Fourier operators in applied harmonic analysis John J. Benedetto Norbert Wiener Center Department of Mathematics University of Maryland, College Park http://www.norbertwiener.umd.edu
Outline Waveform design and optimal ambiguity function behavior on 1 Z / N Z MIMO and a vector-valued DFT on Z / N Z 2 Finite Gabor sums on R 3 Balayage on LCAGs, and Fourier frames and non-uniform 4 sampling on R d STFT frame inequalities on R d 5 Φ DO frame inequalities on R d 6
Outline Waveform design and optimal ambiguity function behavior on 1 Z / N Z MIMO and a vector-valued DFT on Z / N Z 2 Finite Gabor sums on R 3 Balayage on LCAGs, and Fourier frames and non-uniform 4 sampling on R d STFT frame inequalities on R d 5 Φ DO frame inequalities on R d 6
Frames Let H be a separable Hilbert space, e.g., H = L 2 ( R d ) , R d , or C d . F = { x n } ⊆ H is a frame for H if A � x � 2 ≤ |� x , x n �| 2 ≤ B � x � 2 . � ∃ A , B > 0 such that ∀ x ∈ H , Theorem If F = { x n } ⊆ H is a frame for H then � � x , S − 1 x n � x n = � � x , x n � S − 1 x n , ∀ x ∈ H , x = where S : H → H , x �→ � � x , x n � x n is well-defined. Frames are a natural tool for dealing with numerical stability, overcompleteness, noise reduction, and robust representation problems.
Fourier frames, goal, and a litany of names Definition E = { x n } ⊆ R d , Λ ⊆ � R d . E is a Fourier frame for L 2 (Λ) if ∃ A, B > 0 , ∀ F ∈ L 2 (Λ) , � | < F ( γ ) , e − 2 πix n · γ > | 2 ≤ B || F || 2 A || F || 2 L 2 (Λ) ≤ L 2 (Λ) . n Goal Formulate a general theory of Fourier frames and non-uniform sampling formulas parametrized by the space M ( R d ) of bounded Radon measures. Motivation Beurling theory (1959-1960). Names Riemann-Weber, Dini, G.D. Birkhoff, Paley-Wiener, Levinson, Duffin-Schaeffer, Beurling-Malliavin, Beurling, H.J. Landau, Jaffard, Seip, Ortega-Cert` a–Seip. Balayage and the theory of generalized Fourier frames
Balayage Let M ( G ) be the algebra of bounded Radon measures on the LCAG G . Balayage in potential theory was introduced by Christoffel (early 1870s) and Poincar´ e (1890). Definition (Beurling) Balayage is possible for ( E, Λ) ⊆ G × � G , a LCAG pair, if ∀ µ ∈ M ( G ) , ∃ ν ∈ M ( E ) such that ˆ µ = ˆ ν on Λ . We write balayage ( E, Λ) . The set, Λ , of group characters is the analogue of the original role of Λ in balayage as a collection of potential theoretic kernels. Kahane formulated balayage for the harmonic analysis of restriction algebras. Balayage and the theory of generalized Fourier frames
Spectral synthesis Definition (Wiener, Beurling) Closed Λ ⊆ � G is a set of spectral synthesis (S-set) if ∀ µ ∈ M ( G ) , ∀ f ∈ C b ( G ) , � supp ( � f ) ⊆ Λ and ˆ µ = 0 on Λ = ⇒ G f dµ = 0 . ( ∀ T ∈ A ′ ( � G ) , ∀ φ ∈ A ( � G ) , supp ( T ) ⊆ Λ and φ = 0 on Λ ⇒ T ( φ ) = 0 . ) Ideal structure of L 1 ( G ) - the Nullstellensatz of harmonic analysis T ∈ D ′ ( � R d ) , φ ∈ C ∞ c ( � R d ) , and φ = 0 on supp ( T ) ⇒ T ( φ ) = 0 , with same result for M ( � R d ) and C 0 ( � R d ) . S 2 ⊆ � R 3 is not an S-set (L. Schwartz), and every non-discrete � G has non-S-sets (Malliavin). Polyhedra are S-sets. The 1 3 -Cantor set is an S-set with non-S-subsets. Balayage and the theory of generalized Fourier frames
Strict multiplicity Definition Γ ⊆ � G is a set of strict multiplicity if ∃ µ ∈ M (Γ) \{ 0 } such that ˇ µ vanishes at infinity in G . Riemann and sets of uniqueness in the wide sense. Menchov (1916): ∃ closed Γ ⊆ � R / Z and µ ∈ M (Γ) \{ 0 } , µ ( n ) = O ((log | n | ) − 1 / 2 ) , | n | → ∞ . | Γ | = 0 and ˇ 20th century history to study rate of decrease: Bary (1927), Littlewood (1936), Salem (1942, 1950), Ivaˇ sev-Mucatov (1957), Beurling. Assumption ∀ γ ∈ Λ and ∀ N ( γ ) , compact neighborhood, Λ ∩ N ( γ ) is a set of strict multiplicity. Balayage and the theory of generalized Fourier frames
A theorem of Beurling Definition E = { x n } ⊆ R d is separated if ∃ r > 0 , ∀ m, n, m � = n ⇒ || x m − x n || ≥ r. Theorem R d be a compact S-set, symmetric about 0 ∈ � Let Λ ⊆ � R d , and let E ⊆ R d be separated. If balayage (E, Λ ), then E is a Fourier frame for L 2 (Λ) . Equivalent formulation in terms of PW Λ = { f ∈ L 2 ( R d ) : supp ( ˆ f ) ⊆ Λ } . F = � ∀ F ∈ L 2 (Λ) , x ∈ E < F, S − 1 ( e x ) > Λ e x in L 2 (Λ) . For R d and other generality beyond Beurling’s theorem in R , the result above was formulated by Hui-Chuan Wu and JB (1998), see Landau (1967). Balayage and the theory of generalized Fourier frames
Outline Waveform design and optimal ambiguity function behavior on 1 Z / N Z MIMO and a vector-valued DFT on Z / N Z 2 Finite Gabor sums on R 3 Balayage on LCAGs, and Fourier frames and non-uniform 4 sampling on R d STFT frame inequalities on R d 5 Φ DO frame inequalities on R d 6
Short time Fourier transform (STFT) � f ( t ) g ( t − x ) e − 2 π it · ω dt . STFT V g f ( x , ω ) = � g � 2 = 1 . Vector-valued inversion, � � f = V g f ( x , ω ) e ω τ x g d ω dx ′ V g f ( x , ω ) = e − 2 π ix · ω V G F ( ω, − x ) , � f = F and � g = G . � � � V g f � L 2 ( R 2 d ) = � g � 2 � f � 2 . Quantum mechanics and Moyal’s formula (1949) for the cross-Wigner distribution W ( f , g )( x , ω ) . STFT as ( X , µ ) -frames. See Ali, Antoine, and Gazeau (1993 and 2000) , Gabardo and Han (2003), and Fornasier and Rauhut (2005).
The Feichtinger algebra g 0 ( γ ) = 2 d / 4 e − π � γ � 2 and Let g 0 ( x ) = 2 d / 4 e − π � x � 2 . Then G 0 ( γ ) = � � g 0 � 2 = 1. The Feichtinger algebra , S 0 ( R d ) , is S 0 ( R d ) = { f ∈ L 2 ( R d ): � f � S 0 = � V g 0 f � 1 < ∞} . The Fourier transform of S 0 ( R d ) is an isometric isomorphism onto itself, and, in particular, f ∈ S 0 ( R d ) if and only if F ∈ S 0 ( � R d ) . Feichtinger On a new Segal algebra 1981; Wiener amalgam spaces; modulation spaces; Feichtinger and Gr¨ ochenig, W. Sun, Walnut, Zimmermann; Gr¨ ochenig Foundations of Time-Frequency Analysis 2001. Norbert Wiener Center Balayage and short time Fourier transform frames
Gr¨ ochenig’s non-uniform Gabor frame theorem Theorem Given any g ∈ S 0 ( R d ) . There is r = r ( g ) > 0 such that if E = { ( s n , σ n ) } ⊆ R d × � R d is a separated sequence satisfying � ∞ B (( s n , σ n ) , r ( g )) = R d × � R d , n = 1 then the frame operator, S = S g , E , defined by � ∞ S g , E f = n = 1 � f , τ s n e σ n g � τ s n e σ n g , is invertible on S 0 ( R d ) . Further, if f ∈ S 0 ( R d ) , then � ∞ n = 1 � f , τ s n e σ n g � S − 1 f = g , E ( τ s n e σ n g ) , where the series converges unconditionally in S 0 ( R d ) . E depends on g . Norbert Wiener Center Balayage and short time Fourier transform frames
Balayage and a non-uniform Gabor frame theorem Theorem Let E = { ( s n , σ n ) } ⊆ R d × � R d be a separated sequence; and let R d × R d be an S-set of strict multiplicity that is compact, convex, Λ ⊆ � R d × R d . Assume balayage is possible for and symmetric about 0 ∈ � ( E , Λ) . Given g ∈ L 2 ( R d ) , such that � g � 2 = 1 . Then supp ( � ∀ f ∈ S 0 ( R d ) , ∃ A , B > 0 , such that for which V g f ) ⊆ Λ , � ∞ n = 1 | V g f ( s n , σ n ) | 2 ≤ B � f � 2 A � f � 2 2 ≤ 2 . Norbert Wiener Center Balayage and short time Fourier transform frames
Balayage and a non-uniform Gabor frame theorem (continued) Theorem Consequently, the frame operator, S = S g , E , is invertible in L 2 ( R d ) –norm on the subspace of S 0 ( R d ) , whose elements f have the property, supp ( � V g f ) ⊆ Λ . Further, if f ∈ S 0 ( R d ) and supp ( � V g f ) ⊆ Λ , then � ∞ n = 1 � f , τ s n e σ n g � S − 1 f = g , E ( τ s n e σ n g ) , where the series converges unconditionally in L 2 ( R d ) . E does not depend on g . Norbert Wiener Center Balayage and short time Fourier transform frames
The support hypothesis To show supp ( � V g f ) ⊆ Λ . We compute that if f , g ∈ L 1 ( R d ) ∩ L 2 ( R d ) , then V g f ( ζ, z ) = e − 2 π iz · ζ f ( − z ) ˆ � g ( − ζ ) . Let d = 1 , Λ = [ − Ω , Ω] × [ − T , T ] ⊆ � R × R , g ∈ PW [ − Ω , Ω] , g ∈ L 1 ( R ) , ˆ g ( t ) = ˆ g ( − t ) . For this window g , we take any even f ∈ L 2 ( R ) that is supported in [ − T , T ] . Norbert Wiener Center Balayage and short time Fourier transform frames
Outline Waveform design and optimal ambiguity function behavior on 1 Z / N Z MIMO and a vector-valued DFT on Z / N Z 2 Finite Gabor sums on R 3 Balayage on LCAGs, and Fourier frames and non-uniform 4 sampling on R d STFT frame inequalities on R d 5 Φ DO frame inequalities on R d 6
DFT Frames Definition Let N ≥ d and let s : Z / d Z → Z / N Z be injective. The rows { E m } N − 1 m = 0 of the N × d matrix � e 2 π ims ( n ) / N � m , n form an equal-norm tight frame for C d which we call a DFT frame .
DFT FUNTFs Let N ≥ d and form an N × d matrix using any d columns of the N × N DFT matrix ( e 2 π ijk / N ) N − 1 j , k = 0 . The rows of this N × d matrix, up to 1 , form a FUNTF for C d . multiplication by √ d John J. Benedetto and Jeffrey J. Donatelli Frames and a vector-valued ambiguity function
Recommend
More recommend
