Shift Invariant Spaces and BMO Morten Nielsen Department of Mathematical Sciences Aalborg University mnielsen@math.aau.dk Joint work with H. ˇ Siki´ c Fourier Talks, University of Maryland, February 20-21, 2014 M. Nielsen Shift Invariant Spaces and BMO
(Integer) Shifts of a Fixed Function Given a function ψ ∈ L 2 ( R d ), one of the most basic operations we can consider is translation: T k ψ := ψ ( · − k ), k ∈ Z d . Translation is a fundamental operator in harmonic analysis since it is ”simple” and behaves well under the Fourier transform: � F ( T k ψ ) = e − 2 π ik · ˆ f ( x ) e − 2 π ix · ξ dx . ψ, where F ( f )( ξ ) := R M. Nielsen Shift Invariant Spaces and BMO
Shift Invariant Spaces A finitely generated shift-invariant (FSI) subspaces of L 2 ( R d ) is a subspace S ⊂ L 2 ( R d ) for which there exists a finite family Ψ of L 2 ( R d )-functions such that S = S (Ψ) := span { ψ ( · − k ) : ψ ∈ Ψ , k ∈ Z d } . Remark To keep the notation simple, we only consider the most basic case: d = 1 and #Ψ = 1 [PSI space]. Applications FSI/PSI subspaces are used in several applications. Wavelets and other multi-scale methods are based on PSI subspaces FSI/PSI subspaces play an important role in multivariate approximation theory such as spline approximation and approximation with radial basis functions. M. Nielsen Shift Invariant Spaces and BMO
Generating Sets Stable generating set Given the structure of S , it is natural to consider a generating sets of integer translates. That is, a system with the following structure, { ϕ ( · − k ) : k ∈ Z } , Often we take ϕ = ψ , but ϕ may be different from ψ . However, we always require that S ( ϕ ) = S ( ψ ). M. Nielsen Shift Invariant Spaces and BMO
Basic Fourier Analysis of S ( ψ ) It can easily be deduced from the identity, c k e − 2 π ik · ˆ � c k ψ ( · − k ) ⇒ ˆ � f = f = ψ k k 2 � � � � ψ ( ξ + j ) | 2 d ξ ⇒ � ˆ f � 2 � c k e − 2 π ik ξ | ˆ � � 2 = � � � � T k j that J ψ m := ( m · ˆ ψ ) ∨ is an isometry from L 2 ( T ; p ψ ) onto S ( ψ ), where p ψ is the periodization of | ˆ ψ | 2 , given by � | ˆ ψ ( ξ + k ) | 2 , p ψ ( ξ ) := ξ ∈ R . k ∈ Z Observation The system { e 2 π ik ξ } k in L 2 ( T ; p ψ ) is mapped by J ψ to { ψ ( · − k ) } k . M. Nielsen Shift Invariant Spaces and BMO
Some well-known classical results Orthonormal and Riesz bases Let ψ ∈ L 2 ( R ) and consider B := { ψ ( · − k ) : k ∈ Z } . We let � | ˆ ψ ( ξ + k ) | 2 , p ψ ( ξ ) := ξ ∈ R . k ∈ Z Then B forms an orthonormal basis for S ( ψ ) provided p ψ ≡ 1. B forms a Riesz basis for S ( ψ ) provided that p ψ ≍ 1. Extension to FSI spaces The above result can be extended to FSI spaces using the Grammian for the generating set. Question Is stability of B possible even if p ψ �≍ 1? M. Nielsen Shift Invariant Spaces and BMO
Weaker notion of stability: Schauder bases Definition A family B = { x n : n ∈ N } of vectors in a Hilbert space H is a Schauder basis for H if there exists a unique dual sequence { y n : n ∈ N } ⊂ H such that for every x ∈ H , N � lim � x , y n � x n = x (norm convergence) . N →∞ n =1 Ordering of the system The Schauder basis convergence may not be unconditional so the ordering of the system becomes important. M. Nielsen Shift Invariant Spaces and BMO
Schauder bases of translates and Muckenhoupt weights The Muckenhoupt A 2 -class A measurable, 1-periodic function w : R → (0 , ∞ ) is an A 2 ( T )-weight provided that � 1 �� 1 � � � w ( ξ ) − 1 d ξ [ w ] A 2 := sup w ( ξ ) d ξ < ∞ , | I | | I | I ∈I I I where I is the collection of intervals (arcs) on T . Proposition [Sikic and N., ACHA (2008)] Let ψ ∈ L 2 ( R ) \{ 0 } . The system B := { ψ ( · − k ) : k ∈ Z } forms a Schauder basis for S ( ψ ), with Z ordered the natural way as 0 , 1 , − 1 , 2 , − 2 , . . . , if and only if the periodization function p ψ satisfies the A 2 ( T ) condition. Remark The result is based on the well-known Hunt-Muckenhoupt-Wheeden Theorem. Similar results for Gabor systems were obtained by Heil and Powell [J. Math. Phys. (2006)]. The PSI result can be generalized to multivariate FSI spaces using a theory of product A 2 -matrix weights [N., JFAA (2010)]. M. Nielsen Shift Invariant Spaces and BMO
Conditional Schauder bases of integer translates Examples Define ψ ∈ L 2 ( R ) by � ˆ � ln(2 + | ξ | − 1 ) � ψ ( ξ ) = ln · χ [0 , 1) ( ξ ) . � ln(2 + | ξ | − 1 ) � It follows that p ψ ( ξ ) = ln , ξ ∈ [ − 1 / 2 , 1 / 2). A direct calculation shows that p ψ ∈ A 2 ( T ), so B := { ψ ( · − k ) : k ∈ Z } forms a Schauder basis for S ( ψ ). However, p ψ is not bounded and consequently B fails to be an unconditional Riesz basis for S ( ψ ). Another example is provided by ψ ∈ L 2 ( R ) defined by ψ ( ξ ) = | ξ | α · χ [0 , 1) ( ξ ) , ˆ with α ∈ ( − 1 / 2 , 1 / 2) M. Nielsen Shift Invariant Spaces and BMO
Integer translates, A 2 , and BMO The A 2 class is closely related to the functions of bounded mean oscillation. Definition Let f ∈ L 1 , loc ( R ) be 1-periodic, and let I be the collection of intervals (arcs) on T . We say that f ∈ BMO ( T ) provided that 1 � � f � BMO ( T ) := sup | f ( x ) − f I | dx < ∞ , | I | I ∈I I where f I := 1 � I f ( x ) dx . | I | One can verify that log( A 2 ( T )) ⊂ BMO ( T ). Conversely, for f ∈ BMO ( T ) there is some α > 0 such that e α f ∈ A 2 ( T ) [by the John-Nirenberg inequality]. It is also easy to check that L ∞ ( T ) ֒ → BMO ( T ). M. Nielsen Shift Invariant Spaces and BMO
Integer translates and the role played by L ∞ ⊂ BMO All of this is related to stability of integer translates by the fact that { ψ ( · − k ) : k ∈ Z } forms a Riesz basis ⇐ ⇒ log( p ψ ) ∈ L ∞ Question Can we use the distance to L ∞ of log( p ψ ) ∈ BMO ( T ) to quantify the “quality” of a conditional Schauder basis? Distance to L ∞ For f ∈ BMO ( T ) we let dist( f , L ∞ ( T )) := g ∈ L ∞ ( T ) � f − g � BMO ( T ) . inf M. Nielsen Shift Invariant Spaces and BMO
The distance to L ∞ in BMO One additional observation It is known that L ∞ is not a closed subset of BMO . In fact, � f ∈ BMO : e mf ∈ A 2 , m ∈ Z � { f ∈ BMO ( T ) : dist( f , L ∞ ) = 0 } = . This follows from the celebrated result by Garnett and Jones that asserts that dist( f , L ∞ ) and ε ( f ) := inf { λ > 0 : [ e f /λ ] A 2 ( T ) < ∞} are in fact equivalent independent of f ∈ BMO ( T ). Theorem (Garnett and Jones) There exist positive constants C 1 and C 2 such that for f ∈ BMO ( T ), C 1 ε ( f ) ≤ dist( f , L ∞ ( T )) ≤ C 2 ε ( f ) . M. Nielsen Shift Invariant Spaces and BMO
The BMO subset dist( f , L ∞ ( T )) = 0 Example An example of an unbounded BMO function in { f : dist( f , L ∞ ) = 0 } is given by ln(2 + | x | − 1 ) � � f ( x ) = ln , x ∈ T . This is a consequence of the fact that ln N (2 + | x | − 1 ) ∈ A 2 ( T ) for any N ∈ N , which follows by direct calculation. M. Nielsen Shift Invariant Spaces and BMO
Improved stability: The coefficient space Let B = { x n } n ∈ N be a Schauder basis for H with dual system { y n } n ∈ N . The coefficient space associated with B is the sequence space given by � � C ( B ) := {� x , y n �} n ∈ N : x ∈ H . Controlling C ( B ) For a Riesz basis B , we have C ( B ) = ℓ 2 . For a normalized conditional Schauder basis B in H one can find 2 ≤ p < ∞ (possibly very large) such that C ( B ) ֒ → ℓ p . [Gurari˘ ı and Gurari˘ ı, 1971] M. Nielsen Shift Invariant Spaces and BMO
Improved conditioning based on dist(ln( p ψ ) , L ∞ ( T )) Theorem [ˇ Siki´ c and N. JFA (2014)] Let ψ ∈ L 2 ( R ) and suppose that p ψ ∈ A 2 ( T ). We let C ( E ) denote the coefficient space for the Schauder basis E = { ψ ( · − k ) } k for S ( ψ ). Define ε = ε (ln p ψ ) := inf { λ > 0 : [ p 1 /λ ] A 2 < ∞} . Then the ψ following inclusion holds 2 � C ( E ) ⊂ ℓ p ( Z ) , p 0 := 1 − ε. p 0 < p < ∞ In particular, if dist(ln( p ψ ) , L ∞ ( T )) = 0 then � C ( E ) ⊂ ℓ p ( Z ) . 2 < p < ∞ M. Nielsen Shift Invariant Spaces and BMO
Sketch of proof i. The A 2 condition implies that L 2 ( T , p ψ ) ֒ → L 1 ( T ) | k |≤ N � f , ˜ ii. Take f = lim N →∞ � ψ ( · − k ) � ψ ( · − k ) ∈ S ( ψ ) and let m f = J − 1 ψ ( f ) ∈ L 2 ( T , p ψ ). k ∈ Z � m f , e k � L 2 ( T ) e 2 π ikx . iii. Using i., verify that m f = � iv. Now use the Reverse H¨ older Inequality for p ψ and the H¨ older inequality to estimate � m f � L r for r ≈ 2. v. Conclude using the Hausdorff-Young inequality. M. Nielsen Shift Invariant Spaces and BMO
An example Example Recall the previous example with ψ ∈ L 2 ( R ) defined by � ˆ � ln(2 + | ξ | − 1 ) � ψ ( ξ ) = ln · χ [0 , 1) ( ξ ) , � ln(2 + | ξ | − 1 ) � and p ψ ( ξ ) = ln , ξ ∈ [ − 1 / 2 , 1 / 2). A direct calculation shows that p N ψ ∈ A 2 ( T ) for any N ∈ N , so E = { ψ ( · − k ) } k forms a conditional Schauder basis for S ( ψ ) with coefficient space for E controlled by � C ( E ) ⊂ ℓ p ( Z ) . 2 < p < ∞ M. Nielsen Shift Invariant Spaces and BMO
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