Đang tải... (xem toàn văn)
. We study certain configurations of points on the unit sphere in R N. As an application, we prove that the sequence of Lagrange interpolation polynomials of holomorphic functions at certain ChungYao lattices converge uniformly to the interpolated functions.
ON CONFIGURATIONS OF POINTS ON THE SPHERE AND APPLICATIONS TO APPROXIMATION OF HOLOMORPHIC FUNCTIONS BY LAGRANGE INTERPOLANTS PHUNG VAN MANH A BSTRACT. We study certain configurations of points on the unit sphere in RN . As an application, we prove that the sequence of Lagrange interpolation polynomials of holomorphic functions at certain Chung-Yao lattices converge uniformly to the interpolated functions. 1. I NTRODUCTION Let Pd (CN ) denote the space of polynomials of degree at most d in N complex variables. A subset A of CN that consists of N+d distinct points is said to be unisolvent of degree d if, d for every function f defined on A, there exists a unique polynomials P ∈ Pd (CN ) such that P(z) = f (z) for all z ∈ A. This polynomial is called the Lagrange polynomial interpolation of f at A and is denoted by L[A; f ]. We are concerned with the problem of approximation of holomorphic functions. Problem 1. Let F be a subclass of entire functions in CN and Ad a unisolvent set of degree d for d = 1, 2, . . .. Under what conditions does L[Ad ; f ] converge to f uniformly on every compact subset of CN for every f ∈ F ? It is well-known that if N = 1 then a sufficient condition is the boundedness of ∪∞ d=1 Ad . This is an immediate consequence of the Hermite Remainder Formula (see [12, p. 59]). Moreover, when the interpolation sets are unbounded, there exists a function f ∈ H(C) for which convergence does not hold. In this case, the problem is valid for a subclass F of H(C) in which the modulus of the interpolation points are controlled by the order of f (see [1] for more details). It is also proved in [1] that the same results also hold true for Kergin interpolation in CN , a natural generalization of the univariate Lagrange interpolation. In contrast to the univariate case, Bloom and Levenberg showed in [5] that the boundedness of the interpolation array (Ad ) does not guarantee the uniform convergence of every entire function as soon as N ≥ 2. The trouble here is that the interpolation operator has bad behavior when interpolation points tend to an algebraic hypersurface of degree d. Problem 2. Let E be a compact subset of CN and F the class of functions which are holomorphic in a neighborhood of E (which can depend on the functions). Let Ad ⊂ CN be a unisolvent set of degree d for d = 1, 2, . . .. Under what conditions does L[Ad ; f ] converge to f uniformly on E for every f ∈ F ? We mention that tools from (pluri)potential theory can be used to solve Problem 2. The sufficient conditions are related to the Lebesgue constants, the transfinite diameter and global extremal functions, etc. We refer the reader to [2, 4, 18] and the references therein. The aim of this paper is to give answers to the above problems when Ad is a Chung-Yao lattice generated by hyperplanes in CN . Under natural conditions on (normal) vectors defining the hyperplanes, we prove in Theorem 3.8 that the interpolation polynomials of any holomorphic functions in a sufficiently large domain converge uniformly on a compact subset of CN to the 2000 Mathematics Subject Classification. Primary 41A05, 41A63, 52C35. Key words and phrases. Lagrange interpolation, Chung-Yao lattices. Configurations on spheres. 1 interpolated functions, provided the boundedness of the interpolation points. From Theorem 3.8, we obtain a result regarding the uniform convergence of any entire function on every compact set of CN . Under the same assumptions on the vectors, we prove in Theorem 3.11 that the convergence in Problem 1 is valid for entire functions of finite order and for the interpolation points that may be unbounded. Moreover, the additional conditions regarding the location of interpolation points are similar to those given in [1, Theorem 2.3]. In Theorem 3.6, we give explicit Chung-Yao lattices satisfying Theorems 3.8 and 3.11. We mention the equally spaced points in the simplex introduced in [3] and which also satisfy the above two theorems. More generally, explicit interpolation points can be obtained by interwining sequences in the complex plane, see for instance [18]. Now the problem turns to find of set of vectors which satisfy certain conditions. For simplicity, we work with unit vectors of real coordinates. They can be viewed as points on the unit sphere SN−1 in RN . Let Vd = {n1 , . . . , nd } be a set of points on SN−1 . We associate Vd to a number hVd that measures the thickness of Vd (see Section 2 for precise definitions). If each n ∈ Vd is regarded as a normal vector of a hyperplane, then the condition on vectors in Theorems 3.8 and 3.11 become lim infd→∞ hVd > 0. In Theorem 2.6, we construct a set Vd such that hVd has this asymptotic behavior. We finally note that hVd is well-studied when N = 2. We obtain precise estimates in this case. Notations. The product of z = (z1 , . . . , zN ), w = (w1 , . . . , wN ) in RN or CN is defined by z, w = 1 ∑Nj=1 z j w j . Let us denote by z = (∑Nj=1 |z j |2 ) 2 the norm of z. The set B(z, R) (resp. B(z, R)) is the open ball (resp. closed ball) of center z ∈ CN and radius R > 0. Let A be a nonempty set. Let An denote the class of subsets of A containing n elements. 2. C ONFIGURATIONS OF POINTS ON THE SPHERE 2.1. Distance on the sphere. In RN , N ≥ 2, each point can be regarded as a vector. Let e := {e j = (0, . . . 0, 1, 0, . . . , 0) : j = 1, . . . , N} stand for the standard basis for RN . We denote by SN−1 the unit sphere in RN , SN−1 = {x ∈ RN : x = 1}. Let V = {n1 , . . . , nN−1 } be a set of linear independent unit vectors and nk = (nk1 , nk2 , . . . , nkN ), 1 ≤ k ≤ N − 1. We can easily check that the vector nV defined by e1 e2 ··· eN n11 n12 · · · n1N nV = det(e, n1 , . . . , nN−1 ) := det (2.1) .. .. .. ... . . . nN−11 nN−12 · · · nN−1N is nonzero and orthogonal to V since n, nV = det(n, n1 , . . . , nN−1 ) for n ∈ RN . Here the determinant in (2.1) is taken pointwisely according to the first row. Therefore, it is a normal vector of the hyperplane HV which passes through V and the origin. We thus get HV = {x ∈ RN : nV , x = 0} and | n, nV | , n ∈ RN . (2.2) dist(n, HV ) = nV Note that dist(n, HV ) ≤ 1 for all n ∈ SN−1 . Now let Vd = {n1 , . . . , nd } be a set of d ≥ N points on SN−1 . First, we make the assumption that Vd is N-independent, that is, any N vectors in Vd are linearly independent, or equivalently Vd det(nk1 , nk2 , · · · , nkN ) = 0 for all 1 ≤ k1 , k2 , . . . , kN ≤ d pairwise distinct. For each V ∈ N−1 ,a 2 subset consisting of N − 1 vectors of Vd , we write hV,Vd for the geometric mean of the distances from all points in Vd \V to HV . Due to (2.2), we have hV,Vd := ∏ dist(n, HV ) 1 d−N+1 n∈Vd \V ∏n∈Vd \V | n, nV | = nV 1 d−N+1 . (2.3) Let us define Vd . (2.4) N −1 On the other hand, if Vd is not N-independent, then we set hVd = 0. By definition, hVd is independent of the ordering of the vectors in Vd . In the result below, we fix an order of Vd and identify Vd with (n1 , . . . , nd ). hVd := min hV,Vd : V ∈ Proposition 2.1. The map (n1 , . . . , nd ) → h{n1 ,...,nd } is continuous on (SN−1 )d . Proof. We first prove the continuity of the map at a N-independent tuple Vd = (n1 , . . . , nd ). By definition, we can choose a neighborhood Ω of Vd in (SN−1 )d such that every Xd ∈ Ω is also N-independent. Thus formulas (2.3) and (2.4) are well-defined for Xd ∈ Ω and imply the continuity. We now turn to the case in which Vd is not N-independent. Hence, there exist N dependent vectors, say n1 , . . . , nN ∈ Vd . We can assume that nN = ∑N−1 j=1 a j n j where a1 , . . . , aN−1 ∈ R. Since hVd = 0, we only need to verify that lim hXd = 0, Xd →Vd where Xd is N-independent. Write Xd = (x1 , . . . , xd ). We denote by H the hyperplane H{x1 ,...,xN−1 } . Using (2.2) we have dist(nN , H ) ≤ N−1 ∑ |a j |dist(n j , H ) ≤ j=1 N−1 ∑ |a j | nj −xj . j=1 It is easily seen that dist(xN , H ) ≤ nN − xN + dist(nN , H ) ≤ nN − xN + N−1 ∑ |a j | nj −xj . j=1 As xk goes to nk for k = 1, . . . , N, we have dist(xN , H ) → 0. Since all perpendicular distances appearing in the definition of hXd are bounded by 1, hXd ≤ dist(xN , H ), which implies the desired limit. Let A be a subset of SN−1 . If A contains at least d ≥ N points, then we define hd (A) := sup{hVd : Vd ⊂ A}. (2.5) Otherwise, we set hd (A) = 0. The supremum in (2.5) is attained when A is compact by PropoN−1 . We define the asymptotic sition 2.1. Now, let V = (Vd )∞ d=N be an array of points on S behavior of hVd as follows. hV := lim inf hVd . (2.6) d→∞ We state a few simple properties which follow immediately from the definitions. Proposition 2.2. (1) 0 ≤ hVd , hd (A) ≤ 1 for all Vd and A. (2) hVd and hd (A) are invariant under linear isometries of RN . (3) hVd keeps its value when we replace some elements of Vd by their additive inverses. (4) hd (A) = hd (A ∪ (−A)), where −A = {−a : a ∈ A}. 3 Vd Proof. The first two assertions are trivial. Observe that for all V ∈ N−2 , HV coincides with N H(V \{n})∪{−n} for n ∈ X and dist(a, HV ) = dist(−a, HV ) for a ∈ R . The third assertion follows. The last conclusion is an immediate consequence of the third one. 2.2. The two-dimensional case. We treat the case N = 2. Each point on S1 can be regarded as a complex number. Let Vd = {n1 , . . . , nd }, d ≥ 2 and set nk = eiθk for 1 ≤ k ≤ d. If V = {nk }, then HV is the straight line passing through nk and the origin. Hence dist(n j , H{nk } ) = | sin(θ j − θk )| and hnk ,Vd = d ∏ | sin(θ j − θk )| 1 d−1 . (2.7) j=1, j=k We now recall the notation of the d-th diameter and transfinite diameter of a plane compact set A. For details we refer the reader to [16, p. 152-158]. The d-th diameter of A, denoted by diamd (A), is defined by diamd (A) := sup{ |xk − x j | ∏ 2 d(d−1) : x1 , . . . , xd ∈ A}. (2.8) 1≤ j 0. It is easily seen that Hεd is the image of Hd under the dilation of center 0 and ratio ε. Thus, when ε > 0 is small enough, ΘHεd ⊂ B(0, R). N ∞ Theorem 3.8. Let (Θd )∞ d=N be a β -regular array of Chung-Yao lattices in C and ∪d=N Θd ⊂ B(0, R). Let R1 > 0 and f holomorphic in a neighborhood of B(0, R2 ) with R2 = max(R, R1 ) + e(R+R1 ) . Then the sequence (L[Θd ; f ])∞ d=N converges to f uniformly on B(0, R1 ). β 13 Proof. By hypothesis, f is holomorphic in a neighborhood of B(0, R3 ) for some R3 > R2 . We 1) choose 0 < ε < β small enough such that R3 > max(R, R1 ) + e(R+R β −ε . N Let (Hd )∞ d=N be the array of hyperplanes in C defined as in Definition 3.4. Suppose that Hd = { 1 , . . . , d }, j (z) = n j , z +c j , n j = 1 for j = 1, . . . , d. By hypothesis, lim infd→∞ hHd = β . We can find α > 0 such that hd−N+1 ≥ α(β − ε)d−N+1 , Hd ∀d ≥ N. (3.18) Consequently, j | ˜j (nK )| ≥ α(β − ε)d−N+1 , ∏ n K ∈H \K ∀K ∈ d Hd , d ≥ N. N −1 (3.19) Now using Lemma 3.3 we obtain f − L[Θd ; f ] B(0,R1 ) ≤ 1 d α N −1 e(R + R1 ) (β − ε)(R3 − max(R, R1 )) The right hand side of (3.20) tends to 0 as d → ∞ since follows. d−N+1 f e(R+R1 ) (β −ε)(R3 −max(R,R1 )) B(0,R3 ) , (3.20) < 1 and the theorem Corollary 3.9. Under the same assumptions of Theorem 3.8, for any entire function f in CN , N the sequence (L[Θd ; f ])∞ d=N converges to f uniformly on every compact subset of C . Remark 3.10. 1) We do not know whether the radius R2 in Theorem 3.8 is optimal. 2) Roughly speaking, the boundedness of the array of interpolation points is a necessary condition for the convergence of entire functions. Indeed, suppose that N ≥ 2. Let Ad ⊂ CN be unisolvent of degree d such that ∪∞ d=1 Ad has no limit points. By Theorem 7.2.11 in [13], there exists an entire function f = 0 such that f (a) = 0 for all a ∈ ∪∞ d=1 Ad . It follows that L[Ad ; f ] = 0 for all d ≥ 1. Hence L[Ad ; f ](z) does not tend to f (z) whenever f (z) = 0. We recall the following measures of the rate of growth of entire functions in CN , following Boas [6, p. 8]. Let f be an entire function in CN . The order of f is defined by log(log f B(0,r) ) . log r The order is a non-negative real number or infinite. If 0 < µ < ∞, then we can define the type of f as follows. log f B(0,r) σ = lim sup . rµ r→∞ If f is of order 0 < µ < ∞ and type σ , then for every δ > 0 there exists r0 > 0 such that µ = lim sup r→∞ log f B(0,r) ≤ (σ + δ )r µ , ∀r > r0 . Theorem 3.11. Let f be an entire function in CN of order µ ∈ (0, ∞). Let (Θd )∞ d=N be a regular 1 N array of Chung-Yao lattices in C . Suppose that there exist λ ∈ (0, µ ) and 0 < ρd ≤ d λ , limd→∞ ρd = ∞ such that Θd ⊂ B(0, ρd ) for sufficiently large d. Then the sequence of Lagrange N polynomials (L[Θd ; f ])∞ d=N converges to f uniformly on every compact subset of C . Proof. Suppose that (Θd )∞ d=N is β -regular. Then relation (3.19) still holds true. We fix R1 > 0. Using Lemma 3.3 in which R and β are replaced by ρd and β − ε respectively we have f − L[Θd ; f ] B(0,R1 ) ≤ 1 d α N −1 e(ρd + R1 ) (β − ε)(R2 − max(ρd , R1 )) 14 d−N+1 f B(0,R2 ) , (3.21) where R2 > max(R1 , ρd ) and 0 < ε < β . We want to show that there is a sequence R2 = R2 (d) such that the logarithm of right hand side of (3.21) tends to −∞ as d → ∞. Given δ > 0, since f is of order µ, we have log f B(0,r) ≤ (σ + δ )r µ , ∀r > r0 , (3.22) e where σ is the type of f . Choose C > 1 such that log( (β −ε)(C−1) ) < 0 and then take R2 = R2 (d) = C(R1 + ρd ). Obviously, R2 (d) → ∞ as d → ∞. Now we see at once that e e(ρd + R1 ) ≤ log( ). (β − ε)(R2 − max(ρd , R1 )) (β − ε)(C − 1) On the one hand, using (3.22) we obtain log log f µ B(0,R2 ) ≤ (σ + δ )R2 = (σ + δ )C µ (R1 + ρd )µ ≤ (σ + δ )C µ (R1 + d λ )µ ≤ (σ + δ )(2C)µ d λ µ . On the other hand, log that log f − L[Θd ; f ] d N−1 B(0,R1 ) ≤ (N − 1) log d. From what has already proved, we may conclude ≤ − log α + (N − 1) log d + (d − N + 1) log( e ) + (σ + δ )(2C)µ d λ µ (3.23) (β − ε)(C − 1) e Since log( (β −ε)(C−1) ) < 0 and λ µ < 1, the right hand side of (3.23) tends to −∞ as d → ∞. The proof is now complete. Remark 3.12. 1) We have proved more, namely that the sequence of Lagrange polynomials in Corollary 3.9 and Theorem 3.11 converges geometrically on every compact subset of CN to the interpolated function, that is lim d→∞ f − L[Θd ; f ] 1 d−N B(0,R) < 1, R > 0. 2) In [17], Sauer and Xu constructed beautiful bi-dimensional Chung-Yao lattices ΘH2d+1 located in the real unit disk in R2 . Moreover, Proposition 3 in [17] shows that the array (ΘH2d+1 )∞ d=1 is regular. Thus it gives an explicit array of points having the convergence property. N Remark 3.13. Let (Hd )∞ d=N be an array of regular Chung-Yao lattices in C in which the set of normal vectors of Hd is Vd defined in Lemma 2.10 and ΘHd ⊂ B(0, ηd ), limd→∞ ηd = 0. We write Hd = { 1 , . . . , d } where k (z) = nk , z + ck , nk = vk / vk , vk = (1,tk , . . . ,tkN−1 ). Looking at the proof of Lemma 3.3 we have |ck | ≤ ηd for all 1 ≤ k ≤ d. On the other hand, for H ∈ HNd , ϑH ∈ B(0, ηd ). It follows that | k (ϑH )| ≤ nk · ϑH + |ck | ≤ 2ηd , ∀1 ≤ k ≤ d, H ∈ Hd . N Given R > 0, we have k (R, 0, · · · , 0) = R R + ck ≥ − ηd , vk C ∀1 ≤ k ≤ d, 2j where we use the inequality vk ≤ ∑N−1 j=0 a := C in the last estimate. We conclude from the formula of the FLIP in Theorem 3.1 that | k (z)| R/C − ηd d−N sup |l(ΘHd , ϑH ; z)| = sup ≥ . ∏ 2ηd z∈B(0,R) z∈B(0,R) ∈H \H | k (ϑH )| k d 15 Hence we can not find a real-valued continuous function h defined in B(0, R) with the following property: Given ε > 0 there exists an integer d1 (depending on ε) such that, for all d ≥ d1 and H ∈ HNd , 1 log |l(ΘHd , ϑH ; z)| ≤ ε. z∈B(0,R) d − N h(ϑH ) + sup The condition (2.7.1) in [2] does not hold. Therefore the assumptions in Theorem 3.8 are not stronger than those given in [2]. Remark 3.14. We have proved that the convergence results hold if the arrays of Chung-Yao lattices are regular in the sense of Definition 3.4. The following question arises naturally: Is there any array of Chung-Yao lattices without the separation condition of hyperplanes but the convergence still holds? Acknowledgements. The author wishes to express his thanks to Professor Jean-Paul Calvi for suggesting this problem and for stimulating conversations. The author would like to thank the referees for a careful reading of the manuscript. This work has been partially done during a visit of the author at the Vietnam Institute for Advanced Mathematics in 2014. He wishes to thank this institution for financial support and the warm hospitality that he receive. This work was supported by the NAFOSTED program. R EFERENCES [1] T. Bloom, Kergin interpolation of entire functions on CN , Duke Math. J., 48:69-83, 1981. [2] T. Bloom, On the convergence of multivariate Lagrange interpolants, Constr. Approx., 5:415-435, 1989. [3] T. Bloom, The Lebesgue constant for Lagrange interpolation in the simplex, J. Approx. Theory, 54:338-353, 1988. [4] T. Bloom, L. Bos, C. Christensen and N. Levenberg, Polynomial interpolation of holomorphic function in C and Cn , Rocky Mountain J. Math., 22:441-470, 1992. [5] T. Bloom and N. Levenberg, Lagrange interpolation of entire functions in C2 , New Zealand J. Math., 22:6583, 1993. [6] R. P. Boas, Entire functions, Academic Press, 1954. [7] C. de Boor , The error in polynomials tensor-product, and Chung-Yao, interpolation, in "Suface Fitting and Multisolution Methods", Vanderbilt University press, Nashville, 1997. Available online at: ftp://ftp.cs.wisc.edu/Approx/chamonix.pdf [8] L. Bos and M. Vianello, Subperiodic trigonometric interpolation and quadrature, Appl. Math. Comput. 218:10630-10638, 2012 [9] J.-P. Calvi, Interwining unisolvent arrays for multivariate Lagrange interpolation, Adv. Comp. Math., 23:393414, 2005. [10] J.-P. Calvi and Phung V. M., On the continuity of multivariate Lagrange interpolation at natural lattices, L.M.S. J. Comput. Math., 16:45-60, 2013. [11] K. C. Chung and T. H. Yao, On lattices admitting unique Lagrange interpolation, SIAM J. Numer. Anal., 14:735-743, 1977. [12] D. Gaier, Lectures on complex approximation, Birkhauser, 1987. [13] L. Hörmander, An introduction to complex analysis in several variables, North-Holland Publishing company, 1973. [14] C. A. Micchelli, A constrictive approach to Kergin interpolation in Rn : multivariate B-spline and Lagrange interpolation, Rocky Mountain J. Math., 10(3):485-497, 1980. [15] L. Nachbin, Topology on spaces of holomorphic mappings, Springer-Verlag, 1969. [16] T. Ransford, Potential theory in the complex plane, Cambridge University Press, 1996. [17] T. Sauer and Y. Xu, Regular points for Lagrange interpolation on the unit disk, Numer. Algorithms., 12:287296, 1996. [18] J. Siciak, On some extremal functions and their applications in the theory of analytic functions of several complex variables, Trans. Amer. Math. Soc., 105:322-357, 1962. 16 D EPARTMENT OF M ATHEMATICS , H ANOI STREET, C AU G IAY, H ANOI , V IETNAM NATIONAL U NIVERSITY OF E DUCATION , 136 X UAN T HUY E-mail address: ♠❛♥❤❧t❤❅❣♠❛✐❧✳❝♦♠ 17 [...]... < 1 and the theorem Corollary 3.9 Under the same assumptions of Theorem 3.8, for any entire function f in CN , N the sequence (L[Θd ; f ])∞ d=N converges to f uniformly on every compact subset of C Remark 3.10 1) We do not know whether the radius R2 in Theorem 3.8 is optimal 2) Roughly speaking, the boundedness of the array of interpolation points is a necessary condition for the convergence of entire... [15] L Nachbin, Topology on spaces of holomorphic mappings, Springer-Verlag, 1969 [16] T Ransford, Potential theory in the complex plane, Cambridge University Press, 1996 [17] T Sauer and Y Xu, Regular points for Lagrange interpolation on the unit disk, Numer Algorithms., 12:287296, 1996 [18] J Siciak, On some extremal functions and their applications in the theory of analytic functions of several complex... hold if the arrays of Chung-Yao lattices are regular in the sense of Definition 3.4 The following question arises naturally: Is there any array of Chung-Yao lattices without the separation condition of hyperplanes but the convergence still holds? Acknowledgements The author wishes to express his thanks to Professor Jean-Paul Calvi for suggesting this problem and for stimulating conversations The author... set of normal vectors of Hd Roughly speaking, the condition lim infd→∞ hHd > 0 guarantees ΘHd not to tend to a hyperplane It is a quite natural assumption (see the discussions in Section 1) Theorem 3.6 Let N ≥ 2 and 2a ≥ b > 0 For each d ≥ N, let Td = {t1 , ,td } be a set of real numbers in [−a, a] such that |t j − tk | ≥ b/d for j = k We denote by Θd the subset of CN d consisting of d−N points and. .. Bloom, On the convergence of multivariate Lagrange interpolants, Constr Approx., 5:415-435, 1989 [3] T Bloom, The Lebesgue constant for Lagrange interpolation in the simplex, J Approx Theory, 54:338-353, 1988 [4] T Bloom, L Bos, C Christensen and N Levenberg, Polynomial interpolation of holomorphic function in C and Cn , Rocky Mountain J Math., 22:441-470, 1992 [5] T Bloom and N Levenberg, Lagrange. .. like to thank the referees for a careful reading of the manuscript This work has been partially done during a visit of the author at the Vietnam Institute for Advanced Mathematics in 2014 He wishes to thank this institution for financial support and the warm hospitality that he receive This work was supported by the NAFOSTED program R EFERENCES [1] T Bloom, Kergin interpolation of entire functions on. .. ΘH ∩ ∩K, the points in ΘH on the line ∩K = ∩ PK (z) = z ∈ Ω, ∏ ∈H\K ∈K , and (z) ˜(nK ) 3.3 Convergence of Lagrange interpolants at Chung-Yao lattices Let f be a holomorphic function in an open subset Ω of CN The norm of the k-th total derivative of f at z ∈ Ω is defined by Dk f (z) = sup{|Dk f (z)(z1 , , zk )| : z j ∈ CN , z j ≤ 1, j = 1, , k} We set f E := supz∈E | f (z)| for E ⊂ Ω The following... N Consequently, ∩N k=1 k = z = ϑU ∈ Θd Next, we consider the system of N + 1 equations, z1 + tk z2 + · · · + tkN−1 zN = tkN , k = 1, 2, , N + 1 (3.17) Since the rank of the coefficient matrix is equal to N and the rank of its augmented matrix equals N + 1, Kronecker-Capelli’s theorem shows that the system (3.17) has no solution It / Consequently, Hd = { 1 , · · · , d } is in general position in... k = 1, 2, , N (3.13) Evidently, the determinant of the coefficient matrix is equal to D = ∏1≤ j ... Sauer and Y Xu, Regular points for Lagrange interpolation on the unit disk, Numer Algorithms., 12:287296, 1996 [18] J Siciak, On some extremal functions and their applications in the theory of analytic... is the open ball (resp closed ball) of center z ∈ CN and radius R > Let A be a nonempty set Let An denote the class of subsets of A containing n elements C ONFIGURATIONS OF POINTS ON THE SPHERE. .. N −1 On the other hand, if Vd is not N-independent, then we set hVd = By definition, hVd is independent of the ordering of the vectors in Vd In the result below, we fix an order of Vd and identify