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ETNA Electronic Transactions on Numerical Analysis. Volume 19, pp. 84-93, 2005. Copyright  2005, Kent State University. ISSN 1068-9613. Kent State University etna@mcs.kent.edu LOCALIZED POLYNOMIAL BASES ON THE SPHERE NOEMÍ LAÍN FERNÁNDEZ Abstract. The subject of many areas of investigation, such as meteorology or crystallography, is the reconstruction of a continuous signal on the -sphere from scattered data. A classical approximation method is polynomial interpolation. Let denote the space of polynomials of degree at most on the unit sphere . As it is well known, the so-called spherical harmonics form an orthonormal basis of the space . Since these functions exhibit a poor localization behavior, it is natural to ask for better localized bases. Given "!$# %& , we consider the spherical polynomials 0 + 3 465 ' )( +*-, . /0 +7 8:9 ( ;)<=>*? 0 0 21 where 9 denotes the Legendre polynomial of degree 3 normalized according to the condition 9 ( 5*2.@5 . In this paper, we present systems of ( A4%5* points on that yield localized polynomial bases of the above form. Key words. fundamental systems, localization, matrix condition, reproducing kernel. AMS subject classifications. 41A05, 65D05, 15A12. FUTWV denote the unit sphere em1. Introduction. Let BDC%EGFIHKJMLINPOQESR+JPR C NXO2^kml^noeqp gji bedded in the Euclidean space NXO and let YZE\[ ])^_`bac[ ])^d_fehgji kmrs$tulwv>xyr&n)^zr;s$tulwrst{n)^zv>xyr&l|e be its parameterization in spherical coordinates k}lj^;noe . Corresponding to the surface element ~okme , we have the {CkmBDC;e -inner product and norm m ^buEFy k}e kme;~ok}e # Cz F cr;s$t{l kYkml^noe;e kYk}lj^;noee)~olw~on)^ R R C EF m ^ + Furthermore, let Harm=k"NO|e denote the space of harmonic homogeneous polynomials of degree in three variables. Restricting these functions to BC , we obtain the so-called spherical harmonics of order . Throughout this paper, we will focus on the space -%EFH # EhL -k"NXOye+V of spherical polynomials of degree at most . It can be shown that Harm kmB C e+^ (1.1) F where this direct sum decomposition has to be understood in the sense that any spherical polynomial of degree at most is the restriction of a harmonic polynomial of degree less or equal to to the sphere. Since dim Harm kmBDCe¡F¢d£¤T , it follows that ¥ EF dim F§¦ kdWy£¨T e©Fkmª£¨T«e;C . An wCk"BfC;e -orthonormal basis of that is not localized on the sphere is given by ® ¬w® ®{¶ d)£°T ¼ ± ² ² kmv³xor&l&e-´ µ ^X·¸FgA=^³K³>^z=^XF¹]2^³K³>^uº»^ (1.2) kYkml^noe;euEFq¯ _ Received November 30, 2002. Accepted for publication May 10, 2003. Communicated by R. Alvarez-Nodarse. Supported by the DFG Graduiertenkolleg 357 Effiziente Algorithmen und Mehrskalenmethoden. GSF – National Research Center for Environment and Health Institute of Biomathematics and Biometry, Neuherberg, Germany. E-mail:noemi.lain@gsf.de. 84 ETNA Kent State University etna@mcs.kent.edu Localized polynomial bases on the sphere F IG . 1.1. Spherical harmonic where ® kmÊ;eEFÌË 85 ½A¾P¿ÁÀwmÂ 1 (Ã ?Ä>*}Å and reproducing kernel Æ m1 ( «?=< * with X. (mÇ 5+?È«?È>*}É . k"g¡·»e+Í k"2£Î·»e+ÍÏÐÑ C ® ® k=TugÒÊ C e C ~ ® D;kmÊ;e+^Ô·¸FÕ])^K³³³^=^DÊwLÖ[g×T^³T³`^ ~ Ê Ó Ñ denote the associated Legendre functions and stands for the Legendre polynomial of degree normalized according to the condition =k=T e{FcT . From now on, this basis will be referred to as the basis of spherical harmonics. ® A way of constructing better localized bases is by means of the reproducing kernel of EÙ·UFÚT^K³³³^zdWX£hTo^ÛFÜ])^³K³>^V be an arbitrary the underlying space. Let H«Ø ÓCokmBfCe -orthonormal basis of . It is straightforward to check that the reproducing kernel of Harm kmBPC;e is given by ® ® ß Cà Ý;k}&^Þ&euEF ® Ø kme=Ø kmÞ&e>^ &^;Þ%L@B C Ð Combining now the direct sum decomposition of in (1.1) and the addition theorem for Ð HarmkmBPC;e (see Müller [4], page 10), one comes up with the following result. L EMMA 1.1. The unique reproducing kernel of j is given by à à d&£¹T á km&^;Þ)e{EGF ± (1.3) k}&^Þ)eÓF k}ÝâKÞ&e©FªEyã kmâKÞ)e+^ &^;ÞÛL»B C _ á It should be observed that -k}&^;Þ)eÓFÕãfk}ÓâÞ)e , as a zonal function, only depends on the Euclidean product of the vectors and Þ . Therefore, it is invariant with respect to rotations, i.e., transformations of the group ä©åæk"çoe . ® 1.1. Scaling Functions. In contrast to the spherical harmonics introduced in (1.2), á the function -k}&^³âe»EuBDC»gjièNZk}°LqBDC;e defined in (1.3) is the spherical polynomial with minimal wCk"BfCe -norm among all spherical polynomials of degree at most that attain the same value at the prescribed point . The following lemma establishes this localization property. L EMMA 1.2. Let L@BDC . Then é á é é km&^³âe éé é (1.4) é á km&^;e é F°êæs$tÝHyR>REcL»^2kmeÓFëTV© ETNA Kent State University etna@mcs.kent.edu 86 N. Laı́n Fernández The proof is a direct consequence of applying Cauchy-Schwarz’s inequality and the addition theorem (see Laı́n Fernández [2]). Our aim is to study the problem of characterizing sets of points HK V BDC á µ µ k} ^Kâ e+V such that the functions H«ñ EF constitute a basis of the space . The µ µ Ðìííí ì îðï functions ñ ^Xk}òXFëT^K³K>^¥e will be calledµ scaling functions. The linear independence of the µ ìíían í ì î ¥óa¥ matrix. Given HK V scaling functions is reflected in the regularity Ðof BDC , µ µ we can construct the interpolation matrix Ð+ìííí ì îï ú³û õö ø æ ø æ ø ø û ö m k e m k e m k e ³ ³ m k e ö û C O ö k} e k} e k} e ³³ k} e ûû ö C O ö û kmÐ Ð e kmÐ e kmÐ e ³³ kmÐ î e ö û î Ð km e Ð Ð km C e Ð km O e Ð ö û ³³ km e ö û C O ô Ð î Ð Ð Ð Ð ö û L»ý .. Ðø .. Ðø . . .. Ðø EF ö .. Ðø (1.5) û . .Ð .Ð .Ð î ö÷ . Ð Ð û îbþ&î km e km e km e km O eù³³ C .. .. .. . . .. ü . . . . . Ð î ³³ k} e k} C e k} O e k} e By virtue of the addition theorem, one can directly see that the symmetric positive semidefiî ô ô Ð nite matrix ÿ EGF has the entries ø ® ® à à á á á ® ø k}Þ^;Þ+ewF k}Þo^³âe+^ kmÞ ^Kâ e- ÿ k^ eÓF k}ÞKe kmÞ>e©F Therefore, the matrix ÿ is the Gram matrix of the scaling functions and will be positive definite, in particular regular, if and only if the scaling functions are linearly independent. As ô ô det ÿ F det C , we can study the regularity of either or ÿ in order to determine whether the scaling functions constitute a basis of or not. Since the regularity or singularity ô of the matrix is independent of the basis of on hand, we will assume from now on that the underlying basis is the basis (1.2) of spherical harmonics. D EFINITION 1.3. A set of points HK V BDC for which the interpolation matrix ô µ µ is nonsingular or equivalently the associated scaling functions H«ñ V constitute a µ µ + Ð ì í í í ì î ï basis of , is called a fundamental system for . Ðìííí ì î 2. Construction of fundamental systems. With growing , the analysis of the regularô ity of the interpolation matrices becomes a very difficult task in general, so we have to restrict our analysis to specific choices of point constellations. A possible way of constructing fundamental systems for is due to v. Golitschek and Light [6] and Sünderman [5]. The key idea of their construction is to locate the ¥ nodes on w£6T parallel circles, such that the ã th k"ãF°]2^³K³>^fe circle contains doã£6T points. Another description of specific sets of points, which admit unique polynomial interpolation, is given by Xu [7, 8]. In his construction, the fundamental systems arise from relating an interpolation problem on the unit disc to an interpolation problem on the unit sphere: one starts with k}6£§T e>km%£ de d points lying on [ d`z£@T concentric circles inside the unit disc, such that one of the circles is the unit circle B itself and each circle contains £ T equidistantly distributed nodes. Finally, the points areÐ projected onto the northern and southern hemispheres of BDC , yielding thereby a symmetric distribution of ¥ points on the sphere. Unfortunately, this construction only works in the case of even polynomial degree . Here and from now on, we will assume that is an odd natural number and consider accordingly that the kmw£6T«e;C nodes are distributed on w£6T parallel circles, where each of them ETNA Kent State University etna@mcs.kent.edu 87 Localized polynomial bases on the sphere T v>xyr&l _ Ðì Ð C ìÐ d O ìÐ C O C ì ì Ðì C ì Ð+ì C O ì O Ð+ì C C C Ð ] WCzß ] O O O ì O ì ìÐ ì ì ì ] _ d_ F IG . 2.1. Case Ò . and Ö .q5 : distribution of the points in a local-coordinates-grid: the points are g×T _ distributed equidistantly on four symmetric latitudinal circles. bears u£ðT equidistantly distributed points. Moreover, the latitudinal circles are symmetric with respect to the equator and the points on them exhibit a rotational symmetry with respect to the axis that joins the north and south poles. However, as the next lemma shows, a distribution with km%£§T«e;C points lying simultaneously on {£»T meridians and on u£@T latitudes, does not yield a fundamental system for . L EMMA 2.1. Let L be an odd positive integer. Furthermore, denote with n óEGF d_fã k} £QT«e@kãIFZT^K³³³^;£ T«e and l LSkm])^_feðk ÌF T^K³K>^;£°T e pairwise different longitudinal and latitudinal angles, respectively. Then the system of points HYkml ^;n e+V ß does not constitute a fundamental system for . matrix ô . The proof follows directly from constructing the associated interpolation Wß ì Ð+ìííí ì Ð ø is straightforward C and It to check that the rows corresponding to the functions Wß C , k"fFkm£¹T«e d)^³³K>^;fe are linearly dependent. Ð Ñ In Ðview of the preceding lemma we have to consider different longitudes on the chosen Ñ circles HykmJ ^;J ^;J ebL BfCEJ Fcv³xor& l WVª!k »F To^³³K>^;u£@T e in order to obtain fundamenC O O tal systems. Based on this idea, the following theorem establishes a possible construction Ð principle for fundamental systems when is odd. T HEOREM 2.2. Let be an odd integer and let # ] " l "l "¤³³$ "l Wß " ø C C C and l ß ÝEGF¹_ÛgÒ l k PFT^K³K>^Kkm£°T«e de denote symmetric latitudinal angles. Then the C Ð Ð Ñ set of points %Ùk &e{EFëHK' ªEGF§Ykml W^n e+( ^ o^zãFT^K³³³^;£°TV , where ¬ ì if is odd, Czß ø ^ n F ß ) * C ß ^ if is even, Ð Ð Ð with & LÖk"])^de , constitutes a fundamental system for . Proof. In order to establish the linear independence of the scaling functions correspondô ing to the point set %Ù+k &Pe , we have to study the regularity of the interpolation matrix in (1.5). The trick of the proof is to reduce our kmw£6T«eC -dimensional problem to w£6T problems of dimension £@T . Using the structure of the spherical harmonics as functions of separated ô variables, we can convert our original interpolation matrix into an equivalent block diag® onal one consisting of £°T blocks , k}·¸FÕ])^K³³³^;fe of dimension km£¹T«eaÖk}£¹T«e by ETNA Kent State University etna@mcs.kent.edu 88 N. Laı́n Fernández regular transformations. Multiplying from the left hand side by a proper permutation matrix ô the rows of so that the matrix attains the form ® õö ú³û õö ú³û ® ö û ö E / · 2Õ] mod km£¹T«e>^P$ 3 ·Î ûû L%ý / 1 ö÷ . û ö÷ 0 E / · 2T mod km£¹T«e>^P$ 3 ·Î L%ý . 0 EF .. . ® . ü ü .Ð . L»ý E ·/2¹ mod k}£¹T«e+^$3ë ·Î 0 ® . , we can reorder Ð Wß ß W Ð þ)î ß Ð þ)î W ® Ð þ)î Here, 0 denotes the row vector containing the evaluation of harmonic at ® the spherical Wß the nodes HK' |^(o^zãFT^³K³>^Ý£ÎTWV . Within the matrices % L ý km·¸F¹]2^³K³>^fe ® . the functions – each of them determines one row of – are ordered in the following way Ð ) þ î ® ® ® ø ø .® ø ø ® ø ø ì ®® ® ø ® ø ® ^ ^ Wß ^ÓK³³^ ß ^³K³^ ^ Wß C Ð Ð Ð Ð the distributionÐ of the points on the parallel Furthermore, bearing in mind circles lF°l k ªF ® T^K³K>^;%£ëT«e , we can split the k}%£T ea ¥ -dimensional matrices . k}· Fc])^³K³>^fe into £°T square matrices of dimension k}w£6T«ewaÖk}w£6T e ® ® ® ® ß C ^³K³>^ q F k ^ e>^ ·QF¹]2^³³K>^;X^ . . . . ® Ð Ð ß Wß where . L ý contains the information relative to the points lying on the th latitudinal circle. Ð þ Ð Moreover, let 4 EG6 F 5798Dk"d_-ò k}6£T e;e and consider the km6£T«eaªkm6£T«e -dimensional Fourier matrix : Wß with entries ø ø ø ø ø ø < < >= = T T # ?CFA E @ Ð BD ; 4 F ; ´ :uWß !k o^zã)euEF (2.1) £¹T £¹T Ð Ð Ð Ð Ð Multiplication from the right hand side by the ¥ aª¥ -dimensional block diagonal matrix :§EF diag +k :uWß G^ :uWß ^³K³KG^ :uWß e yields õö ú û Wß Ð Ð Ð : ß û ö C : Wß ³ ³ : Wß û ö÷ . . . Wß :uß C :uWß ³³ . ô Ð :uWß . Ð . Ð Ð : F H (2.2) .. Ð .. Ð .. Ð ... ü . Ð Ð Ð Ð . Ð ÐWß . Ð C :uWß ³³ . :uWß :uWß . . ® Ð Ð :uWß Let us now focus our attention on the Ð k}Ö£hT«eDa6kmÐ Ò£hT e -dimensionalÐ matrices . and compute their entries. Since the longitudinal angles of the points on hand vary from latitude to latitude, it is necessary to examine the cases of odd and even separately. Also theÐ ® structure of the functions involved in each of the matrices . recommends the distinction ® of two cases. On the one hand, we study the entries of the bg·Õ£ÒT first rows, i.e., the ® ® ø ø e , and, on the other hand, we rows corresponding to the functions ß I @k JF ]2^³³K>^;»gÎø ·» * ® analyze the remaining · rows relative to the functions Wß ß I +k JÛF¹])^³K³>^· g T«e . * Ð Ð (I) Let be odd. a) For TLKMNK¨%gÒ·h£°T , we obtain O . ® :uWß P Ð kQ^R eÓF ; à ß ® ® ø T £°T Ð Û Ð Ð ß ËYcË|l d _fã ^ £°TÏAÏ ø 4 Ð ø Ð ETNA Kent State University etna@mcs.kent.edu 89 Localized polynomial bases on the sphere ® á F (2.3) where (2.4) á ® F ® ® ø á ® kmv³xor2l«e ß ® ® ø Ð ß kmv³xor2l«eF4 ® à ß 4 Ð ø Ð Wà ß Ð Ð denotes the constant ® á duk}· g Tw£UeD£°T ± ET F S ^ _ðk}Û£¹T«e ø ß ø Ð ß > ®Ð ø 4 ß ^ Ð Ð ÝFT^K³K>^;%g¡·¤£°To g 2£»T«V e 2c] mod k}{£»T«e . However, Note that the sum in (2.3) is nonzero if and only if km·°N since · L H«]2^³³K>^;V and ÛL HoTo^³K³+^;u£@TV , this implies that ©F·¨£»T . Accordingly, we ® ® ® obtain ® ø á ® O P km£¹T«e ^ if F ·¤£°To^ ß kmv³xor&l«eF4 :uWß k ^ e©X F W . ]2^ otherwise Ð b) Similarly, ifÐ @gÒ·h£¨dNKMNK¨£°T , then ø ® ® á Wø ß ® O P kmÛ£°T e C kmv>xyr)l F e 4 ^ if F¹·h£¹T^ : Wß k ^ eÓX F W . ]2^ otherwise Ð Ð Ð (II) Now let be even. T KMNK¨%g¡·Ù£°T , it holds a) For L ø ø Wà ß ® ® ø ® O P T k"d2k"ãg T«eM £ &Pe=_ ; : Wß k ^ e©F l ^ 4 ß ËYcË| . £°T ÏAÏ w£6T Ð Ð Ð ® ® ® ® ø ø ® ø Ð Wà ß Ð ® ø á ) ß ß ß > Ð C F l «e 4 ß kmv³xor& Ð Ñ Ð ® ® Ð ® ø ® ø Wß ® ø ® ø à á ) ß ß F e 4 Ð C 4 ^ (2.5) ß kmv³xor&l F Ð ® Ñ Ð Ð á Ð is nonzero if and only if where was already introduced in (2.4). Again, the sum in (2.5) Ð F°·Ù£¹T . Accordingly, it follows that ® ® ® ø á ® O P kmÛ£°T e l «F e 4ZY\# [ ^ if F¹·h£¹T^ ß kmv³xor& W : Wß k ^ eÓF . ]2^ otherwise Ð Ð b) In a similar way, if 6gÒ·h£¨dNK]ZK¨ø £¹ ® T , then ø øß á W ) ® O P kmÛ£°T e kmv³xor2 l «_ e 4 Y# [ ´ µ ^ if F¹·h£¹T^ C :uWß k ^ eÓ^ F W . otherwise ]2^ Ð ® Ð Ð these calculations, we realize that each of the matrices : Wß possesses In view of . only one nonzero column, namely the k}·¹£¡T«e st one. Moreover, bearing in mind the structure Ð multiply of the matrix in (2.2), we observe that we can obtain a block diagonal matrix if we from the right hand side by the following permutation matrix EGF/`ba ^ca Wß ^da C Wß ß ^³K³>^(a Wß ß ^ca ^Ga Wß ^da ß ß W C O C C C C K³Ð K^Ga W ß ß ^³KÐ³K^dÐ a ß ^ca Wß Ð ^ca Ð Wß ³^ K³>^ca ß Ð #fe C C O ô :gÐ Thereby, the product attains theÐ form diag k , ^, Ð ³^ K³³^, e , Ð where the matrix ® Ð C Wß Wß , L%ý k}·óF°])^K³K>^;fe has the following entries: Ð Ð Ð þ Ð - ETNA Kent State University etna@mcs.kent.edu 90 h N. Laı́n Fernández If TLK]ZK , ® ® ® ® ø á ® km£¹T«e ® ® ø á km£¹T«e ® 6gÒ·h£¹T , then k^@|e©F W ß ß kmv³xor2lKeF4 kmv³xor2 l KF e 4 Y\# [ ^ ^ if odd ^ if even Ð ø ® If %g¡·¤£\dZKZK¨Û£°T , then ® Ð ø á ß ® ø ® km£¹T«e C ø k"v>xyr&l«e94 ^ ø if odd ^ á , Qk ^i|eÓX F W ß ) if even. km£¹T«e C Ð k"v>xyr&l«e94 Y# [ ´ µ^ Ð Ð Summarizing, we have obtained a simpler representation of the ¥ëau¥ -dimensional maÐ ô trix as the block diagonal matrix diag +k , ^ , ^³K³³^ , e via multiplication by regular ô matrices. Hence, we have reduced the study of the regularity of to the analysis of the ® regularity of each of the lower dimensional matricesÐ , k}·°F6])^³K³K^;fe . In order to face this problem, we will make use of some properties of the calculus of determinants. Note ø ® also that it is enough® to restrict our analysis to the cases · F6]2^³³K³^«k}£ is equivalent to® , for ·¹F»])^³K³+^«k}Û£ÕT«e d . Indeed, the latter one is T«e d , since , Wß a permuted and scaled version of , . ® Ð constants appearing in the entries of each of the matrices , k}·èF Extracting the ])^K³K>^Kkm £ T«e de , first rowwise, then columnwise, and introducing the notation J EGF v>xyr&l !k F To^³³K>^;£¹T«e , we obtainø ® ® ® l ß Wl ß ® á á ?A ß ø ® C m ´ µ ´ Y[ # a Fqk}£¹T«e det , Zm n Ð Ð ß Ð j ® k j C ® ® ® ® ® o o ® ® ® Ð o o ® k}J e ® kmJ e ³K ® kmJ Wß e o o C o o ß } k J e } k J e ³ K } k J e ß ß ß o o C Ð Ð o o . . . . ® .. ® .. ® .. .. o o Ð Ð Ð Ð Ð o o o o ø k}® J e kmJ ø e ® ³K kmJ øWß ® e o o ø ø C o o ® ® ® ø ø ø Wß ß Wß o o ) ) (2.6) a o Wß ø ® Ð k}J e ´ ø µ Wß ø ® kmJ C e ³KÌ´ ø µ Wß ø ® Ð kmJ Wß e o o o ® ® ® ø ø ø o o Wß Ð ß Wß ) ) o Wß Ð k}J Ð e ´ µ Wß Ð Ð kmJ e ³KÌ´ µ Wß Ð Ð kmJ Wß Ð e o C o o C C C o o Ð . Ð Ð. o o . . Ð Ð .. .. .. .. o o ø ® ø ø ® ø ø ® o o o o ) ß ) Wß o Wß k}J e ´ µ kmJ e ³KÌ´ ø µ kmJ Wß e o C ) Ð Ð Ð appears in the even Note that in the lower half µ only Ð of the above matrix, the factor ´ Ð ) columns. For the sake of symmetry, let us multiply the last · rows by ´ µ C . Thereby, it ® ® remains to show that the determinant ® ® Ñ ® ® o o ® ® ® o o ß } k J e } k J e K ³ } k J e ® ® ® o o C o o W ß } k J e m k J e K ³ } k J e ß ß ß o o C Ð Ð . . . o o . ® .. ® .. ® .. .. o o Ð Ð Ð Ð Ð o o o o k}J ø ® e k}J ø e ® K³ k}J ø ß ® e ø ø o o C ® ® ® ø ø ø o o ?A Wß ?A Wß ?A ß o Ðø ® kmJ e ´ ø Y # ß ø ® k}J e K³Ì´ ø Y # Wß ø ® Ð kmJWß e oo (2.7) o ´ Y # Wß C o o o o ?A ?A ?A Wß Ð ø ® Wß Ð ø ® ß Ð ø ® o p ´ Y # Wß Ð kmJ Ð e ´ Y # ß Ð k}J e K³Ì´ Y # Wß Ð kmJ Wß Ð e o C o o C C C o o Ð Ð ..Ð . . o o .. Ð Ð .. .. o o . . ø ® ø ø ® ø ø ® o o o o ?A ?A ?A W ß W ß ß o ´ Y # p kmJ e ´ Y # k}J e K³Ì´ Y # kmJ Wß e o C Ð Ð Ð Ð Ð h ETNA Kent State University etna@mcs.kent.edu 91 Localized polynomial bases on the sphere is nonzero. Exploiting the parity of the associated Legendre functions, we can transform the ®sr r , where is the km@£T ejakm6£T«e above matrix by elementary row operations into q dimensional diagonal matrix ® ® ® r and q ® O F diag k=TugÒJ C e is given by õö ö ö ö J ö ö ö ö ö ö ö ^«k=TugÒJ C e C ^³³K>^Kk;Tug¡J CWß Ð e Ð T JC ..ø Ð ® . Ð J T ø C JjC ..ø C ® . ³³ ³³ ³³ .. . ³³ ³³ ³³Ú´ .. . P L@N Wß Ð þ Wß ^ Ð ú û T JWß JCWß ..ø ® Ð . Ð J ß ?A ´vu Y # k=TugÒJCWß ewt ?A Ð Y# J Wß k;TAgÒJCWß ie t Ð ® ø ... ü Ð Ð ?A Y# J Wß k;Tug¡JjW C ß ie t û û û û û û û û û û û C û ´ Y # k=TugÒJC eit =k TugÒJC eit Y # ö û C ö÷ û ?A ?A Ð ´ Y # J k;Tug¡JjC ewt ´ Y # J k;Tug¡JjC ewt u C C Ð ® ø ... ® ø ... ø Ð Ð ?A ?A ´ Y # J k;TAg¡JjC we t ´ Y # J k=TugÒJC ewt ³³ ´vu C C Ð Ð Ð Ð Ð with xÖFhkm¡Ð £qT«e dæg ·Ð . Consequently, in order to conclude the proof of Theorem 2.2, ® it remains to establish the regularity of the -dimensional matrices } k Ù £ T e » a k } Ò ¤ £ « T e q k}·¸FÕ])^K³K>^Kkm£¹T«e de . ® As it is shown in Theorem 2.7 of Laı́n Fernández [1], all these matrices q k}· F ])^K³³³^Kk}£¹T«e de are regular, which completes the proof. ô 2.1. Conditions of the matrices . According to Lemma 2.1, the point distribution described in Theorem 2.2 does not yield a fundamental system for when &×F6] or &ÝF%d . In order to understand better the influence of the parameter & on the stability of the interpolation ô problem, let us now study the asymptotic behavior of the condition number cond k &e when & approximates the extremal values ] or d . On account of the symmetric distribution of the ô ô Ð assume points in Theorem 2.2, we have that cond +k &PeoF cond kdAy g &Pe . Hence, we will without loss of generality that & LÖkm]2^³T>` . L EMMA 2.3. Let FT and let HK ^; ^; ^; V BDC be a fundamental system of the form C O described in Theorem 2.2. Then Ð ï ô z{| êæst-H v>xoc t } k &e{~ E &ÎLÒk"])^KT>`V FT^ ö ö ?A J ø´ J ?A ; ô Ð and cond latitude v³xor&l FqT k=T«e|FÒT if and only if the underlying ç is the positive zero of the Legendre polynomial . Moreover, C ø Ð Ð O P ô as &¡iM]2 v>xoc t } k &PeÓFÕå & + Ð Ð ô Proof. In order to estimate the condition number of the matrix k+&Pe in dependence on the shift & , let us compute the eigenvalues of the Gram matrix ÿ +&Pe . For the sake of k Ð simplicity, let AEFðv>xorKk &D_ doe and EGF»v³xorCl . It is straightforward to check that the matrix Ð ÿ k &e attains the form Ð ÷ õ ú 7 ' Ð 5j 4 ( ( Ç 5 * Ç * 5j 4 ( ( 5 Ç * Ç * Ç ' 7 T 5j 4 ( ( 5 7 Ç * Ç * 5j 4 ( '( Ç 5* Ç * ü Ç ± 5j4 (W( Ç 5 * Ç *S5j4 (W( 5 Ç * Ç * _ ' 7Ç 5j 4 ( W( 5 Ç * Ç *S5j 4 ( W( Ç 5* Ç * Ç ETNA Kent State University etna@mcs.kent.edu 92 N. Laı́n Fernández 7 35 10 Legendre nodes Tschebyscheff nodes equidistant nodes n=1 n=3 n=5 n=7 6 10 30 5 10 4 10 log cond2α Condition 25 20 3 10 15 2 10 10 5 0.5 1 10 0 1 angle shift 10 −5 10 1.5 α −4 −3 10 −2 10 10 −1 10 0 10 log α F IG . 2.2. On the left hand side: Condition number of in dependence of the shift for different latitudinal heights. Note that because of the symmetry of the sphere, cond ( )*K. cond ( Ç )* . The minimal condition number is achieved for .×5 . On the right hand side: log-log plot of cond ( )* , when tends to È . Since this matrix is block circulant with circulant blocks, one can compute its eigenvalues via diagonalization by Fourier matrices ± k &eÓF + ^ k |ewFëT« + d ^ k &eÓ + F k bg T«e³+k g T«e>^ y+k |e©F g 2k;TÓU £ &e>@k g¨T e+ C O Ð Considering now the cases UK and separately, it can be seen that the quotient cf cf O O its minimum kÿ k &e;e kÿ k+&Pee attains Ð Ð at the intersection point of the lines O k+&e and o+k |e , i.e. at the point F6] or equivalently &¡FT . Ð Accordingly, for &ÝÐ FT , the Gram matrix ÿ k=T e possesses the eigenvalues ± F ^ F 2k=Tw£\v>xyr2dl e>^ Ð FgAç2k=g×TÓ£\v>xyr2dl e> C O Ð Ð ì Ð only if v³xor&l F It is; now straightforward to prove that the matrix has equal eigenvalues if and T ç . For this value of l , the condition number of the matrix is equal to one and the scaling Ð functions form an orthogonal basis of . ø Ð Having computed the eigenvalues of ÿ +k &Pe , we can immediately conclude that O P Ð ^ & iÚ] by analyzing cond +k &Pe Fqå & the cases v>xyrC-l KhT ç and v>xyr;Cfl hT ç Ð separately. Ð Ð Ð ÐO ; P At this point it should be mentioned that for the special value of l F z{ v³v³xor T ç , the points constituting the set %Ùk;T«e , i.e. Ð F g ¯ j Ðì Ð C F j ]2^ ¯ d ^]2^ ç ç d ^³g T ; ^ ; m ç F ¯ C j Ðì T ç m ^ C C F d ç ^]2^ j ])^Kg ¯ ; d ç T ç ^Kg ^ m ; T ç m ^ ìÐ ì are the vertices of a regular tetrahedron inscribed in the sphere. Motivated by the results obtained for the case FóT , it is natural to ask whether it is possible to make similar assertions for higher values of . Unfortunately, so far it was only possible to conduct numerical experiments, which indeed support the conjecture that for ETNA Kent State University etna@mcs.kent.edu Localized polynomial bases on the sphere ø ø 93 ô 3 d the condition number cond k+&Pe behaves like åæk+& ] e , resp. åæk;k"dgU&e e , when the shift & approximates the singular values ] or d . Ð Ð On the right hand side of Figure 2.2, we display the behavior of the condition number ô cond k &e for small values of & and for the odd values of between T and . In order ô to bring out the asymptotic behavior of cond k &e , we have used logarithmic scales in the plot. According to our expectation, we obtain a straight line with slope equal to g×T . It can also be observed that with increasing the intersection of the graph with the -axis moves upwards. In the numerical tests, we have chosen equidistantly distributed latitudinal angles for our calculations. Remember that the considered fundamental systems in Theorem 2.2 allowed arbitrary heights of the latitudinal circles. ô k &e for the On the left hand side of Figure 2.2, we illustrate the behavior of cond O values of &qL [T d&^ç d` . Note that for 6F§T the minimal condition number is attained at &ÝFT , which corresponds to Lemma 2.3. On account of Lemma 2.3, for ÖFcT orthogonality of the scaling functions could be achieved by considering the zeros of the Legendre polynomial of the next higher degree as latitudes. Motivated by this fact, we now choose as heights for our parallel circles the zeros of orthogonal polynomials of degree four. In particular we consider the zeros of Legendre and Tschebyscheff polynomials of the first kind. This choice fits into the approach pursued by Fischer and Prestin [3], where polynomial wavelets on the interval were studied and orthogonality of the scaling functions was achieved by employing the zeros of the underlying orthogonal ô polynomials as nodes. As we might have expected, the condition number of the matrix k &Pe + O improves when we choose the zeros of the Legendre polynomials as heights of the latitudinal circles. The choice of equidistantly distributed heights ¡FQv³xorkzk"doãg T«e&_ k"dAk}æ£ÕT e;e;e yields similar condition numbers to the case of the Legendre nodes. REFERENCES [1] N. L A ÍN F ERN ÁNDEZ , Polynomial Bases on the Sphere, PhD thesis, in preparation. [2] N. L A ÍN F ERN ÁNDEZ , Polynomial Bases on the Sphere, in Proceedings IDoMAT, 2001, Int. Dortmund Meeting in Approximation Theory, Witten-Bommerholz, Birkhäuser, 2001. [3] B. F ISCHER AND J. P RESTIN , Wavelets based on orthogonal polynomials, Math. Comp, 66 (1997), pp. 1593– 1618. [4] C. M ÜLLER , Spherical Harmonics, Springer, Berlin, 1966. [5] B. S ÜNDERMANN , Projektionen auf Polynomräumen in mehreren Veränderlichen, PhD thesis, Diss. Dortmund, 1983. [6] M. V. G OLITSCHEK AND W. L IGHT , Interpolation by Polynomials and Radial Basis Functions on Spheres, Constr. Approx., 17 (2001), pp. 1–18. [7] Y. X U , Polynomial interpolation on the unit sphere, SIAM J. Numer. Anal., to appear. [8] Y. X U , Polynomial interpolation on the unit sphere and on the unit ball, Adv. in Com. Math., to appear.

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