Continuous Frames and Sampling

Transcription

1 NuHAG University of Vienna, Faculty for Mathematics Marie Curie Fellow within the European network HASSIP HPRN-CT SampTA05, Samsun July 2005 Joint work with Massimo Fornasier

2 Overview 1 Continuous and discrete frames Relation to reproducing kernel Hilbert spaces Associated Banach spaces (coorbit spaces) General construction of discretizations (sampling theorems)

3 Continuous Frames 2 H: Hilbert space, X: locally compact space µ: Radon measure on X with supp µ = X A family F := {ψ x, x X} H is called a continuous frame for H if there exist constants 0 < C 1, C 2 < such that C 1 f 2 F is called tight if C 1 = C 2. X f, ψ x 2 dµ(x) C 2 f 2. Frame operator (defined weakly) Sf = X f, ψ x ψ x dµ(x). S is a self-adjoint, positive, boundedly invertible operator. If F is tight then S = c Id.

4 Reconstruction of f H: f = SS 1 f = = S 1 Sf = X X f, S 1 ψ x ψ x dµ(x) f, ψ x S 1 ψ x dµ(x) 3 F = {S 1 ψ x } x X is another frame, called the canonical dual frame. Transforms V f(x) = f, ψ x W f(x) = f, S 1 ψ x The frame property ensures that V, W : H L 2 (X, µ) are bounded and boundedly invertible on their image.

5 Reproducing formula 4 Define the kernel R(x, y) := ψ y, S 1 ψ x, x, y X A kernel acts on Functions via R(F )(x) = Reproducing formula X R(x, y)f (y)dµ(y). V f = R(V f), W f = R(W f) for all f H. Thus, the range of V and W is the reproducing kernel Hilbert space {F L 2 (X, µ), R(F ) = F }.

6 Reproducing Kernel Hilbert Spaces 5 H L 2 (X, µ) reproducing kernel Hilbert space with reproducing kernel K x (t), x, t X, i.e., f(x) = f, K x for all f H. It holds X f, K x 2 dµ(x) = X f(x) 2 dµ(x) = f 2 for all f H. Thus, {K x } x X is a tight continuous frame. Note that V f(x) = f, K x = f(x), i.e., V = Id.

7 Discrete Frames 6 X = I: discrete index set µ: counting measure on I Frame condition: C 1 f 2 X i I f, ψ i 2 C 2 f 2 for all f H Frame expansion: f = X i I f, S 1 ψ i ψ i = X i I f, ψ i S 1 ψ i

8 Short Time Fourier Transform 7 Let H = L 2 (R d ) and 0 g L 2 (R d ). Then Rd R d f, M ξ T x g 2 dxdξ = g 2 f 2, i.e., {T x M ξ g, (x, ξ) R d R d } is a tight continuous frame. The associated transform is the short time Fourier transform: V g f(x, ξ) = f, M ξ T x g = R d f(t)g(t x)e 2πix ξ dt = (ft x g)(ξ)

9 Further examples 8 Continuous and discrete wavelet frames Square-integrable group representations Square-integrable group representations modulo subgroups Wavelet frames on the sphere (Antoine / Vandergheynst) Gabor transform on the sphere (Torresani) mixed Gabor / wavelet transform on R d (α-transform) (Folland, Cordoba/Fefferman, Torresani, Holschneider/Nazaret, Hogan/Lakey, Bros/Iagolnitzer, Fornasier) Shift invariant spaces...

10 Problems 9 Is it possible to sample a discrete frame from a continuous one? Let {ψ x, x X} be a continuous frame. Which conditions ensure that one can find {x i, i I} X such that {ψ xi, i I} is a discrete frame? (Ali, Antoine, Gazeau) Extension to Banach spaces? Characterizations of Banach spaces by discrete and continuous frames? Simultaneous validity of frame expansions in a class of Banach spaces? Simultaneous treatment of these questions by a generalization of coorbit space theory.

11 The discretization problem in reproducing kernel Hilbert spaces 10 Let H L 2 (X, µ) be a reproducing kernel Hilbert space with kernel K x (t). If it is possible to determine (x i ) such that {K xi } i I is a discrete frame for H then f(t) = X i I f, K xi (S 1 K xi )(t) = X i I f(x i )(S 1 K xi )(t) for all f H, i.e., we have a sampling theorem in H.

12 Banach spaces associated to a frame 11 Idea: Let {ψ x, x X} be a continuous frame. Suppose Y is Banach space of functions on X, then define the coorbit spaces with norms CoY := {f, V f Y }, e CoY := {f, W f Y } f CoY := V f Y, f e CoY = W f Y (1) (Recall that V f(x) = f, ψ x and W f(x) = f, S 1 ψ x.) Note that CoL 2 = e CoL 2 = H

13 Conditions on the continuous frame 12 A natural condition on the frame and the function space Y for a useful definition of coorbit spaces is that the kernel R(x, y) := ψ y, S 1 ψ x acts continuously on Y via R(F )(x) = X R(x, y)f (y)dµ(y). This condition relies on the reproducing formulae R(V f) = V f and R(W f) = W f.

14 If Y = L p w (X) then continuity of R on Lp w is ensured if R is contained in 13 A m := {K : X X C, K A m := Km A 1 < } with K A 1 := max ess sup x X and X m(x, y) = max K(x, y) dµ(y), ess sup y X w(x) w(y), w(y). w(x) X K(x, y) dµ(x) General assumptions: R is contained in A m ; m and Y are related via K(F ) Y K A m F Y for all K A m, F Y.

15 Example: Modulation Spaces 14 Let g(x) := e x 2 be a Gaussian function on R d, {M ξ T x g, (x, ξ) R d R d } the associated continuous frame and V g the short time Fourier transform. Further let Y = L p,q s, 1 p, q, s R the mixed norm space on Rd R d whose norm is defined by q/p 1/q F L p,q s := F (x, ξ) p dx (1 + ξ 2 ) dξ! qs/2 R d R d The modulation space M p,q s is defined as the collection of tempered distributions f S such that f M p,q s := V g f L p,q s is finite. Hence, it can be identified with a coorbit space! Note that M s 2,2 coincides with a Sobolev space W s 2.

16 15 Equality of CoY and CoY Define K 1 (x, y) := ψ y, ψ x and K 2 (x, y) := S 1 ψ y, S 1 ψ x If both K 1 A m and K 2 A m then R A m (recall R(x, y) = ψ y, S 1 ψ x ) and it holds CoY = e CoY. Under certain conditions on a subalgebra A A m (A and B(L 2 (X, µ)) form a Wiener pair) K 1 A already implies K 2 A A m and R A A m. = Localization of frames and Banach algebra techniques

17 Discretization 16 Idea: Cover X by sufficiently small subsets (U i ) i I and choose x i U i to obtain a (discrete) Banach frame {ψ xi, i I} for CoY resp. e CoY. Definition 1. A family U = (U i ) i I of relatively compact subsets of X is called a moderate covering of X if the following conditions are satisfied: X = i I U i. sup j I #{i I, U i U j } N < (finite overlap property) There exists a constant C such that µ(u i ) Cµ(U j ) for all i, j such that U i U j. There exists a constant D > 0 such that µ(u i ) D for all i I.

18 Definition 2. A frame F is said to possess property D[δ, m] if there exists a moderate covering U = U δ = (U i ) i I of X with the property such that the kernel osc U defined by m(x, y) C for all x, y U i, for all i I, osc U (x, y) := sup S 1 ψ x, ψ y ψ z = sup R(x, y) R(x, z), z Qy z Qy where Q y := i,y Ui U i, satisfies osc U A m < δ. 17

19 Main Theorem 18 Theorem 1. Suppose the frame F = {ψ x, x X} possesses property D[δ, m] for some sufficiently small δ > 0. Let U δ denote the corresponding moderate covering of X and choose points (x i ) i I X such that x i U i. Then F d := {ψ xi } i I is a Banach frame for CoY. This means ( f, ψ xi ) i I Y (I) f CoY and there exists an operator S : Y (I) CoY, such that S( f, ψ xi ) i I = f for all f CoY. F d is an atomic decomposition for e CoY. This means that there exist linear bounded functionals λ i such that f = P i I λ i(f)ψ xi for all f e CoY and (λ i (f)) i I Y (I) f e CoY. The theorem also holds when replacing F d by F d = {S 1 ψ xi } i I and interchanging the roles of CoY and e CoY.

20 If Y = L p w then 19 (λ i ) i I (L p w ) (I) = (λ i ) i I (L p w ) (I) = X i I X i I λ i p µ(u i )w(x i ) p! 1/p λ i p µ(u i ) 1 p w(x i ) p! 1/p. Corollary 2. Under the assumptions of the previous theorem with m 1 the family is a discrete (Hilbert) frame for H. { q µ(u i )ψ xi } i I

21 Application to examples 20 Let F = {ψ x } x X be some continuous frame indexed by (X, µ). (1) Show R(x, y) = ψ y, S 1 ψ x A m. coorbit spaces (2) Find suitable covering of X, such that F possesses property D[δ, m] for sufficiently small δ. Recall osc U (x, y) := sup S 1 ψ x, ψ y ψ z = sup R(x, y) R(x, z), z Qy z Qy where Q y := i,y Ui U i, osc U A m < δ. At least for m 1.

22 Consequences: 21 (irregular) Gabor frames for modulation spaces (irregular) wavelet frames for homogeneous Besov and Triebel-Lizorkin spaces (irregular) wavelet frames for inhomogeneous Besov spaces radial wavelet frames radial Gabor frames Gabor frames and modulation spaces on the 2-sphere (Dahlke, Steidl, Teschke) Characterization of α-modulation spaces as coorbit spaces and associated mixed Gabor/wavelet frames (translates, modulates and dilations of a single atom), joint work with Dahlke, Fornasier, Steidl, Teschke Open: wavelet frames on the sphere and construction of associated function spaces Sampling theorems for particular reproducing kernel Hilbert spaces...

23 Conclusions 22 Relation between (continuous) frames and function spaces Important function spaces can be described with frames General method for the construction of atomic decompositions and Banach frames in function spaces General discretization method for sampling discrete frames from continuous ones Expected: Characterizations and atomic decompositions of other relevant function spaces, for examples on manifolds.

24 23 The Art of Frames by Helmut Rauhut