Uncertainty Principles for Fourier Multipliers Michael Northington V School of Mathematics Georgia Tech 6/6/2018 With Shahaf Nitzan (Ga Tech) and Alex Powell (Vanderbilt) Uncertainty Principles for Fourier Multipliers M. Northington V (mcnv3@gatech.edu) June 6, 2018
Table of Contents Exponentials in Weighted Spaces Restrictions on Fourier Multipliers Applications to Balian-Low Type Theorems Uncertainty Principles for Fourier Multipliers M. Northington V (mcnv3@gatech.edu) June 6, 2018
Exponentials in Weighted Spaces � e 2 πik · x � Let E = E ( d ) = k ∈ Z d . Uncertainty Principles for Fourier Multipliers M. Northington V (mcnv3@gatech.edu) June 6, 2018
Exponentials in Weighted Spaces � e 2 πik · x � Let E = E ( d ) = k ∈ Z d . ◮ E is an orthonormal basis for L 2 ( T d ) = L 2 ( R d / Z d ) . Uncertainty Principles for Fourier Multipliers M. Northington V (mcnv3@gatech.edu) June 6, 2018
Exponentials in Weighted Spaces � e 2 πik · x � Let E = E ( d ) = k ∈ Z d . ◮ E is an orthonormal basis for L 2 ( T d ) = L 2 ( R d / Z d ) . ◮ For a weight w satisfying w ( x ) > 0 almost everywhere, consider L 2 w ( T d ) with norm, � � g � 2 T d | g ( x ) | 2 w ( x ) dx. w ( T d ) = L 2 Uncertainty Principles for Fourier Multipliers M. Northington V (mcnv3@gatech.edu) June 6, 2018
Exponentials in Weighted Spaces � e 2 πik · x � Let E = E ( d ) = k ∈ Z d . ◮ E is an orthonormal basis for L 2 ( T d ) = L 2 ( R d / Z d ) . ◮ For a weight w satisfying w ( x ) > 0 almost everywhere, consider L 2 w ( T d ) with norm, � � g � 2 T d | g ( x ) | 2 w ( x ) dx. w ( T d ) = L 2 ◮ Question 1: What basis properties does E have in L 2 w ( T d ) and can these be characterized in terms of w ? Uncertainty Principles for Fourier Multipliers M. Northington V (mcnv3@gatech.edu) June 6, 2018
Exponentials in Weighted Spaces � e 2 πik · x � Let E = E ( d ) = k ∈ Z d . ◮ E is an orthonormal basis for L 2 ( T d ) = L 2 ( R d / Z d ) . ◮ For a weight w satisfying w ( x ) > 0 almost everywhere, consider L 2 w ( T d ) with norm, � � g � 2 T d | g ( x ) | 2 w ( x ) dx. w ( T d ) = L 2 ◮ Question 1: What basis properties does E have in L 2 w ( T d ) and can these be characterized in terms of w ? ◮ Question 2: Why do we care about this setting? Uncertainty Principles for Fourier Multipliers M. Northington V (mcnv3@gatech.edu) June 6, 2018
Example 1: Gabor Systems and the Zak Transform ◮ Gabor System: For g ∈ L 2 ( R ) , G ( g ) := { e 2 πimx g ( x − n ) } m,n ∈ Z = { M m T n g } m,n ∈ Z Uncertainty Principles for Fourier Multipliers M. Northington V (mcnv3@gatech.edu) June 6, 2018
Example 1: Gabor Systems and the Zak Transform ◮ Gabor System: For g ∈ L 2 ( R ) , G ( g ) := { e 2 πimx g ( x − n ) } m,n ∈ Z = { M m T n g } m,n ∈ Z ◮ Zak Transform: Zg ( x, y ) := � k ∈ Z g ( x − k ) e 2 πiky Uncertainty Principles for Fourier Multipliers M. Northington V (mcnv3@gatech.edu) June 6, 2018
Example 1: Gabor Systems and the Zak Transform ◮ Gabor System: For g ∈ L 2 ( R ) , G ( g ) := { e 2 πimx g ( x − n ) } m,n ∈ Z = { M m T n g } m,n ∈ Z ◮ Zak Transform: Zg ( x, y ) := � k ∈ Z g ( x − k ) e 2 πiky ◮ Converts TF-shifts to exponentials: Z ( M m T n g ) = e 2 πi ( mx − ny ) Zg Z ( G ( g )) = { e 2 πi ( mx − ny ) Z ( g ) } n,m ∈ Z Uncertainty Principles for Fourier Multipliers M. Northington V (mcnv3@gatech.edu) June 6, 2018
Example 1: Gabor Systems and the Zak Transform ◮ Gabor System: For g ∈ L 2 ( R ) , G ( g ) := { e 2 πimx g ( x − n ) } m,n ∈ Z = { M m T n g } m,n ∈ Z ◮ Zak Transform: Zg ( x, y ) := � k ∈ Z g ( x − k ) e 2 πiky ◮ Converts TF-shifts to exponentials: Z ( M m T n g ) = e 2 πi ( mx − ny ) Zg Z ( G ( g )) = { e 2 πi ( mx − ny ) Z ( g ) } n,m ∈ Z ◮ Leads to an isometric isomorphism: ◮ L 2 ( R ) → L 2 w ( T 2 ) , for w = | Zg | 2 ◮ G ( g ) → E = E (2) . Uncertainty Principles for Fourier Multipliers M. Northington V (mcnv3@gatech.edu) June 6, 2018
Example 2: Shift-Invariant Spaces and Periodization ◮ Integer Translates: For f ∈ L 2 ( R d ) , T ( f ) = { f ( · − l ) } l ∈ Z d Uncertainty Principles for Fourier Multipliers M. Northington V (mcnv3@gatech.edu) June 6, 2018
Example 2: Shift-Invariant Spaces and Periodization ◮ Integer Translates: For f ∈ L 2 ( R d ) , T ( f ) = { f ( · − l ) } l ∈ Z d L 2 ( R d ) ◮ Shift-Invariant Space: V ( f ) = span ( T ( f )) Uncertainty Principles for Fourier Multipliers M. Northington V (mcnv3@gatech.edu) June 6, 2018
Example 2: Shift-Invariant Spaces and Periodization ◮ Integer Translates: For f ∈ L 2 ( R d ) , T ( f ) = { f ( · − l ) } l ∈ Z d L 2 ( R d ) ◮ Shift-Invariant Space: V ( f ) = span ( T ( f )) f ( ξ ) = � ◮ Periodization: P � k ∈ Z d | � f ( ξ − k ) | 2 Uncertainty Principles for Fourier Multipliers M. Northington V (mcnv3@gatech.edu) June 6, 2018
Example 2: Shift-Invariant Spaces and Periodization ◮ Integer Translates: For f ∈ L 2 ( R d ) , T ( f ) = { f ( · − l ) } l ∈ Z d L 2 ( R d ) ◮ Shift-Invariant Space: V ( f ) = span ( T ( f )) f ( ξ ) = � ◮ Periodization: P � k ∈ Z d | � f ( ξ − k ) | 2 ◮ If h ∈ V ( f ) , there exists a Z d -periodic m , so that � h = m � f . Uncertainty Principles for Fourier Multipliers M. Northington V (mcnv3@gatech.edu) June 6, 2018
Example 2: Shift-Invariant Spaces and Periodization ◮ Integer Translates: For f ∈ L 2 ( R d ) , T ( f ) = { f ( · − l ) } l ∈ Z d L 2 ( R d ) ◮ Shift-Invariant Space: V ( f ) = span ( T ( f )) f ( ξ ) = � ◮ Periodization: P � k ∈ Z d | � f ( ξ − k ) | 2 ◮ If h ∈ V ( f ) , there exists a Z d -periodic m , so that � h = m � f . ◮ Leads to an isometric isomorphism: w ( T d ) , for w = P � ◮ V ( f ) → L 2 f ◮ h → m ◮ T ( f ) → E = E ( d ) Uncertainty Principles for Fourier Multipliers M. Northington V (mcnv3@gatech.edu) June 6, 2018
Spanning and Independence Properties Let H be a Hilbert space, and H = { h n } ∞ n =1 ⊂ H . Complete ◮ H is complete if span H = H . Frame
Spanning and Independence Properties Let H be a Hilbert space, and H = { h n } ∞ n =1 ⊂ H . Complete ◮ H is complete if span H = H . ◮ H is a frame if it’s complete, and there exist constants 0 < A ≤ B < ∞ with � ∞ |� h, h n �| 2 ≤ B � h � 2 A � h � 2 ∀ h ∈ H , H ≤ H . n =1 Frame
Spanning and Independence Properties Let H be a Hilbert space, and H = { h n } ∞ n =1 ⊂ H . Complete ◮ H is complete if span H = H . ◮ H is a frame if it’s complete, and there exist constants 0 < A ≤ B < ∞ with � ∞ |� h, h n �| 2 ≤ B � h � 2 A � h � 2 ∀ h ∈ H , H ≤ H . n =1 ◮ Every frame is complete, with the additional bonus that there exist a choice of coefficients such that h = � c n h n with Frame � c n � l 2 ≍ � h � H . Uncertainty Principles for Fourier Multipliers M. Northington V (mcnv3@gatech.edu) June 6, 2018
Spanning and Independence Properties ◮ H is a minimal system if for each n , Complete Exact h n / ∈ span { h m : m � = n } . Frame Riesz Basis
Spanning and Independence Properties ◮ H is a minimal system if for each n , Complete Exact h n / ∈ span { h m : m � = n } . ◮ H is exact if it is complete and minimal. Frame Riesz Basis
Spanning and Independence Properties ◮ H is a minimal system if for each n , Complete Exact h n / ∈ span { h m : m � = n } . ◮ H is exact if it is complete and minimal. ◮ H is a Riesz basis if there is an orthonormal basis { e n } ∞ n =1 and a bounded invertible operator T on H such that Te n = h n . Frame Riesz Basis
Spanning and Independence Properties ◮ H is a minimal system if for each n , Complete Exact h n / ∈ span { h m : m � = n } . ◮ H is exact if it is complete and minimal. ◮ H is a Riesz basis if there is an orthonormal basis { e n } ∞ n =1 and a bounded invertible operator T on H such that Te n = h n . ◮ Riesz basis = ⇒ frame; Riesz basis ⇐ ⇒ Frame minimal frame. Riesz Basis Uncertainty Principles for Fourier Multipliers M. Northington V (mcnv3@gatech.edu) June 6, 2018
( C q ) -systems (Olevskii, Nitzan ’07) ◮ Fix 2 ≤ q ≤ ∞ . { h n } ∞ n =1 ⊂ H is a ( C q ) -system if for each h ∈ H , h can be approximated to arbitrary accuracy by a finite sum � a n h n such that � a n � l q ≤ C � h � H . Uncertainty Principles for Fourier Multipliers M. Northington V (mcnv3@gatech.edu) June 6, 2018
( C q ) -systems (Olevskii, Nitzan ’07) ◮ Fix 2 ≤ q ≤ ∞ . { h n } ∞ n =1 ⊂ H is a ( C q ) -system if for each h ∈ H , h can be approximated to arbitrary accuracy by a finite sum � a n h n such that � a n � l q ≤ C � h � H . ◮ Equivalently, { h n } ∞ n =1 is a ( C q ) -system if and only if � ∞ � 1 /q ′ � |� h, h n �| q ′ � h � H ≤ C n =1 Uncertainty Principles for Fourier Multipliers M. Northington V (mcnv3@gatech.edu) June 6, 2018
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