Gegenbauer polynomials and positive deniteness Christian Berg University of Copenhagen, Denmark Orsay, Paris November 27, 2015 Based on joint work with Emilio Porcu, University Federico Santa Maria Valparaíso, Chile Christian Berg Gegenbauer polynomials Overview 1. Presentation of the problem and main results 2. Reminder about Gegenbauer polynomials 3. The connection to spherical harmonics 4. Positive denite functions on locally compact groups G d d 5. The class P(S ; G) of positive denite functions on S × G 6. The class P(S1; G) and its representations 7. Some indications of proof 8. Applications to some homogeneous spaces Christian Berg Gegenbauer polynomials Presentation of the problem In Geostatistics one examines measurements depending on the location on the earth and on time. This leads to Random Fields of 2 stochastic variables Z(ξ; u) indexed by (ξ; u) belonging to S × R, 2 where S the 2-dimensional sphereis a model for the earth and R is a model for time. If the variables are real-valued, one considers a basic probability space (Ω; F; P), where all the random variables Z(ξ; u) are dened as measurable mappings from Ω to R. The covariance of two stochastic variables X ; Y is by denition cov(X ; Y ) = E((X − E(X ))(Y − E(Y )): For n variables X1;:::; Xn the covariance matrix cov X X n [ ( k ; l )]k;l=1 is symmetric and positive semi-denite. Christian Berg Gegenbauer polynomials Isotropic and stationary covariance kernels One is interested in isotropic and stationary random elds 2 Z(ξ; u); (ξ; u) 2 S × R, i.e., the situation where there exists a continuous function f :[−1; 1] × R ! R such that the covariance kernel is given as 2 cov(Z(ξ; u); Z(η; v)) = f (ξ · η; v − u); ξ; η 2 S ; u; v 2 R: Here ξ · η = cos(θ(ξ; η)) is the scalar product equal to cosine of the length of the geodesic arc (=angle) between ξ and η. We require with other words that the covariance kernel only depends on the geodesic distance between the points on the sphere and on the time dierence. Christian Berg Gegenbauer polynomials First main result 2 We shall characterize the class P(S ; R) of continuous functions f :[−1; 1] × R ! R which are positive denite in the following sense: 2 For any n 2 N and any (ξ1; u1);:::; (ξn; un) 2 S × R the matrix f u u n [ (ξk · ξl ; l − k )]k;l=1 is symmetric and positive semi-denite. Theorem (B-Porcu 2015) 2 The functions f 2 P(S ; R) are precisely the functions 1 1 X X f (x; u) = 'n(u)Pn(x); 'n(0) < 1; n=0 n=0 where ('n) is a sequence of real-valued continuous positive denite functions on R and Pn are the Legendre polynomials on [−1; 1] normalized as Pn(1) = 1. The series is uniformly convergent. Christian Berg Gegenbauer polynomials Generalizations This Theorem can be generalized in various ways: 2 d The sphere S can be replaced by S ; d = 1; 2;:::. The additive group R can be replaced by any locally compact group G. 2 1 The sphere S can be replaced by the Hilbert sphere S . d We shall characterize the set P(S ; G) of continuous functions f :[−1; 1] × G ! C such that the kernel f (ξ · η; u−1v) d 2 is positive denite on (S × G) . Here d = 1; 2;:::; 1: If G = feg is the trivial group we get a classical Theorem of Schoenberg from 1942 about positive denite functions on spheres. Christian Berg Gegenbauer polynomials Gegenbauer polynomials To formulate these generalizations we need to recall: (λ) The Gegenbauer polynomials Cn for λ > 0 are given by the generating function 1 2 −λ X (λ) n (1 − 2xr + r ) = Cn (x)r ; jrj < 1; x 2 C: (1) n=0 For λ > 0, we have the classical orthogonality relation: Z 1 n 2 21−2λ 1 x2 λ−1=2C (λ) x C (λ) x dx πΓ( + λ) (2) ( − ) n ( ) m ( ) = 2 δm;n: −1 Γ (λ)(n + λ)n! Christian Berg Gegenbauer polynomials Chebyshev polynomials For λ = 0 we use the generating function 1 1 − xr X (0) n = Cn (x)r ; jrj < 1; x 2 : (3) 1 − 2xr + r 2 C n=0 It is well-known that (0) Cn (x) = Tn(x) = cos(n arccos x); n = 0; 1;::: are the Chebyshev polynomials of the rst kind. Z 1 π if n 0 1 x2 −1=2T x T x dx 2 δm;n > (4) ( − ) n( ) m( ) = if n 0 −1 πδm;n = ; (λ) Warning: λ ! Cn (x) is discontinuous at λ = 0. Christian Berg Gegenbauer polynomials More on Gegenbauer polynomials Putting x = 1 in the generating functions yields (λ) Cn (1) = (2λ)n=n!; λ > 0; Tn(1) = 1: Recall that for a 2 C (a)n = a(a + 1) ··· (a + n − 1); n ≥ 1; (a)0 = 1: It is of fundamental importance that (λ) (λ) jCn (x)j ≤ Cn (1); x 2 [−1; 1]; λ ≥ 0: The special value λ = (d − 1)=2 is relevant for the d-dimensional sphere d d+1 S = fx 2 R j jjxjj = 1g; d 2 N (d−1)=2 because of the relation of Cn to spherical harmonics. Christian Berg Gegenbauer polynomials Spherical harmonics d d A spherical harmonic of degree n for S is the restriction to S of a d+1 real-valued harmonic homogeneous polynomial in R of degree n. d Hn(d) = fspherical harmonics of degree ng ⊂ C(S ) d is a nite dimensional subspace of the continuous functions on S . We have (d)n−1 Nn(d) := dim Hn(d) = (2n + d − 1); n ≥ 1; N0(d) = 1: n! The surface measure of the sphere is denoted !d , and it is of total mass 2π(d+1)=2 jj!d jj = : Γ((d + 1)=2) The spaces Hn(d) are mutual orthogonal subspaces of the Hilbert 2 d space L (S ;!d ), which they generate. Christian Berg Gegenbauer polynomials Ultraspherical polynomials 2 d This means that any F 2 L (S ;!d ) has an orthogonal expansion 1 1 X 2 X 2 F = Sn; Sn 2 Hn(d); jjF jj2 = jjSnjj2; n=0 n=0 2 d where the rst series converges in L (S ;!d ), and the second series is Parseval's equation. Here Sn is the orthogonal projection of F onto Hn(d) given as Nn(d) Z Sn(ξ) = cn(d; ξ · η)F (η) d!d (η); d jj!d jj S where ((d−1)=2) ((d−1)=2) cn(d; x) = Cn (x)=Cn (1): These polynomials are called ultraspherical polynomials or d-dimensional Legendre polynomials. The 2-dimensional Legendre polynomials are the classical Legendre polynomials previously denoted Pn. Christian Berg Gegenbauer polynomials Orthogonality relation for ultraspherical polynomials Specializing the orthogonality relation for the Gegenbauer polynomials to λ = (d − 1)=2: Z 1 2 d=2−1 jj!d jj (1 − x ) cn(d; x)cm(d; x) dx = δm;n: −1 jj!d−1jjNn(d) (Dene jj!0jj = 2). Note that jcn(d; x)j ≤ 1; x 2 [−1; 1]: Christian Berg Gegenbauer polynomials Schoenberg's Theorem from 1942 d Let P(S ) denote the class of continuous functions f :[−1; 1] ! R d such that for any n 2 N and for any ξ1; : : : ; ξn 2 S the n × n symmetric matrix f n [ (ξk · ξl )]k;l=1 is positive semi-denite. Theorem (Schoenberg 1942) d A function f :[−1; 1] ! R belongs to the class P(S ) if and only if 1 X f (x) = bn;d cn(d; x); x 2 [−1; 1]; n=0 for a non-negative summable sequence b 1 given as ( n;d )n=0 N d Z 1 jj!d−1jj n( ) 2 d=2−1 bn;d = f (x)cn(d; x)(1 − x ) dx: jj!d jj −1 Christian Berg Gegenbauer polynomials Positive denite functions on groups Consider an arbitrary locally compact group G, where we use the multiplicative notation, and in particular the neutral element of G is denoted e . In the representation theory of these groups the following functions play an crucial role. A continuous function f : G ! C is called positive denite if for any n 2 N and any u1;:::; un 2 G the n × n-matrix f u−1u n [ ( k l )]k;l=1 is hermitian and positive semi-denite. By P(G) we denote the set of continuous positive denite functions on G. Christian Berg Gegenbauer polynomials A generalization d We shall characterize the set P(S ; G) of continuous functions f :[−1; 1] × G ! C such that the kernel −1 d f (ξ · η; u v); ξ; η 2 S ; u; v 2 G (5) is positive denite in the sense that for any n 2 N and any d (ξ1; u1);::: (ξn; un) 2 S × G the n × n-matrix f u−1u n (6) (ξk · ξl ); k l ) k;l=1 is hermitian and positive semi-denite. d Note that for G = feg we can identify P(S ; G) with P(Sd ). Christian Berg Gegenbauer polynomials Simple properties Proposition d (i) For f1; f2 2 P(S ; G) and r ≥ 0 we have rf1; f1 + f2, and d f1 · f2 2 P(S ; G). d (ii) For a net of functions (fi )i2I from P(S ; G) converging pointwise to a continuous function f :[−1; 1] × G ! C, we d have f 2 P(S ; G).