Reproducing formulas associated with symbols

Reproducing formulas associated with symbols Filippo De Mari Ernesto De Vito Università di Genova, Italy Modern Methods of Time-Frequency Analysis II Workshop on Applied Coorbit space theory September 1th to December 15th, 212 Erwin Schroedinger Institute Vienna Ernesto De Vito (Genova) Reproducing formulas associated and symbols 1 / 26

Signal analysis and unitary representations a Hilbert space H H is the space of signals a locally compact group G with a left Haar measure dg (G, dg) is the space of parameters a unitary representation U = {U g } g G of G acting on H with a vector η satisfying the reproducing formula f 2 = f, U g η 2 dg f H G {U g η} g G is a continuous tight frame with associated voice transform V : H L 2 (G, dg) Vf (g) = f, U g η H V intertwines U with the (left) regular representation λ VU g f (h) = Vf (g 1 h) = (λ g Vf )(h) U is called a reproducing representation square integrable if U is irreducible η is called an admissible vector Ernesto De Vito (Genova) Reproducing formulas associated and symbols 2 / 26

Wavelets - I the Hilbert space is H = L 2 (R, dx) the group is G = {g = (b, a) b, a R, a } = a x + b (b, a)(b, a ) = (b + ab, aa ) dg = a 2 dbda the representation U = {U b,a } (b,a) G is U b,a : L 2 (R, dx) L 2 (R, dx) U b,a f (x) = a 1 x b 2 f ( a ) U is square integrable (irreducible and reproducing) a function η : R C is admissible if and only if η 2 2 = ˆη(ξ) 2 ˆη(ξ) 2 dξ < + and dξ = 1 ξ R R } {{ } Calderon condition where ˆη is the Fourier transform of η Ernesto De Vito (Genova) Reproducing formulas associated and symbols 3 / 26

Wavelets - II The corresponding voice transform is given by Vf (b, a) = a 1 2 = R R f (x)η( x b a ) dx f (x)η a,b (x) dx where η b,a is a translation of a dilation of η Ernesto De Vito (Genova) Reproducing formulas associated and symbols 4 / 26

Usual setting for d-dimensional signals H = L 2 (R d, dx) where dx is the Lebesgue measure G = R d H g = (t, h) [ t1. ] R d is the group of all translations: t = H is a group of dilations: h H is a d d invertible matrix the multiplication in G is (t, h)(t, h ) = (t + ht, hh ) i.e. G is the semi-direct product of H and the normal abelian factor R d the representation U is the quasi-regular representation on L 2 (R d, dx) U t,h f (x) = det h 1/2 f (h 1 (x t)) t d the infinitesimal generators of translations U (,...,tj,...,,i T j f := lim d )f f tj t j = f x j j = 1,..., d Ernesto De Vito (Genova) Reproducing formulas associated and symbols 5 / 26

Admissible vectors if U is irreducible - I since U is irreducible on L 2 (R d, dx), the action of H in the frequency domain R d is transitive (up to a negligible set) for almost all row vectors ξ R d there exists h(ξ) H such that ξ h(ξ) = ξ for some fixed origin ξ R d (the map ξ h(ξ) is called a a section) in general the measure dg is left invariant, but it is not right invariant F(g 1 h) dh = F(h) dh G G F(hg) dh = G (g 1 ) G G F(h) dh where G : G (, + ) is a character (called the modular function) the stability subgroup at ξ is H = {h H ξ = ξ h} Ernesto De Vito (Genova) Reproducing formulas associated and symbols 6 / 26

Admissible vectors if U is irreducible - I Theorem 1 (Kleppner-Lipsmann, 1972; Bernier-Taylor, 1996) the representation U is reproducing if and only if the stability subgroup H is compact a function η : R d C is an admissible vector if and only if η 2 2 = Wavelet group With the choice ξ = 1 R d η(ξ) 2 dξ < + the stability subgroup is H = {1} the section is h(ξ) = ξ 1 and the modular function is G (b, a) = a 1 R d η(ξ) 2 G (h(ξ)) dξ = 1 Ernesto De Vito (Genova) Reproducing formulas associated and symbols 7 / 26

Admissible conditions if U is not irreducible (Führ-Mayer, 22) Necessary and sufficient conditions based on Plancherel non-commutative formula. See H. Führ, Abstract harmonic analysis of continuous wavelet transforms, Springer-Verlag 25. For example, η is admissible if and only if for almost all ξ R d H η(ξh) 2 dh = 1 Ernesto De Vito (Genova) Reproducing formulas associated and symbols 8 / 26

Symbols (elementary definition) take a (smooth) function τ : R d R the symbol D(τ) is the (unbounded) skew-adjoint operator defined by D(τ) : D L 2 (R d, dx) D(τ) f (x) = 2πi τ(ξ) ˆf (ξ) e 2πi ξ x dξ R d D ={f L 2 (R d, dx) R d τ(ξ)ˆf (ξ) 2 dξ < + } Examples if τ(ξ) = τ(ξ 1,..., ξ d ) = ξ j, then D(τ) = x j = T j if τ is a polynomial, then D(τ) is a differential operator, for example τ(ξ) = ξ t ξ = ξ 2 = D(τ) = 1 2πi Ernesto De Vito (Genova) Reproducing formulas associated and symbols 9 / 26

Unitary representation associated with symbols - I [ t1. ] a group R n of n translations: t = n can be different from d t n a group H of dilations: h H is a d d invertible matrix a linear action M = {M h } h H of H on R n : M h is a n n invertible matrix n symbols τ 1,..., τ n : R d R Two pairs of dual variables M Id = I n M hh = M h M h τ(ξ) = (τ 1 (ξ),..., τ n (ξ)) R n i) (x, ξ) d-dimensional vectors: d is the dimension of the configuration space ii) (t, τ) n-dimensional vectors: n is the number of generalized translations x h x and H acts on each variable: namely ξ ξ h a covariant condition on the symbols and t M h t τ τ M h τ(ξ) M h = τ(ξ h) h H ξ R d Ernesto De Vito (Genova) Reproducing formulas associated and symbols 1 / 26

Unitary representation associated with symbols - II H = L 2 (R d, dx) the group G is R n H with multiplication (t, h)(t, h ) = (t + M h t, hh ) the Haar measure and the modular function of G are dt dh dg = G (t, h) = H(h) det M h det M h dt is n-dimensional Lebesgue measure dh and H are the Haar measure and the modular function of H the representation U acts on L 2 (R d, dx) by U t,h f (x) = det h 1 2 e 2πi τ(ξ) t ˆf (ξ h) e 2πi ξx dξ the infinitesimal generators of translations are the symbols R d T j = D(τ j ) j = 1,..., n Ernesto De Vito (Genova) Reproducing formulas associated and symbols 11 / 26

Motivation: metaplectic reproducing groups G is a triangular subgroup of the symplectic group Sp(d, R) [ t h G (t, h) 1 ] σ t t h 1 Sp(d, R) h where Σ is an n-dimensional subspace of symmetric d d-matrices [ t1. ] Σ = {t 1 σ 1 +... + t n σ n t = R n } H is a subgroup of GL(d, R) G is a subgroup of Sp(d, R) if and only if M h σ := hσ t h Σ σ Σ, h H (1) t n the symbols are the quadratic forms associated to σ 1,..., σ n τ j (ξ) = 1 2 ξ σ j t ξ Û is the restriction to G of the metaplectic representation of Sp(d, R) Ernesto De Vito (Genova) Reproducing formulas associated and symbols 12 / 26

Theorem 2 (De Mari, D., ACHA 212) If representation U is reproducing, then i) G is non-unimodular (provided that τ(ξ) is homogeneous in ξ) ii) n d iii) the set of critical points of the map ξ τ(ξ) is negligible Furthermore, if n = d, then iv) for almost every τ = τ(ξ) R n the stability subgroup at τ is compact If i), ii), iii) and iv) hold true (without assuming homogeneous τ and n = d), then U is a reproducing representation a critical point ξ is such that the n d-matrix D ξ = [ τ i(ξ) ξ j ] ij has not full rank, i.e. Jτ(ξ) = det (D t ξ D ξ ) = there are examples with n < d and with non compact stability subgroup a technical assumption is needed to avoid pathological groups. A sufficient condition is that the H-orbits in R n are locally closed (see Führ 21 for a complete discussion) Ernesto De Vito (Genova) Reproducing formulas associated and symbols 13 / 26

Example n < d: Schrödinger-lets two dimensional signals: H = L 2 (R 2, dx) d = 2 one translation: t R n = 1 the dilations are rotations and homogeneous dilations H = {h a,θ := [ ] cos θ sin θ a a >, θ < 2π} sin θ cos θ with Haar measure dh = da dθ a 2π and modular function H(a, θ) = 1 the linear action M = {M a,θ } of H is M a,θ t = at = det M a,θ = a the symbol is the Laplacian τ : R 2 R τ(ξ 1, ξ 2 ) = ξ 2 1 + ξ 2 2 τ(ξ ha,θ ) = τ(ξ) M a,θ f (t) := U(t,I) f satisfies the 2D-Schrödinger equation 2πi f 1 (x, t) = f (x, t) = T = t 2πi Ernesto De Vito (Genova) Reproducing formulas associated and symbols 14 / 26

The representation U is reproducing G is non-unimodular: G (a, θ) = det M a,θ 1 H (a, θ) = a 1 n d the Jacobian is Jτ(ξ) = 2 ξ the set of critical points is the singleton {}, which is negligible {τ(ξ) R ξ } = (, + ) and τ M a,θ = τa for all τ > the stability subgroup is the compact group SO(2) then the representation U is reproducing U provides a voice transform for L 2 (R 2, dx) the space G has three parameters: (t, a, θ) (directional wavelets and shearlets have four parameters) U is not irreducible Ernesto De Vito (Genova) Reproducing formulas associated and symbols 15 / 26

Admissible vectors A function η : R 2 C is an admissible vector if and only if k Z + ˆη k (r) 2 r dr = η 2 2 < + and + ˆη k (r) 2 dr = 1 r ˆη k (r) = 1 2π ˆη(r cos ω, r sin ω)e i kω dω 2π k Z The Schrödinger voice transform: t R, a R +, θ [, 2π) Vf (t, a, θ) = + ˆfk ( τ) a ˆη k ( aτ)e 2πi tτ dτ e ikθ τ = r 2 2 k Z Vf (t, a) k = 2π + ˆfk (r) ( η a,t) k (r) }{{} r dr Schrödinger evolution of a dilation of η Ernesto De Vito (Genova) Reproducing formulas associated and symbols 16 / 26

Example of admissible vectors take ψ L 2 (R, dx) satisfying the 1D-Calderon condition + ˆψ(r) 2 dr = 1 = r + ˆψ(a 1 r) 2 dr = 1 a > r take a sequence (a k ) k Z of positive dilations such that a k < + define η : R 2 C by k Z ˆη(r cos ω, r sin ω) = 1 2π k Z ˆψ(a 1 k r)e i kω = ˆη k (r) = ψ(a 1 k r) The kernel is not in L 1 (G, dg), but it is in L p (G, dg) for all p > 1 Ernesto De Vito (Genova) Reproducing formulas associated and symbols 17 / 26

Open problems Is this kind of voice transform useful for signal analysis? Coorbit space theory and smoothness spaces Discretization and sparsity Computable algorithm for the voice transform The standard theory (H.G. Feichtinger, K. Gröchenig) is based on a L 1 condition on the kernel. Some recent extensions Führ, H.; Gröchenig, K. Sampling theorems on locally compact groups from oscillation estimates. Math. Z. 255 (27) Christensen, J.; Ölafsson, G. Coorbit spaces for dual pairs. Appl. Comput. Harmon. Anal. 31 (211) Christensen, J. Sampling in reproducing kernel Banach spaces on Lie groups. J. Approx. Theory 164 (212) Führ, H., Coorbit spaces and wavelet coefficient decay over general dilation groups (Preprint- 212) Ernesto De Vito (Genova) Reproducing formulas associated and symbols 18 / 26

Admissible vectors: a glance to the theory Write U as a sum of its (non-equivalent) irreducible components H = k m k H k U = k m k U k where U k is irreducible on H k and m k is the multiplicity of U k in U H k L 2 (O, µ; C N ) where O = {τ M h R n h H} is an H-orbits in R n with origin τ µ is an H-quasi invariant measure on O U k is an irreducible representation acting on H k U k t,h F(τ) = α(h, τ) e 2πi τt D s(h,τ) F(τ M h ) D = {Ds } s H is an irreducible representation acting on C N of the (compact) stability subgroup at τ H = {h H τ M h = τ } s(h, τ) H is the Mackey cocycle and α(h, τ) > is a positive number (if µ is invariant is 1) Ernesto De Vito (Genova) Reproducing formulas associated and symbols 19 / 26

Abstract characterization of admissible vectors Theorem 3 (De Mari, D. 212) A vector η L 2 (R d, dx) is admissible if and only if for any index k det M h(τ) F kl (τ), F kl (τ) C m H (h(τ)) dµ(τ) = δ N ll vol(h ) O where for all τ O, h(τ) H is such that τ M h(τ) = τ and l, l = 1,..., m k L 2 (R d, dx) = k m k H k f = k m k F kl F kl H k L 2 (O, µ; C N ) l=1 This result is a concrete realization of Führ abstract machine If n = d and τ(ξ) = ξ is in Führ, Mayer 22 Ernesto De Vito (Genova) Reproducing formulas associated and symbols 2 / 26

First step: translation group R The space H is L 2 (R 2, dx) L 2 ( R 2, dξ) The dual space of the translation group is R The translations are not in canonical form Û t ˆf (ξ) = e 2πi τ(ξ)t ˆf (ξ) τ(ξ) = ξ 2 1 + ξ2 2 Jτ(ξ) = 2 ξ {ξ R 2 Jτ(ξ) } = R 2 \ {} = {} = {τ(ξ) Jτ(ξ) } = (, + ) for each τ (, + ) the level set Sτ 1 = {ξ τ(ξ) = τ} = {ξ1 2 + ξ2 2 = τ} is a Riemannian manifold: a circle with radius r = τ Ernesto De Vito (Genova) Reproducing formulas associated and symbols 21 / 26

Coarea formula applied to ξ τ(ξ) (Federer - 1969) + ˆf (ξ) dξ = ˆf (ξ)dντ (ξ) dτ R 2 \{} Sτ 1 ν τ = Riemannian measure of the manifold of S1 τ Jτ(ξ) For us it is the change of variables with polar coordinates R 2 ˆf (ξ1, ξ 2 )dξ 1 ξ 2 = + ( 2π ˆf ( τ cos ω, τ sin ω) dω 2 Direct integral decomposition of the Hilbert space of signals L 2 (R 2, dx) Fourier L 2 ( R 2, dξ) = 1 2 + L 2 (S 1 τ, dω)dτ ) dτ Ernesto De Vito (Genova) Reproducing formulas associated and symbols 22 / 26

Second step: dilation group H Recall h a,θ := [ ] cos θ sin θ a sin θ cos θ M a,θ = a the dual action of H on (, + ) is transitive τ M a,θ = τ a there is only one orbit O with measure µ = dτ and origin τ = 1 the stability [ subgroup at ] τ = 1 is the compact group of rotations cos θ sin θ H = { θ [, 2π)} = SO(2) sin θ cos θ with volume vol(h ) = 1 based on Weil formula ψ(h) det M h 1 dh = ψ(h(τ)s)ds dτ τ M h(τ) = τ H H O Ernesto De Vito (Genova) Reproducing formulas associated and symbols 23 / 26

Consequences for all τ (, + ) set h(τ) = h τ 1, = τ 1 2 I 2 H, so that τ M h(τ) = τ S 1 τ h(τ) = S 1 τ = S 1 pull-back L 2 (Sτ 1, dω) onto L 2 (S 1, dω) + the direct integral L 2 (Sτ 1, dω)dτ becomes a direct product the following identification holds true L 2 (R 2, dx) = 1 + L 2 (Sτ 1, ω)dτ = L ((, 2 + ), dτ ) 2 2 ; L2 (S 1, dω) f det M h(τ) det h(τ) 1 ˆf (ξ h(τ) 1 ) = ˆf ( τ cos ω, τ sin ω) τ (, + ) ξ = (cos ω, sin ω) S 1 Ernesto De Vito (Genova) Reproducing formulas associated and symbols 24 / 26

Third step the unit circle S 1 is invariant under rotations (it is not H-invariant) U SO(2) leaves invariant L 2 (S 1, dω) and acts as the regular representation Û θ ˆf (cos ω, sin ω) = ˆf (cos(ω θ), sin(ω θ)) L 2 (S 1, dω) decomposes into sum of irreducible non-equivalent subspaces by means of the Fourier series L 2 (S 1, dω) = k Z C{e ikω } U θ = k Z e ikθ hence the multiplicity m k and the dimension N k are equal to 1 for all k the identification is f (x 1, x 2 ) 1 F k (τ)e i kω 2π k F k (τ) = 1 2π ˆf ( τ cos ω, τ sin ω)e ikω dω = ˆf k ( τ) 2π Ernesto De Vito (Genova) Reproducing formulas associated and symbols 25 / 26

Final step A function η L 2 (R 2, dx) is admissible if and only if for all k Z + + F k (τ) 2 τ dτ 2 = 1 τ = r 2 ˆη k (r) = F k (r 2 ) ˆη k (r) 2 dr = 1 r Remember F kl (τ), F kl (τ) C N O det M h(τ) H (h(τ)) dµ(τ) = δ N ll vol(h ) l, l = 1,..., m k Ernesto De Vito (Genova) Reproducing formulas associated and symbols 26 / 26