Bounds for the Green Energy on SO (3) Damir Ferizovi c joint work - PowerPoint PPT Presentation

Bounds for the Green Energy on SO (3) Damir Ferizovi´ c joint work with Carlos Beltr´ an (Universidad de Cantabria) Institute of Analysis and Number Theory Graz University of Technology December 21, 2018

Green Functions Definition A Green function G ( x , y ) for a linear differential operator L is given as the distributional solution to L x G ( x , y ) = δ 0 ( x − y ); or put differently, if we want to solve Lu ( x ) = f ( x ) we set � u ( x ) = f ( y ) G ( x , y ) d y . *It follows that � � Lu ( x ) = f ( y ) L x G ( x , y ) d y = f ( y ) δ 0 ( x − y ) d y = f ( x ) .

f-Energies of N-Point Sets Definition Given a non-empty set X , N ∈ N and a function f : X × X → R ∪ {±∞} ; the (discrete) f -energy of X is given by N � E ( f , N ) = inf f ( x j , x k ) . { x 1 ,..., x N }⊂ X j � = k

f-Energies of N-Point Sets Definition Given a non-empty set X , N ∈ N and a function f : X × X → R ∪ {±∞} ; the (discrete) f -energy of X is given by N � E ( f , N ) = inf f ( x j , x k ) . { x 1 ,..., x N }⊂ X j � = k Example (Riesz potential) Regard the unit sphere S 2 ⊂ R 3 and for s > 0, let 1 f ( x , y ) = � x − y � s . For some s < 0 the problem also makes sense (Fejes-Toth potential). The case s = 0 sometimes refers to the logarithmic potential.

Why Care? Theorem (Beltr´ an, Corral, Del Cray) Let M be a compact Riemannian manifold of dimension n > 1 and let G be the (normalized) Green function for its Laplace-Beltrami operator. The unique probability measure minimizing the continuous Green energy �� I G [ µ ] = G ( x , y ) d µ ( x ) d µ ( y ) , M is the uniform measure on M. Moreover, for each N > 1 , let w ∗ N = { x 1 , . . . , x N } be a set of minimizers for the Green energy, then 1 ∗ � δ x ⇀ λ. N x ∈ w ∗ N *“Discrete and Continuous Green Energy on Compact Manifolds” by C. Beltrn, N. Corral, and J. G. Criado Del Rey, (2017).

Lemma The Green function for the Laplace-Beltrami operator on SO (3) is � ω ( α − 1 β ) � π − ω ( α − 1 β ) � � G ( α, β ) = cot − 1 . 2

Enter Determinantal Point Processes We will have for any measurable function f : M × M → [0 , ∞ ],   �  = f ( x i , x j ) E i � = j �� � K H ( x , x ) K H ( y , y ) − |K H ( x , y ) | 2 � f ( x , y ) d µ ( x ) d µ ( y ) , M where H ⊆ L 2 ( M ) is any N –dimensional subspace in the set of square–integrable functions and K H is the projection kernel onto H .

Let’s Start Simple A simple point process on a locally compact polish space Λ with reference measure µ is a positive Radon measure � χ = δ x j , j =1 with x j � = x s for j � = s . One usually identifies χ with a discrete subset of Λ.

Let’s Start Simple A simple point process on a locally compact polish space Λ with reference measure µ is a positive Radon measure � χ = δ x j , j =1 with x j � = x s for j � = s . One usually identifies χ with a discrete subset of Λ. The joint intensities of χ w.r.t. µ , if they exist, are functions ρ k : Λ k → [0 , ∞ ) for k > 0, such that for pairwise disjoint { D s } k s =1 ⊂ Λ � k � � � χ ( D s ) = ρ k ( y 1 , . . . , y k ) d µ ( y 1 ) . . . d µ ( y k ) , E D 1 × ... × D k s =1 and ρ k ( y 1 , . . . , y k ) = 0 in case y j = y s for some j � = s .

Putting Determinant into Determinantal Point Processes A simple point process is determinantal with kernel K , iff for every k ∈ N and all y j ’s � � ρ k ( y 1 , . . . , y k ) = det K ( y j , y s ) 1 ≤ j , s ≤ k . If the kernel is a projection kernel, then one speaks of a determinantal projection process . Hence if N � φ j ( x )¯ K ( x , y ) = φ j ( y ) j =1 for some orthonormal system of φ j ’s, then N � � | φ j ( y ) | 2 d µ ( y ) = N . � � � χ (Λ) = K ( y , y ) d µ ( y ) = E Λ Λ j =1

Putting Determinant into Determinantal Point Processes A simple point process is determinantal with kernel K , iff for every k ∈ N and all y j ’s � � ρ k ( y 1 , . . . , y k ) = det K ( y j , y s ) 1 ≤ j , s ≤ k . If the kernel is a projection kernel, then one speaks of a determinantal projection process . Hence if N � φ j ( x )¯ K ( x , y ) = φ j ( y ) j =1 for some orthonormal system of φ j ’s, then N � � | φ j ( y ) | 2 d µ ( y ) = N . � � � χ (Λ) = K ( y , y ) d µ ( y ) = E Λ Λ j =1 It follows from the Macchi–Soshnikov theorem that a simple point process with N points, associated to the projection on a finite subspace exists in Λ.

Class Functions and Integrals on SO (3) Definition (Rotation Angle Distance) For α, β ∈ SO (3), we set � Trace ( α − 1 β ) − 1 � ω ( α − 1 β ) = arccos . 2

Class Functions and Integrals on SO (3) Definition (Rotation Angle Distance) For α, β ∈ SO (3), we set � Trace ( α − 1 β ) − 1 � ω ( α − 1 β ) = arccos . 2 Lemma If we are given a function f ∈ L 1 ( SO (3)) such that we can find f ∈ L 1 ([0 , π ]) with f ( x ) = ˜ ˜ f ( ω ( x )) , then � π � f ( x ) d µ ( x ) = 2 f ( t ) sin 2 � t � ˜ d t . π 2 SO (3) 0 *“Surface Spline Approximation on SO(3)” by T. Hangelbroek, D. Schmid; Appl. Comput. Harmon. Anal. Volume 31, Issue 2, 169-184 (2011).

Eigen- Values and Vectors for the Laplacian on SO (3) Lemma The eigenvalues of ∆ in SO (3) are λ ℓ = ℓ ( ℓ + 1) for ℓ ≥ 0 . Moreover, if H ℓ is the eigenspace associated to λ ℓ , then the dimension of H ℓ is √ (2 ℓ + 1) 2 and an orthonormal basis of H ℓ is given by 2 ℓ + 1 D ℓ m , n where − ℓ ≤ m , n ≤ ℓ and D ℓ m , n are Wigner’s D–functions. It is known that l l � ω ( α − 1 β ) � �� � � D l m , n ( α ) D l m , n ( β ) = U 2 l cos , 2 m = − l n = − l where U 2 l ( x ) is the Chebyshev polynomial of second kind.

Calculating the Projection Kernel for SO (3) Thus a projection kernel on the space H L = ⊕ L ℓ =1 H ℓ is given by L l l � � � D l K ( α, β ) = (2 l + 1) m , n ( α ) D l m , n ( β ) l =0 m = − l n = − l L � ω ( α − 1 β ) � �� � = (2 l + 1) U 2 l cos 2 l =0 L = d � � T 2 l +1 ( x ) � dx � cos( ... ) l =0 = d 1 � 2 U 2 L +1 ( x ) � dx � cos( ... ) � ω ( α − 1 β ) � �� = C (2) cos . 2 L 2 Here, C (2) 2 L , L ≥ 0, is the sequence of Gegenbauer (ultraspherical) polynomials.

Series Representation of Green’s Functions Theorem Given a compact Riemannian manifold ( M , g ) , then a system of orthonormal eigenfunctions { φ k } ∞ k =1 of the Laplacian on M with corresponding eigenvalues { λ k } ∞ k =1 forms a basis for the Hilbert space L 2 ( SO (3)) ; “the” Green function is given by φ k ( x ) φ k ( y ) � G ( x , y ) = . λ k k ≥ 1

Series Representation of Green’s Functions Theorem Given a compact Riemannian manifold ( M , g ) , then a system of orthonormal eigenfunctions { φ k } ∞ k =1 of the Laplacian on M with corresponding eigenvalues { λ k } ∞ k =1 forms a basis for the Hilbert space L 2 ( SO (3)) ; “the” Green function is given by φ k ( x ) φ k ( y ) � G ( x , y ) = . λ k k ≥ 1 Lemma The Green function for the Laplace-Beltrami operator on SO (3) can be written in terms of the metric ω : � ω ( α − 1 β ) � π − ω ( α − 1 β ) � � G ( α, β ) = cot − 1 . 2

An Upper Bound for the Green Energy   � f ( x i , x j ) E  i � = j �� � K H ( x , x ) K H ( y , y ) − |K H ( x , y ) | 2 � = f ( x , y ) d µ ( x ) d µ ( y ) M �� ��� 2 � � 2 − � ω ( α − 1 β ) �� � � C (2) C (2) = SO (3) 2 G ( α, β ) 2 L (1) cos d µ ( α ) d µ ( β ) 2 L 2 � � ω ( α − 1 ) ��� 2 � � 2 − �� � � C (2) C (2) = G ( α, 1) 2 L (1) cos d µ ( α ) 2 L 2 SO (3) � π = − 2 ��� 2 � t � 2 � t � �� � C (2) ( π − t ) cot( t 2 ) − 1 cos sin d t 2 L 2 π 2 0 ...some technicalities occur... � 4 � 3 3 N 4 3 + O ( N ) . = − 4 4

Handling Technicalities Lemma The Gegenbauer polynomials C (2) n − 2 ( x ) satisfy � 1 � 2 ( x 2 − 1) � d x = O ( n 2 log( n )) . C (2) n − 2 ( x ) 0 Lemma The Gegenbauer polynomials C (2) n − 2 ( x ) satisfy � 1 d x = n 4 � 2 � 16 + O ( n 2 log( n )) . C (2) n − 2 ( x ) 0 Actually we have exact formulae.

A Lower Bound for the Green Energy We define for α, β ∈ SO (3) and t > 0: ∞ e − l ( l +1) · t 2 l + 1 � ω ( α − 1 β ) � �� � G t ( α, β ) = l ( l + 1) U 2 l cos . 2 l =1

A Lower Bound for the Green Energy We define for α, β ∈ SO (3) and t > 0: ∞ e − l ( l +1) · t 2 l + 1 � ω ( α − 1 β ) � �� � G t ( α, β ) = l ( l + 1) U 2 l cos . 2 l =1 Lemma (N. Elkies) For all t > 0 and α � = β we have G ( α, β ) ≥ G t ( α, β ) − t .

A Lower Bound for the Green Energy We define for α, β ∈ SO (3) and t > 0: ∞ e − l ( l +1) · t 2 l + 1 � ω ( α − 1 β ) � �� � G t ( α, β ) = l ( l + 1) U 2 l cos . 2 l =1 Lemma (N. Elkies) For all t > 0 and α � = β we have G ( α, β ) ≥ G t ( α, β ) − t . Then for any collection of distinct points { α 1 , . . . , α N } ⊂ SO (3) N N � � G ( α s , α k ) + N ( N − 1)2 t ≥ G 2 t ( α s , α k ) . s � = k s � = k