Uncertainty Principles for Fourier Multipliers Michael Northington V - - PowerPoint PPT Presentation
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
School of Mathematics Georgia Tech
Uncertainty Principles for Fourier Multipliers
June 6, 2018
Uncertainty Principles for Fourier Multipliers
June 6, 2018
k∈Zd.
Uncertainty Principles for Fourier Multipliers
June 6, 2018
k∈Zd. ◮ E is an orthonormal basis for L2(Td) = L2(Rd/Zd).
Uncertainty Principles for Fourier Multipliers
June 6, 2018
k∈Zd. ◮ E is an orthonormal basis for L2(Td) = L2(Rd/Zd). ◮ For a weight w satisfying w(x) > 0 almost everywhere,
w(Td) with norm,
L2
w(Td) =
Uncertainty Principles for Fourier Multipliers
June 6, 2018
k∈Zd. ◮ E is an orthonormal basis for L2(Td) = L2(Rd/Zd). ◮ For a weight w satisfying w(x) > 0 almost everywhere,
w(Td) with norm,
L2
w(Td) =
◮ Question 1: What basis properties does E have in L2 w(Td)
Uncertainty Principles for Fourier Multipliers
June 6, 2018
k∈Zd. ◮ E is an orthonormal basis for L2(Td) = L2(Rd/Zd). ◮ For a weight w satisfying w(x) > 0 almost everywhere,
w(Td) with norm,
L2
w(Td) =
◮ Question 1: What basis properties does E have in L2 w(Td)
◮ Question 2: Why do we care about this setting?
Uncertainty Principles for Fourier Multipliers
June 6, 2018
◮ Gabor System: For g ∈ L2(R),
Uncertainty Principles for Fourier Multipliers
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◮ Gabor System: For g ∈ L2(R),
◮ Zak Transform: Zg(x, y) := k∈Z g(x − k)e2πiky
Uncertainty Principles for Fourier Multipliers
June 6, 2018
◮ Gabor System: For g ∈ L2(R),
◮ Zak Transform: Zg(x, y) := k∈Z g(x − k)e2πiky ◮ Converts TF-shifts to exponentials:
Uncertainty Principles for Fourier Multipliers
June 6, 2018
◮ Gabor System: For g ∈ L2(R),
◮ Zak Transform: Zg(x, y) := k∈Z g(x − k)e2πiky ◮ Converts TF-shifts to exponentials:
◮ Leads to an isometric isomorphism:
◮ L2(R) → L2
w(T2), for w = |Zg|2
◮ G(g) → E = E(2). Uncertainty Principles for Fourier Multipliers
June 6, 2018
◮ Integer Translates: For f ∈ L2(Rd), T(f) = {f(· − l)}l∈Zd
Uncertainty Principles for Fourier Multipliers
June 6, 2018
◮ Integer Translates: For f ∈ L2(Rd), T(f) = {f(· − l)}l∈Zd ◮ Shift-Invariant Space: V (f) = span(T(f)) L2(Rd)
Uncertainty Principles for Fourier Multipliers
June 6, 2018
◮ Integer Translates: For f ∈ L2(Rd), T(f) = {f(· − l)}l∈Zd ◮ Shift-Invariant Space: V (f) = span(T(f)) L2(Rd) ◮ Periodization: P
k∈Zd |
Uncertainty Principles for Fourier Multipliers
June 6, 2018
◮ Integer Translates: For f ∈ L2(Rd), T(f) = {f(· − l)}l∈Zd ◮ Shift-Invariant Space: V (f) = span(T(f)) L2(Rd) ◮ Periodization: P
k∈Zd |
◮ If h ∈ V (f), there exists a Zd-periodic m, so that
Uncertainty Principles for Fourier Multipliers
June 6, 2018
◮ Integer Translates: For f ∈ L2(Rd), T(f) = {f(· − l)}l∈Zd ◮ Shift-Invariant Space: V (f) = span(T(f)) L2(Rd) ◮ Periodization: P
k∈Zd |
◮ If h ∈ V (f), there exists a Zd-periodic m, so that
◮ Leads to an isometric isomorphism:
◮ V (f) → L2
w(Td), for w = P
f
◮ h → m ◮ T(f) → E = E(d) Uncertainty Principles for Fourier Multipliers
June 6, 2018
Complete Frame
n=1 ⊂ H. ◮ H is complete if span H = H.
Complete Frame
n=1 ⊂ H. ◮ H is complete if span H = H. ◮ H is a frame if it’s complete, and there exist
H ≤ ∞
H.
Complete Frame
n=1 ⊂ H. ◮ H is complete if span H = H. ◮ H is a frame if it’s complete, and there exist
H ≤ ∞
H. ◮ Every frame is complete, with the additional
Uncertainty Principles for Fourier Multipliers
June 6, 2018
Complete Exact Frame Riesz Basis
◮ H is a minimal system if for each n,
Complete Exact Frame Riesz Basis
◮ H is a minimal system if for each n,
◮ H is exact if it is complete and minimal.
Complete Exact Frame Riesz Basis
◮ H is a minimal system if for each n,
◮ H is exact if it is complete and minimal. ◮ H is a Riesz basis if there is an orthonormal
n=1 and a bounded invertible
Complete Exact Frame Riesz Basis
◮ H is a minimal system if for each n,
◮ H is exact if it is complete and minimal. ◮ H is a Riesz basis if there is an orthonormal
n=1 and a bounded invertible
◮ Riesz basis =
Uncertainty Principles for Fourier Multipliers
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◮ Fix 2 ≤ q ≤ ∞. {hn}∞ n=1 ⊂ H is a (Cq)-system if for each
Uncertainty Principles for Fourier Multipliers
June 6, 2018
◮ Fix 2 ≤ q ≤ ∞. {hn}∞ n=1 ⊂ H is a (Cq)-system if for each
◮ Equivalently, {hn}∞ n=1 is a (Cq)-system if and only if
Uncertainty Principles for Fourier Multipliers
June 6, 2018
◮ Fix 2 ≤ q ≤ ∞. {hn}∞ n=1 ⊂ H is a (Cq)-system if for each
◮ Equivalently, {hn}∞ n=1 is a (Cq)-system if and only if
◮ (Cq) stands for completeness with lq control of coefficients.
Uncertainty Principles for Fourier Multipliers
June 6, 2018
◮ Fix 2 ≤ q ≤ ∞. {hn}∞ n=1 ⊂ H is a (Cq)-system if for each
◮ Equivalently, {hn}∞ n=1 is a (Cq)-system if and only if
◮ (Cq) stands for completeness with lq control of coefficients. ◮ Bessel (C2)-system ⇐
Uncertainty Principles for Fourier Multipliers
June 6, 2018
◮ Fix 2 ≤ q ≤ ∞. {hn}∞ n=1 ⊂ H is a (Cq)-system if for each
◮ Equivalently, {hn}∞ n=1 is a (Cq)-system if and only if
◮ (Cq) stands for completeness with lq control of coefficients. ◮ Bessel (C2)-system ⇐
◮ (Cq)-system =
Uncertainty Principles for Fourier Multipliers
June 6, 2018
Complete Exact (C2)-system Frame Riesz Basis (C∞)-system
Complete Exact (C2)-system Frame Riesz Basis (C∞)-system
Exact Riesz Basis
w(Td):
Uncertainty Principles for Fourier Multipliers
June 6, 2018
w(Td)?
Uncertainty Principles for Fourier Multipliers
June 6, 2018
w(Td)?
Uncertainty Principles for Fourier Multipliers
June 6, 2018
w(Td)?
Uncertainty Principles for Fourier Multipliers
June 6, 2018
w(Td)?
2
Uncertainty Principles for Fourier Multipliers
June 6, 2018
w(Td)?
2 ◮ The first 3 are well known: de Boor, DeVore, Ron (’92), Ron,
Uncertainty Principles for Fourier Multipliers
June 6, 2018
w(Td)?
2 ◮ The first 3 are well known: de Boor, DeVore, Ron (’92), Ron,
◮ Nitzan, Olsen (’11) gave necessary and sufficient conditions
Uncertainty Principles for Fourier Multipliers
June 6, 2018
2?
c−1 c0 c1 c2−1 2
2?
c−1 c0 c1 c2−1 2
−1 −0.5 0.5 1 −2 −1 1 2
2?
c−1 c0 c1 c2−1 2
−1 −0.5 0.5 1 −2 −1 1 2
−1 −0.5 0.5 1 −2 −1 1 2
2?
c−1 c0 c1 c2−1 2
−1 −0.5 0.5 1 −2 −1 1 2
−1 −0.5 0.5 1 −2 −1 1 2
−1 1 2
2?
c−1 c0 c1 c2−1 2
−1 −0.5 0.5 1 −2 −1 1 2
−1 −0.5 0.5 1 −2 −1 1 2
−1 1 2
Uncertainty Principles for Fourier Multipliers
June 6, 2018
2?
◮ For a periodic function u and a finite sequence c = {cn}n∈Zd,
Uncertainty Principles for Fourier Multipliers
June 6, 2018
2?
◮ For a periodic function u and a finite sequence c = {cn}n∈Zd,
◮ Then, u ∈ Mq 2, if for all such c,
Uncertainty Principles for Fourier Multipliers
June 6, 2018
2?
◮ For a periodic function u and a finite sequence c = {cn}n∈Zd,
◮ Then, u ∈ Mq 2, if for all such c,
◮ Properties and Special Cases:
◮ If q < 2, Mq
2 = {0}
Uncertainty Principles for Fourier Multipliers
June 6, 2018
2?
◮ For a periodic function u and a finite sequence c = {cn}n∈Zd,
◮ Then, u ∈ Mq 2, if for all such c,
◮ Properties and Special Cases:
◮ If q < 2, Mq
2 = {0}
◮ M2
2 = L∞(Td) (Agrees with Riesz basis characterization)
Uncertainty Principles for Fourier Multipliers
June 6, 2018
2?
◮ For a periodic function u and a finite sequence c = {cn}n∈Zd,
◮ Then, u ∈ Mq 2, if for all such c,
◮ Properties and Special Cases:
◮ If q < 2, Mq
2 = {0}
◮ M2
2 = L∞(Td) (Agrees with Riesz basis characterization)
◮ M∞
2 = L2(Td) (Agrees with minimal system characterization)
Uncertainty Principles for Fourier Multipliers
June 6, 2018
Uncertainty Principles for Fourier Multipliers
June 6, 2018
2.
Uncertainty Principles for Fourier Multipliers
June 6, 2018
2.
2, then u ∈ Mq 2 for all q ≥ 2, but if u is
2 for small q.
Uncertainty Principles for Fourier Multipliers
June 6, 2018
2.
2, then u ∈ Mq 2 for all q ≥ 2, but if u is
2 for small q.
2.
Uncertainty Principles for Fourier Multipliers
June 6, 2018
2.
2, then u ∈ Mq 2 for all q ≥ 2, but if u is
2 for small q.
2.
Uncertainty Principles for Fourier Multipliers
June 6, 2018
2 ≤ s ≤ d, and suppose w ∈ Hs(Td) and w has a zero.
Uncertainty Principles for Fourier Multipliers
June 6, 2018
2 ≤ s ≤ d, and suppose w ∈ Hs(Td) and w has a zero.
2 ≤ s < d 2 + 1, then u = 1 w /
2 for any q satisfying
d d−s.
Uncertainty Principles for Fourier Multipliers
June 6, 2018
2 ≤ s ≤ d, and suppose w ∈ Hs(Td) and w has a zero.
2 ≤ s < d 2 + 1, then u = 1 w /
2 for any q satisfying
d d−s. Conversely, for any q > d d−s there exists
w ∈ Mq 2.
Uncertainty Principles for Fourier Multipliers
June 6, 2018
2 ≤ s ≤ d, and suppose w ∈ Hs(Td) and w has a zero.
2 ≤ s < d 2 + 1, then u = 1 w /
2 for any q satisfying
d d−s. Conversely, for any q > d d−s there exists
w ∈ Mq 2.
2 + 1, then u = 1 w /
2 for any q satisfying
2d d−2.
Uncertainty Principles for Fourier Multipliers
June 6, 2018
2 ≤ s ≤ d, and suppose w ∈ Hs(Td) and w has a zero.
2 ≤ s < d 2 + 1, then u = 1 w /
2 for any q satisfying
d d−s. Conversely, for any q > d d−s there exists
w ∈ Mq 2.
2 + 1, then u = 1 w /
2 for any q satisfying
2d d−2. Conversely, there exists w ∈ C∞(Td) with a
2 for any q > 2d d−2.
Uncertainty Principles for Fourier Multipliers
June 6, 2018
2 ≤ s ≤ d, and suppose w ∈ Hs(Td) and w has a zero.
2 ≤ s < d 2 + 1, then u = 1 w /
2 for any q satisfying
d d−s. Conversely, for any q > d d−s there exists
w ∈ Mq 2.
2 + 1, then u = 1 w /
2 for any q satisfying
2d d−2. Conversely, there exists w ∈ C∞(Td) with a
2 for any q > 2d d−2. ◮ Proof relies on Sobolev Embedding Theorem in H¨
Uncertainty Principles for Fourier Multipliers
June 6, 2018
2 ≤ s ≤ d, and suppose w ∈ Hs(Td) and w has a zero.
2 ≤ s < d 2 + 1, then u = 1 w /
2 for any q satisfying
d d−s. Conversely, for any q > d d−s there exists
w ∈ Mq 2.
2 + 1, then u = 1 w /
2 for any q satisfying
2d d−2. Conversely, there exists w ∈ C∞(Td) with a
2 for any q > 2d d−2. ◮ Proof relies on Sobolev Embedding Theorem in H¨
◮ For s > d 2 + 1, we can’t say more than the bound in part 2
Uncertainty Principles for Fourier Multipliers
June 6, 2018
◮ We are also interested in finding a similar result when the zero
u has a larger Hausdorff dimension.
Uncertainty Principles for Fourier Multipliers
June 6, 2018
◮ We are also interested in finding a similar result when the zero
u has a larger Hausdorff dimension. ◮ In this case, our functions may not be continuous, so we
τ→0
Uncertainty Principles for Fourier Multipliers
June 6, 2018
◮ We are also interested in finding a similar result when the zero
u has a larger Hausdorff dimension. ◮ In this case, our functions may not be continuous, so we
τ→0
◮ A similar question was studied by Jiang, Lin (’03) and
Uncertainty Principles for Fourier Multipliers
June 6, 2018
2
Uncertainty Principles for Fourier Multipliers
June 6, 2018
2
2
2 + 1), then u = 1 w /
2 for any q
d d−s−σ/2.
Uncertainty Principles for Fourier Multipliers
June 6, 2018
2
2
2 + 1), then u = 1 w /
2 for any q
d d−s−σ/2.
2 + 1 ≤ d − σ, then u = 1 w /
2 for any q satisfying
2d d−2−σ.
Uncertainty Principles for Fourier Multipliers
June 6, 2018
2
2
2 + 1), then u = 1 w /
2 for any q
d d−s−σ/2.
2 + 1 ≤ d − σ, then u = 1 w /
2 for any q satisfying
2d d−2−σ.
2 + 1, then u = 1 w /
2 for any q.
Uncertainty Principles for Fourier Multipliers
June 6, 2018
2
2
2 + 1), then u = 1 w /
2 for any q
d d−s−σ/2.
2 + 1 ≤ d − σ, then u = 1 w /
2 for any q satisfying
2d d−2−σ.
2 + 1, then u = 1 w /
2 for any q. ◮ Part 3 is sharp, but parts 1 and 2 likely are not.
Uncertainty Principles for Fourier Multipliers
June 6, 2018
2
2
2 + 1), then u = 1 w /
2 for any q
d d−s−σ/2.
2 + 1 ≤ d − σ, then u = 1 w /
2 for any q satisfying
2d d−2−σ.
2 + 1, then u = 1 w /
2 for any q. ◮ Part 3 is sharp, but parts 1 and 2 likely are not. ◮ Based on the results of Jiang, Lin (’03) and Schikorra (’13), I
d−σ d−σ−s.
Uncertainty Principles for Fourier Multipliers
June 6, 2018
2
2
2 + 1), then u = 1 w /
2 for any q
d d−s−σ/2.
2 + 1 ≤ d − σ, then u = 1 w /
2 for any q satisfying
2d d−2−σ.
2 + 1, then u = 1 w /
2 for any q. ◮ Part 3 is sharp, but parts 1 and 2 likely are not. ◮ Based on the results of Jiang, Lin (’03) and Schikorra (’13), I
d−σ d−σ−s. ◮ Proof uses a version of Poincare Inequality from Jiang, Lin
Uncertainty Principles for Fourier Multipliers
June 6, 2018
Uncertainty Principles for Fourier Multipliers
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◮ Multipliers in Mq p for certain ranges of p and q.
Uncertainty Principles for Fourier Multipliers
June 6, 2018
◮ Multipliers in Mq p for certain ranges of p and q. ◮ Matrix-weights where W(x) is a K × K matrix
Uncertainty Principles for Fourier Multipliers
June 6, 2018
◮ Multipliers in Mq p for certain ranges of p and q. ◮ Matrix-weights where W(x) is a K × K matrix ◮ Nonsymmetric verisons where the Sobolev smoothness is
Uncertainty Principles for Fourier Multipliers
June 6, 2018
Uncertainty Principles for Fourier Multipliers
June 6, 2018
R
Uncertainty Principles for Fourier Multipliers
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R
◮ Conclusion rephrased: “either f /
Uncertainty Principles for Fourier Multipliers
June 6, 2018
R
◮ Conclusion rephrased: “either f /
◮ Assume f,
loc(R2).
Uncertainty Principles for Fourier Multipliers
June 6, 2018
R
◮ Conclusion rephrased: “either f /
◮ Assume f,
loc(R2). ◮ Quasiperiodicity of Zf forces it to have a (essential) zero.
Uncertainty Principles for Fourier Multipliers
June 6, 2018
R
◮ Conclusion rephrased: “either f /
◮ Assume f,
loc(R2). ◮ Quasiperiodicity of Zf forces it to have a (essential) zero. ◮ The Riesz basis property forces |Zf| ≥ A > 0, which gives
Uncertainty Principles for Fourier Multipliers
June 6, 2018
R
Uncertainty Principles for Fourier Multipliers
June 6, 2018
R
◮ Follows from single zero multiplier result.
Uncertainty Principles for Fourier Multipliers
June 6, 2018
R
◮ Follows from single zero multiplier result. ◮ Nitzan, Olsen (’11) proved similar result, with an additional ǫ
Uncertainty Principles for Fourier Multipliers
June 6, 2018
R
◮ Follows from single zero multiplier result. ◮ Nitzan, Olsen (’11) proved similar result, with an additional ǫ
◮ The q = ∞ case gives the BLT for exact systems (originally
Uncertainty Principles for Fourier Multipliers
June 6, 2018
◮ For Γ ⊂ Rd, V is Γ-invariant if TγV ⊂ V for all γ ∈ Γ.
Uncertainty Principles for Fourier Multipliers
June 6, 2018
◮ For Γ ⊂ Rd, V is Γ-invariant if TγV ⊂ V for all γ ∈ Γ. ◮ For any lattice Γ ⊃ Zd, there exists f ∈ L2(Rd) such that
◮ d = 1 by Aldroubi, Cabrelli, Heil, Kornelson, Molter (2010) ◮ d > 1 by Anastasio, Cabrelli, Paternostro (2011) Uncertainty Principles for Fourier Multipliers
June 6, 2018
◮ For Γ ⊂ Rd, V is Γ-invariant if TγV ⊂ V for all γ ∈ Γ. ◮ For any lattice Γ ⊃ Zd, there exists f ∈ L2(Rd) such that
◮ d = 1 by Aldroubi, Cabrelli, Heil, Kornelson, Molter (2010) ◮ d > 1 by Anastasio, Cabrelli, Paternostro (2011)
◮ Aldroubi, Sun, Wang (2011), and Tessera, Wang (2014),
Uncertainty Principles for Fourier Multipliers
June 6, 2018
1 N Z-invariant. If T(f) is a minimal (Cq)-system in V (f), then
◮ If T(f) is a minimal system for V (f), then T(f) is a
◮ (Hardin, M.N., Powell) In the q = 2 case, the result holds in
Uncertainty Principles for Fourier Multipliers
June 6, 2018
2 − 1 q) + 1 2, 1). Let
Uncertainty Principles for Fourier Multipliers
June 6, 2018
2 − 1 q) + 1 2, 1). Let
◮ Can be extended to finitely many generators, requires a
Uncertainty Principles for Fourier Multipliers
June 6, 2018
2 − 1 q) + 1 2, 1). Let
◮ Can be extended to finitely many generators, requires a
◮ Probably the sharp s is 1 − 1/q in all dimensions.
Uncertainty Principles for Fourier Multipliers
June 6, 2018
◮ Extra-invarance can be characterized in terms of P
Uncertainty Principles for Fourier Multipliers
June 6, 2018
◮ Extra-invarance can be characterized in terms of P
◮ The condition is somewhat technical, so lets look at an
2Z2-invariance.
Uncertainty Principles for Fourier Multipliers
June 6, 2018
◮ Extra-invarance can be characterized in terms of P
◮ The condition is somewhat technical, so lets look at an
2Z2-invariance.
Uncertainty Principles for Fourier Multipliers
June 6, 2018
◮ Extra-invarance can be characterized in terms of P
◮ The condition is somewhat technical, so lets look at an
2Z2-invariance.
◮ V (f) is 1 2Z2-invariant iff P2(x) and it’s shifts have disjoint
Uncertainty Principles for Fourier Multipliers
June 6, 2018
Uncertainty Principles for Fourier Multipliers
June 6, 2018