Intro to harmonic analysis on groups Risi Kondor . The Fourier - PowerPoint PPT Presentation

Intro to harmonic analysis on groups Risi Kondor

. The Fourier series (1807) Any (sufficiently smooth) function f on the unit circle (equivalently, any 2 π –periodic f ) can be decomposed into a sum of sinusoidal waves ∫ 2 π ∞ ∑ c n = 1 f ( x ) e − ikx dx. c n e ikx f ( x ) = 2 π 0 k = −∞ • Workhorse of much of applied mathematics. • Exact conditions under which it works get messy. E.g., for f ∈ L 2 ([0 , 2 π )) almost everywhere convergence proved only in 1966 (Carleson). 2 / 37 2/37 .

. The Fourier transform ∫ ∫ f ( k ) e 2 πikx dk f ( x ) e − 2 πikx dx � � f ( x ) = f ( k ) = • Duality between time domain and Fourier domain (wave/particle duality in quantum mechanics) • Heisenberg uncertainty principle • Easily generalizes to R p . 3 / 37 3/37 .

. The discrete Fourier transform (DFT) n − 1 n − 1 ∑ ∑ f ( k ) = 1 f ( k ) e 2 πikx/n � � f ( x ) e − 2 πikx/n f ( x ) = n k =0 x =0 • Unitary transform C n → C n (with appropriate normalization). • Can be seen as discretized version of Fourier series, or as the Fourier transform on { 0 , 1 , 2 , . . . , n − 1 } . • Foundation of all of digital signal processing. • Fast Fourier transforms reduce computation time from O ( n 2 ) to O ( n log n ) [Cooley & Tukey, 1965]. 4 / 37 4/37 .

Underlying principles

. 1. Analytic Take a measurable space X , a space of functions on X , say L 2 ( X ) , and a self-adjoint smoothing operator Υ . For example, on X = R p , Υ may be the time t diffusion operator ∫ 1 f ( y ) e −∥ x − y ∥ 2 / (4 t ) dy. (Υ f )( x ) = √ 4 πt Question: How does Υ filter L 2 ( X ) into a nested sequence of spaces W Ω = { f ∈ L 2 ( X ) | |⟨ f, Υ f ⟩ / ⟨ f, f ⟩| ≤ Ω } ? 6 / 37 6/37 .

. 2. Algebraic Now let a group G act on X inducing linear operators T g : L 2 ( X ) → L 2 ( X ) . E.g., on on X = R p , g ∈ R p . ( T g f )( x ) = f ( x − g ) Question: What are the smallest spaces fixed by these operators, T g ( V ) = V ∀ g ∈ G ? 7 / 37 7/37 .

. On R p we are lucky because these two notions match up: • The diffusion operator is e t ∇ 2 , where ∇ 2 is the Laplacian ∇ 2 = ∂ 2 + ∂ 2 + . . . + ∂ 2 . ∂x 2 ∂x 2 ∂x 2 p 1 2 • The e 2 πik · x Fourier basis functions are eigenfunctions of both ∆ and T g : ◦ ∇ 2 e 2 πik · x = − 4 π 2 ∥ k ∥ 2 e 2 πik · x , ◦ T g e 2 πik · x = e 2 πik · g e 2 πik · x . • Therefore f ( k ) = 0 if ∥ k ∥ 2 ≥ Ω } (band-limited functions) ◦ W Ω = { f | � ◦ V κ = { f | � f ( k ) = 0 if k ̸ = κ } (isotypics) Question: Does this correspondence hold more generally? 8 / 37 8/37 .

. Fourier analysis on graphs On a finite graph G , the analog of ∆ is the graph Laplacian   1 i ∼ j  [ L ] i,j = − d i i = j   0 otherwise . It does lead to a natural measure of smoothness: ∑ f ⊤ Lf = − ( f i − f j ) 2 . i ∼ j Analyzing functions in terms of the eigenfunctions of L is called spectral graph theory . However (in general) on graphs there is no analog of translation. 9 / 37 9/37 .

More properties of the FT on R

. • The Fourier transform is ◦ Linear ◦ Invertible ◦ ∫ ∫ � f ( x ) g ( x ) ∗ dx = f ( k ) � g ( k ) ∗ dk (Parseval thm) Therefore, it is essentially a unitary change of basis. 11 / 37 11/37 .

. • Diagonalizes the derivative operator: g ( x ) = d g ( k ) = 2 πik � dx f ( x ) = ⇒ � f ( k ) . • Diagonalizes the Laplacian: g ( x ) = d 2 g ( k ) = − 4 πk 2 � dx 2 f ( x ) = ⇒ � f ( k ) . 12 / 37 12/37 .

. • Translation theorem: g ( k ) = e − 2 πikt � g ( x ) = f ( x − t ) = ⇒ � f ( k ) • Scaling theorem: f ′ ( k ) = | λ | − 1 � � g ( x ) = f ( λx ) = ⇒ f ( k/λ ) 13 / 37 13/37 .

. • Convolution theorem: ∫ ⇒ � f ∗ g ( k ) = � ( f ∗ g )( x ) = f ( x − y ) g ( y ) dy = f ( k ) · � g ( k ) • Cross-correlation theorem: ∫ ⇒ � f ⋆g ( k ) = � f ( y ) ∗ g ( x + y ) dy = f ( k ) ∗ · � ( f ⋆ g )( x ) = g ( k ) ◦ Autocorrelation: ∫ ⇒ � h ( k ) = ∥ � f ( k ) ∥ 2 h ( x ) = f ( y ) ∗ f ( x + y ) dy = 14 / 37 14/37 .

Fourier analysis on compact groups

. Fourier tranform on R ∫ f ( x ) e − 2 πikx dx � f ( k ) = Observation: χ k ( x ) = e − 2 πikx are exactly the characters of R . 16 / 37 16/37 .

. Locally Compact Abelian groups The Fourier transform of a function on an LCA group G with Haar measure µ is ∫ � χ ∈ � f ( χ ) = f ( x ) χ ( x ) dµ G. G • The dual object is itself a group: T ↔ Z , R ↔ R , and for finite groups G ∼ � = G (Pontryagin duality). • This covers the Fourier series and the Fourier transform. 17 / 37 17/37 .

. Compact non-Abelian groups The Fourier transform of a function on a compact group G with Haar measure µ is ∫ � f ( ρ ) = f ( x ) ρ ( x ) dµ ( x ) ρ ∈ R , G where R is a complete set of inequivalent irreducible representations (irreps). • Now the dual object is no longer a group, but a set of representations (Tannaka–Krein duality). • If G is finite, R is finite. If G is compact, R is countable. • Each Fourier component � f ( ρ ) is a matrix . In the following, we will always assume that each ρ is unitary. Every representation is over C . 18 / 37 18/37 .

Properties

. Invertibility Forward transform: ∫ � f ( ρ ) = f ( x ) ρ ( x ) dµ ( x ) ρ ∈ R . G Inverse transform: [ ] ∑ 1 f ( ρ ) ρ ( x − 1 ) � f ( x ) = d ρ tr x ∈ G. µ ( G ) ρ ∈R • Just as before (with respect to the appropriate scaled matrix norms), this transform is unitary. √ • The e ρ i,j ( x ) = d ρ [ ρ ( x )] i,j functions form an orthogonormal basis (Peter-Weyl theorem). 20 / 37 20/37 .

. Left-translation Theorem. Given f : G → C and t ∈ G , define f t ( x ) = f ( t − 1 x ) . Then f t ( ρ ) = ρ ( t ) · � � f ( ρ ) ρ ∈ R . Proof. ∫ ∫ f t ( x ) ρ ( x ) dµ ( x ) = f ( t − 1 x ) ρ ( x ) dµ ( x ) = ∫ ∫ f ( x ) ρ ( t ) ρ ( x ) dµ ( x ) = ρ ( t ) � f ( x ) ρ ( tx ) dµ ( x ) = f ( ρ ) 21 / 37 21/37 .

. Left-translation • Convolution theorem: ∫ ⇒ � f ∗ g ( ρ ) = � f ( xy − 1 ) g ( y ) dµ ( y ) = ( f ∗ g )( x ) = f ( ρ ) · � g ( ρ ) • Cross-correlation theorem: ∫ ⇒ � f ⋆g ( ρ ) = � f ( xy ) g ( y ) ∗ dµ ( y ) = g ( ρ ) † ( f ⋆ g )( x ) = f ( ρ ) · � 22 / 37 22/37 .

. Right-translation Theorem. Given f : G → C and t ∈ G , define f ( t ) ( x ) = f ( xt − 1 ) . Then f ( t ) ( ρ ) = � � f ( ρ ) · ρ ( t ) ρ ∈ R . Proof. ∫ ∫ f ( t ) ( x ) ρ ( x ) dµ ( x ) = f ( xt − 1 ) ρ ( x ) dµ ( x ) = ∫ ∫ f ( x ) ρ ( x ) ρ ( t ) dµ ( x ) = � f ( x ) ρ ( xt ) dµ ( x ) = f ( ρ ) ρ ( t ) 23 / 37 23/37 .

. Invariant subspaces • To left-translation: W ρ,j = span { e ρ i,j | j = 1 , . . . , d ρ } ρ ∈ R j = 1 , . . . , d ρ . • To right-translation: W ρ,i = span { e ρ i,j | i = 1 , . . . , d ρ } ρ ∈ R i = 1 , . . . , d ρ . • To left- and right-translation: V ρ = span { e ρ i,j | i, j = 1 , . . . , d ρ } ρ ∈ R . 24 / 37 24/37 .

. The group algebra The group algebra C [ G ] is a space with orthonormal basis { e x | x ∈ G } and a notion of multipilication defined by e x e y = e xy ∀ x, y ∈ G. Letting ⟨ f, e x ⟩ = f ( x ) and extending to the rest of C [ G ] by linearity, for any f, g ∈ C [ G ] , ∫ f ( xy − 1 ) g ( y ) dµ ( y ) = ( f ∗ g )( x ) . ( fg )( x ) = The group algebra of any compact group is semi-simple, i.e., it decomposes into a direct sum of simple algebras. 25 / 37 25/37 .

. The group algebra • The group algebra decomposes into a sum of simple algebras: ⊕ C [ G ] = V ρ . (1) ρ Each V ρ is called an isotypic , and corresponds to a single Fourier matrix � f ( ρ ) . This decomposition is unique. • Each V ρ further decomposes into a sum of d ρ left G –modules V ρ = W ρ 1 ⊕ W ρ 2 ⊕ . . . ⊕ W ρ (2) d ρ corresponding to each column of � f ( ρ ) . This decomposition is not unique, i.e., it depends on the choice of R . The Fourier transform is a projection of f onto a basis adapted to (1) and (2). 26 / 37 26/37 .

Fourier analysis on homogeneous spaces

. Group actions • So far we have considered: ◦ f is a function on a compact group G . ◦ G acts on G by t : x �→ tx inducing f �→ f t , where f t ( x ) = f ( t − 1 x ) (similarly for the right-action and right-translation). • In practice it is often more common that: ◦ f is a function on a set X . ◦ G acts on X transitively by t : x �→ tx , inducing f �→ f t , where f t ( x ) = f ( t − 1 x ) . Example: The rotation group SO(3) and the sphere S 2 . The symmetric group acting on a matrix by permuting rows/columns. 28 / 37 28/37 .

. Homogeneous spaces Assume that G acts on X transitively by x �→ gx . • Pick some x 0 ∈ X . • The subset of G fixing x 0 is a subgroup H of G . • Each set gH = { gh | h ∈ H } is called a left H –coset. • The set of left H –cosets we denote G/H . • { gH | gH ∈ G/H } forms a partition of G . • yx 0 = y ′ x 0 if and only if y, y ′ belong to the same coset. • Therefore, we have a bijection X ↔ G/H. X is called a homogeneous space of G . Example: S 2 ∼ SO(3) / SO(2) . 29 / 37 29/37 .