarithmic andRiesz s-energy (0 s d 2) configurations on Sd , d 3, are well-separated. The references [8, 10, 27] also provide an explicit constant in the lower estimate (20). It should benoted that [10] uses external fields to derive the desired separation estimates in the field-freesetting. In the hyper-singular case s d, Kuijlaars and Saff [19] establish well-separationof minimal Riesz s-energy configurations on Sd .We now present a generalization of [10, Theorem 1.5] to the case when an external fieldis present and given by a potential.Theorem 14 Let d 2 s d and Q(x) : Usσ (x) for some signed measure σ with σd a.e. finite Riesz s-potential on Sd . Assume the support of the negative part σ in the Jordandecomposition σ σ σ satisfies that supp(σ ) x Rd 1 : x r(21)for some r 1 and thatcσ cσ (r) : 1 σ (r 1)d s 1Ws (Sd ) (r 1)d σ 1.2(22)dThen any sequence (Xn,Q,s ) n 2 of (Q, s)-Fekete sets on S is well-separated; moreprecisely,KQ,sfor all n 2cσ 1,(23)δ(Xn,Q,s ) 1/dnwhere 1/d d s12KQ,s : .(24)Ws (Sd ) cσIt is understood that for d 2 and s log we replace Ws (Sd ) by 1 and s by 0.

Author's personal copyRiesz External Field Problems on the Hypersphere655Since the Riesz s-energy of Sd appearing in the separation constant in (24) is given bythe formula((d 1)/2) ((d s)/2),s 0,(25) Ws (Sd ) 2d 1 sπ (d s/2)we have in the harmonic case s d 1 that 1/dr 1 KQ,d 1 2 1/d 1 σ 1σ(r 1)d(26)and in the limiting case s d 2 (and d 3) 1/d (r 1)21 ((d 1)/2) 1/d 1 σ,1 σ KQ,d 2 dWd 2 (Sd ) (r 1)dπ (d/2)(27)wheras for s log and d 2 1/2(r 1)2 1 σ.(28)KQ,log 2 1 σ (r 1)2Remark 3 Note that whenever the support of σ lies outside of Sd , then both conditions (21) and (22) are satisfied by taking r( 1) sufficiently close to 1. Also observe that asr approaches 1, the constant KQ,s approaches 0.In case of σ 0 and s d 2 0, the above Theorem 14 yields a known resultfor the well-separation of n-point minimal Riesz (d 2)-energy configurations on Sd ([6]but without explicit constants). Also with σ 0, s log and d 2 we recover the sameseparation result as obtained in [8]. Here we prove them separately (with explicit constantsin the former case), since they will be used to establish the separation bounds when anexternal field (σ 0) is given.Proposition 15 For Q 0 and d 2 we haveδ(Xn,d 2 ) κd(n 1)1/d(29)for any n( 3)-point Riesz (d 2)-energy 2 minimizing configuration Xn,d 2 on Sd , where 1 ((d 1)/2) 1/d.(30)κd dπ (d/2)Observe that κd (4/Wd 2 (Sd ))1/d when d 3. The first three values of κd are κ2 2,κ3 (3π/2)1/3 , and κ4 2/31/4 . Curiously, (κd )d is the ratio of the volume of the unit ballin Rd divided by the surface area of the unit sphere in Rd 1 . (This constant also appears asthe coefficient of the leading term in the asymptotic expansion of the n-point minimal Rieszd-energy as n (cf. [19]).)Finally, we present a well-separation result for sequences of (Q, s)-Fekete sets in thehyper-singular case s d. In this case the (strongly repellent) short-range interactionsbetween points on the sphere ensure well-separation of minimizing configurations for any2 Whend 2 we mean logarithmic energy.

Author's personal copy656J.S. Brauchart et al.continuous external field on Sd . In fact, it is enough that Q be integrable on some smallsubset of Sd of positive surface area measure.Theorem 16 Let s d. Suppose there is a subset B Sd such that σd (B) 0 and thefixed external field Q is integrable over B with respect to σd . Then there is a constant Cindependent of n such thatCδ(Xn,Q,s ) 1/d(31)nfor any n-point (Q, s)-Fekete set on Sd .Remark 4 In case of Q Qn varies with n, there may be no single fixed subset B satisfyingthe hypotheses in Theorem 16. However, one can still deduce well-separation by requiringthe following: there is a sequence {Bn } of subsets of Sd such that for some ε 0, σd (Bn ) ε for all n, and for some C 0 independent of n, 1 Qn (x) dσd (x) min{0, M Qn } C ns/d 1for all n,σd (Bn ) Bnwhere M Qn denotes the minimum of Qn over Sd . These conditions are derived from themain inequality (82) and the estimate (83).Remark 5 For large classes of external fields Q (e.g., continuous external fields), inequality(82) and the estimate (83) can be made explicit which, in turn, yields an explicit constant inthe separation estimate (31), as the following example illustrates.Example 17 Let s d. Consider the external field Q(x) q x Rp s , q 0, R 1,due to a point source above the North Pole p. Clearly, Q is continuous and thus integrableon B Sd . Thus Theorem 16 assures well-separation of n-point (Q, s)-Fekete sets on Sd .An explicit lower bound can be easily derived from (82) to (83). We find 1qdσd (x)(Q(x) min{0, Q(xj )}) dσd (x) q min0,sσd (B) D(R 1)sSd x Rp and, consequently, δ(Xn,Q,s ) whereg(n) : γd1 d1 γd 1/sg(n),n1/d 1/s 1βs,d 2/dq n 2n1 s/d q Usσd (Rp) min 0,.s d2(R 1)sNote that g(n) (s d)1/s as n . The constants γd and βs,d are given in (76) andσ(78), respectively, and the representation of Us d appears in (41).4 Negatively Charged External FieldsIn the following we consider external fields Q that are generated by negative sources. Therequired lower semi-continuity of Q : Sd ( , ] implies that no negative singular-

Author's personal copyRiesz External Field Problems on the Hypersphere657ities can be on the sphere but it may support a negative “continuous” charge distribution(with no discrete part relative to Sd ). We give a detailed analysis for the Riesz external fieldQb,s (x) Usσ (x) q x b s ,x Sd ,b Rp (R 1), q 0,(32)where σ qδ(b), that is due to a negative point source at b below the South Pole and whichalso provides the basis for more general axis-supported fields defined by superposition ofpoint source fields. Our analysis thus extends and complements results in [2] where positiveaxis-supported external fields were considered.Intuitively, a negative point source under the South Pole will “pull” charge towards theSouth Pole and if sufficiently strong will cause a negatively charged spherical cap aroundthe North Pole to appear on a grounded sphere. Grounding of the sphere imposes constantweighted potential everywhere on Sd . This naturally leads to the signed equilibrium problemon the whole sphere or on its parts, say, the spherical cap t : {x Sd : x · p t} centeredat the South Pole. On a positively charged isolated sphere a sufficiently strong negative fieldwill produce a spherical cap around the North Pole that is free of charge. We are specificallyinterested in the charge distribution on the remaining part tc , that is the s-extremal measureon Sd associated with the external field Qb,s and its support tc .We will use the methods and results of [2]. An essential concept is the s-balayage of ameasure. Recall that given a measure ν and a compact set K (of the sphere Sd ), the balayagemeasure ν̂ : Bals (ν, K) preserves the Riesz s-potential of ν onto the set K and diminishesit elsewhere (on the sphere Sd ). Let ηt denote the signed s-equilibrium on t associatedwith the external field Qb,s . Then it can be expressed as (33)ηt s (t)/Ws (Sd ) νt q t ,where t t,s : Bals (δb , t ),νt νt,s : Bals (σd , t )(34)are the s-balayage measures onto t of the positive unit point charge at b and the uniformmeasure σd on Sd . The function s (t), defined by s (t) : Ws (Sd )(1 q t )/ νt ,d 2 s d,(35) in terms of the s-energy of Sd , given in (25) and norms t Sd d t and νt Sd dνt ,plays an important role in what follows. Indeed, s (t)dηt νt q t 1dWs (S ) tand using that Usνt (x) Ws (Sd ) and Us t (x) x a s on t by (34), at every x tUsηt (x) Qb,s (x) s (t) νtU (x) q Us t (x) Qb,s (x) s (t).Ws (Sd ) s(36)By Definition 6, G t ,Qb,s ,s s (t) and (13) relates s (t) to the Fs -functional by means of s (t) Fs ( t ), whereas Proposition 8 implies that the latter is minimized by the supportof the s-extremal measure on Sd associated with the external field (32) which turns out tobe a spherical cap tc . We will see that the unique minimum of s (t) in the interval [ 1, 1]will provide this critical parameter tc (see Theorem 20). Moreover, the remark followingTheorem 19 provides the necessary and sufficient conditions (involving s (t) and therefore

Author's personal copy658J.S. Brauchart et al.Fs ( t )) under which the signed s-equilibrium measure ηt on t turns into the s-extremalmeasure on Sd associatedwith the external field (32). a, ba, b; z and 2 F̃1; z denote the Gauss hypergeometric funcThroughout, 2 F1cc3tion and its regularized form with series expansions (a)n(b)n zn(a)n(b)n zna, ba, b,, z 1,;z: ;z: F̃F2 12 1cc(c)n n!(n c) n!n 0n 0(37)where (a)0 : 1 and (a)n : a(a 1) · · · (a n 1) for n 1 is the Pochhammer symbol.We also recall that the incomplete Beta function and the Beta function are defined as xv α 1 (1 v)β 1 dv,B(α, β) : B(1; α, β),(38)B(x; α, β) : 0whereas the regularized incomplete Beta function is given byI(x; a, b) : B(x; a, b)/ B(a, b).(39)First, we give the representation of the signed equilibrium on the whole sphere Sd , whichis well-known from elementary physics (cf. [15, p. 61]) in the classical Coulomb case, andprovide a necessary and suffi

Department of Mathematical Sciences, Indiana University-Purdue University Fort Wayne, Fort Wayne, IN 46805, USA e-mail: [email protected] E. B. Saff ( ) Center for Constructive Approximation, Department of Mathematics, Vanderbilt University, Nashville, TN 37240, USA e-mail: [email protected] Author's personal copy