functions of variable bandwidth · functions of variable bandwidth roza aceska...
TRANSCRIPT
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
Numerical Harmonic Analysis Group
Functions of Variable Bandwidth
Roza [email protected]
2007
Roza Aceska [email protected] Functions of Variable Bandwidth
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
Outline
Definitions and notationsVariable bandwidth strip in the TF domainDilation-invariance of the corresponding spaceShifting the bandwidthReproducing kernel
Roza Aceska http://nuhag.eu
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
Definitions, notations, ideas
Essential TF support concentrated within a strip withvariable bandwidth
Roza Aceska http://nuhag.eu
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
Definitions, notations, ideas
m−variable bandwidthSTm = (x, ξ) ∈ Rd × Rd : |ξ| ≤ m(x), m(x) > 1 .
Roza Aceska http://nuhag.eu
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
Definitions and notations
Weighted modulation spaces M2w(Rd)
‖f‖2 :=∫∫|Vgf(x, ξ)|2w2(x, ξ)dxdξ <∞, (1)
w(x, ξ) is a moderate weight.
STFT of a signal f , given a Schwartz window g:
Vgf(x, ξ) =∫f(t)g(t− x)e2πit·ξdt = 〈f,MξTxg〉
Moderate weight w
w(z1 + z2) ≤ cv(z1)w(z2) for all zi = (xi, ξi) ∈ R2d, (*)v-submultiplicative weight.
Roza Aceska http://nuhag.eu
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
Definitions and notations
Weighted modulation spaces M2w(Rd)
‖f‖2 :=∫∫|Vgf(x, ξ)|2w2(x, ξ)dxdξ <∞, (1)
w(x, ξ) is a moderate weight.
STFT of a signal f , given a Schwartz window g:
Vgf(x, ξ) =∫f(t)g(t− x)e2πit·ξdt = 〈f,MξTxg〉
Moderate weight w
w(z1 + z2) ≤ cv(z1)w(z2) for all zi = (xi, ξi) ∈ R2d, (*)v-submultiplicative weight.
Roza Aceska http://nuhag.eu
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
Definitions and notations
Weighted modulation spaces M2w(Rd)
‖f‖2 :=∫∫|Vgf(x, ξ)|2w2(x, ξ)dxdξ <∞, (1)
w(x, ξ) is a moderate weight.
STFT of a signal f , given a Schwartz window g:
Vgf(x, ξ) =∫f(t)g(t− x)e2πit·ξdt = 〈f,MξTxg〉
Moderate weight w
w(z1 + z2) ≤ cv(z1)w(z2) for all zi = (xi, ξi) ∈ R2d, (*)v-submultiplicative weight.
Roza Aceska http://nuhag.eu
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
Variable bandwidth m(x) ≥ 1, choose s ≥ 1
Weight
wm(x, ξ) =
1, |ξ| ≤ m(x) + 1(|ξ| −m(x))s, |ξ| ≥ m(x) + 1
(2)
Property
The weight (2) is s−moderate under suitable conditions.(eg. m′ is bounded or m has bounded variations)
Roza Aceska http://nuhag.eu
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
Variable bandwidth m(x) ≥ 1, choose s ≥ 1
Weight
wm(x, ξ) =
1, |ξ| ≤ m(x) + 1(|ξ| −m(x))s, |ξ| ≥ m(x) + 1
(2)
Property
The weight (2) is s−moderate under suitable conditions.(eg. m′ is bounded or m has bounded variations)
Roza Aceska http://nuhag.eu
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
Bandwidth shift: cosine-like band
Roza Aceska http://nuhag.eu
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
DefinitionThe space of functions of variable bandwidth is a weightedmodulation space
Bm(x)(Rd) := M2wm(Rd) = f | Vgf ∈ L2
wm. (3)
Example:
Band-limited functions are in Bm(x)(Rd).
Roza Aceska http://nuhag.eu
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
band-limited functions
Roza Aceska http://nuhag.eu
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
Dilation invariance
Proposition
Let b(λ) be such that |b(λ)| ≤ c |λ| , for λ ∈ R and aconstant c. If the moderate weight w fulfills the inequality
|w(λ−1x, λξ)| ≤ b(λ)w(x, ξ), (4)
then the weighted modulation space M2w is dilation
invariant.More precisely,
‖f(λ·)‖ ≤ b(λ)cg,λ‖f‖. (5)
Roza Aceska http://nuhag.eu
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
Sketch of proof:
For f1(t) = f(λt) and gλ(t) = g( tλ):
|Vgf1(x, ξ)| = 1λ |Vgλf1(λx, ξλ)|.
Thus,
|Vgf1(x, ξ)|2w2(x, ξ) = λ−2|Vgλf1(x, ξ)|2w2(xλ−1, λξ).
Suppose |w( xλ , λξ)| ≤ b(λ)w(x, ξ), b2(λ)/λ2 -bounded. Then∫∫
|Vgf1(x, ξ)|2w2(x, ξ)dxdξ
≤ λ−2b2(λ)∫∫|Vgλf(x, ξ)|2w2(x, ξ)dxdξ︸ ︷︷ ︸
≈‖f‖
.
Roza Aceska http://nuhag.eu
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
Sketch of proof:
For f1(t) = f(λt) and gλ(t) = g( tλ):
|Vgf1(x, ξ)| = 1λ |Vgλf1(λx, ξλ)|.
Thus,
|Vgf1(x, ξ)|2w2(x, ξ) = λ−2|Vgλf1(x, ξ)|2w2(xλ−1, λξ).
Suppose |w( xλ , λξ)| ≤ b(λ)w(x, ξ), b2(λ)/λ2 -bounded.
Then∫∫|Vgf1(x, ξ)|2w2(x, ξ)dxdξ
≤ λ−2b2(λ)∫∫|Vgλf(x, ξ)|2w2(x, ξ)dxdξ︸ ︷︷ ︸
≈‖f‖
.
Roza Aceska http://nuhag.eu
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
Sketch of proof:
For f1(t) = f(λt) and gλ(t) = g( tλ):
|Vgf1(x, ξ)| = 1λ |Vgλf1(λx, ξλ)|.
Thus,
|Vgf1(x, ξ)|2w2(x, ξ) = λ−2|Vgλf1(x, ξ)|2w2(xλ−1, λξ).
Suppose |w( xλ , λξ)| ≤ b(λ)w(x, ξ), b2(λ)/λ2 -bounded. Then∫∫
|Vgf1(x, ξ)|2w2(x, ξ)dxdξ
≤ λ−2b2(λ)∫∫|Vgλf(x, ξ)|2w2(x, ξ)dxdξ︸ ︷︷ ︸
≈‖f‖
.
Roza Aceska http://nuhag.eu
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
Sketch of proof:
For f1(t) = f(λt) and gλ(t) = g( tλ):
|Vgf1(x, ξ)| = 1λ |Vgλf1(λx, ξλ)|.
Thus,
|Vgf1(x, ξ)|2w2(x, ξ) = λ−2|Vgλf1(x, ξ)|2w2(xλ−1, λξ).
Suppose |w( xλ , λξ)| ≤ b(λ)w(x, ξ), b2(λ)/λ2 -bounded. Then∫∫
|Vgf1(x, ξ)|2w2(x, ξ)dxdξ
≤ λ−2b2(λ)∫∫|Vgλf(x, ξ)|2w2(x, ξ)dxdξ︸ ︷︷ ︸
≈‖f‖
.
Roza Aceska http://nuhag.eu
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
Dilation invariance
Corollary
If m(x) is a bounded function and wm is a moderate weight,then Bm(x) is a dilation invariant space.
Roza Aceska http://nuhag.eu
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
Bandwidth shift: cosine-like band
Roza Aceska http://nuhag.eu
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
Bandwidth shift
|ξ| −m(x) > 1, but |ξ| −m(x)− h(x) ≤ 1;
wm(x, ξ) = (|ξ| −m(x))s, wm+h(x, ξ) = 1.
As 0 ≤ |ξ| −m(x) ≤ h(x) + 1,
2−swm+h(x, ξ) < 1 < wm(x, ξ) ≤ (h(x) + 1)s ≤ 2s =2swm+h(x, ξ).
Roza Aceska http://nuhag.eu
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
Bandwidth shift
Proposition
Let |h(x)| < 1 for all x ∈ R , and assume that m(x)generates a moderate weight wm. Then m(x) + h(x) alsogenerates a moderate weight wm+h and for all (x, ξ) ∈ R2d
2−swm+h(x, ξ) ≤ wm(x, ξ) ≤ 2swm+h(x, ξ).
s is the power of wm.
Roza Aceska http://nuhag.eu
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
LemmaMoving the variable bandwidth m(x) for a step |h(x)| < 1generates equivalent norms:
2−s‖Vgf · wm+h‖2 ≤ ‖Vgfwm‖2 ≤ 2−s‖Vgfwm+h‖2. (6)
Comment- finite number of shifting the bandwidth
Corollary
B(m+h)(x) ' Bm(x) ' Hs, (7)
Hs(Rd) = f ∈ S′ :∫|f(ξ)|2(1 + |ξ|)sdξ <∞ - Bessel
potential space.
Roza Aceska http://nuhag.eu
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
LemmaMoving the variable bandwidth m(x) for a step |h(x)| < 1generates equivalent norms:
2−s‖Vgf · wm+h‖2 ≤ ‖Vgfwm‖2 ≤ 2−s‖Vgfwm+h‖2. (6)
Comment- finite number of shifting the bandwidth
Corollary
B(m+h)(x) ' Bm(x) ' Hs, (7)
Hs(Rd) = f ∈ S′ :∫|f(ξ)|2(1 + |ξ|)sdξ <∞ - Bessel
potential space.
Roza Aceska http://nuhag.eu
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
LemmaMoving the variable bandwidth m(x) for a step |h(x)| < 1generates equivalent norms:
2−s‖Vgf · wm+h‖2 ≤ ‖Vgfwm‖2 ≤ 2−s‖Vgfwm+h‖2. (6)
Comment- finite number of shifting the bandwidth
Corollary
B(m+h)(x) ' Bm(x) ' Hs, (7)
Hs(Rd) = f ∈ S′ :∫|f(ξ)|2(1 + |ξ|)sdξ <∞ - Bessel
potential space.
Roza Aceska http://nuhag.eu
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
Example
Roza Aceska http://nuhag.eu
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
band-limited functions
Let l > 0, Ωl = [−l, l]d;A function f is said to be band-limited, if supp(f) ⊆ Ωl .
BΩl(Rd) - the space of band-limited functions with spectralsupport in Ωl.
Roza Aceska http://nuhag.eu
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
band-limited functions
Let l > 0, Ωl = [−l, l]d;A function f is said to be band-limited, if supp(f) ⊆ Ωl .
BΩl(Rd) - the space of band-limited functions with spectralsupport in Ωl.
Roza Aceska http://nuhag.eu
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
band-limited functions
Roza Aceska http://nuhag.eu
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
band-limited functions
Let l > 0, Ωl = [−l, l]d and let BΩl(Rd) be the space ofband-limited functions with spectral support in Ωs.
Proposition
BΩl(Rd) ≤ Bm(x)(Rd).
Roza Aceska http://nuhag.eu
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
band-limited functions
Let l > 0, Ωl = [−l, l]d and let BΩl(Rd) be the space ofband-limited functions with spectral support in Ωs.
Proposition
BΩl(Rd) ≤ Bm(x)(Rd).
Roza Aceska http://nuhag.eu
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
ProblemConsider a sequence of bandlimited functions
fn ∈ BΩn , Ωn = [−m(n),m(n)] . (8)
Is f =∑
n fn ∈ Bm(x)?
Not always! How about∑
n fnψn? (ψn form a bupu)
Roza Aceska http://nuhag.eu
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
ProblemConsider a sequence of bandlimited functions
fn ∈ BΩn , Ωn = [−m(n),m(n)] . (8)
Is f =∑
n fn ∈ Bm(x)?
Not always!
How about∑
n fnψn? (ψn form a bupu)
Roza Aceska http://nuhag.eu
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
ProblemConsider a sequence of bandlimited functions
fn ∈ BΩn , Ωn = [−m(n),m(n)] . (8)
Is f =∑
n fn ∈ Bm(x)?
Not always! How about∑
n fnψn? (ψn form a bupu)
Roza Aceska http://nuhag.eu
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
ProblemConsider a sequence of bandlimited functions
fn ∈ BΩn , Ωn = [−m(n),m(n)] . (8)
Is f =∑
n fn ∈ Bm(x)?
Not always! How about∑
n fnψn? (ψn form a bupu)
Roza Aceska http://nuhag.eu
Six signals
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
used tools
Vg (c1f1 + c2f2) = c1Vg (f1) + c2Vg (f2)
for disjointly compactly supported fn,∥∥Vg (∑n∈IN fn)∥∥ =
∑n∈IN ‖Vg (fn)‖
Roza Aceska http://nuhag.eu
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
used tools
Vg (c1f1 + c2f2) = c1Vg (f1) + c2Vg (f2)
for disjointly compactly supported fn,∥∥Vg (∑n∈IN fn)∥∥ =
∑n∈IN ‖Vg (fn)‖
Roza Aceska http://nuhag.eu
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
Theorem
TheoremGiven a bounded uniform partition of unity (ψn)n∈IN and asequence(cn) ∈ l∞, assume that
fn ∈ BΩn , Ωn = [−m(n),m(n)].
Thenf =
∑n∈IN
cnfnψn ∈ Bm (9)
and||∑n∈IN
cnfnψn||Bm ≤ ‖c‖∞∑n∈IN
‖fnψn‖Bm
. (10)
Roza Aceska http://nuhag.eu
Six signals
The sum
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
Reproducing kernel
A reproducing kernel for Bm is a R2d functionΦ(x, y) := φy(x)such that
φ ∈ Bm and
f(y) = 〈f,Φ(·, y)〉Bm (11)
for every f ∈ Bm.
Roza Aceska http://nuhag.eu
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
Example
The reproducing kernel for a Sobolev space Hs is
Φ(x, y) = φ(x− y) = Ty(F−1(w−2m )). (12)
See Feichtinger & Werther, 2002: Robustness of minimalnorm interpolation in Sobolev algebras.
Roza Aceska http://nuhag.eu
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
Example
The reproducing kernel for a Sobolev space Hs is
Φ(x, y) = φ(x− y) = Ty(F−1(w−2m )). (12)
See Feichtinger & Werther, 2002: Robustness of minimalnorm interpolation in Sobolev algebras.
Roza Aceska http://nuhag.eu
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
Variable bandwidth reproducing kernel
Ms(Rd) = f ∈ S′ :∫∫|Vgf(x, ξ)|2wm(x, ξ)2dxdξ ≤ ∞,
weight is equivalent to vs(x, ξ) = (1 + |ξ|)s.
Scalar product: 〈f, g〉M := 〈Vgfwm, Vghwm〉L2 .
What would Φ be so that f(y) = 〈f, φy〉M?
Roza Aceska http://nuhag.eu
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
Variable bandwidth reproducing kernel
Ms(Rd) = f ∈ S′ :∫∫|Vgf(x, ξ)|2wm(x, ξ)2dxdξ ≤ ∞,
weight is equivalent to vs(x, ξ) = (1 + |ξ|)s.
Scalar product: 〈f, g〉M := 〈Vgfwm, Vghwm〉L2 .
What would Φ be so that f(y) = 〈f, φy〉M?
Roza Aceska http://nuhag.eu
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
Variable bandwidth reproducing kernel
Ms(Rd) = f ∈ S′ :∫∫|Vgf(x, ξ)|2wm(x, ξ)2dxdξ ≤ ∞,
weight is equivalent to vs(x, ξ) = (1 + |ξ|)s.
Scalar product: 〈f, g〉M := 〈Vgfwm, Vghwm〉L2 .
What would Φ be so that f(y) = 〈f, φy〉M?
Roza Aceska http://nuhag.eu
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
Variable bandwidth reproducing kernel
Ms(Rd) = f ∈ S′ :∫∫|Vgf(x, ξ)|2wm(x, ξ)2dxdξ ≤ ∞,
weight is equivalent to vs(x, ξ) = (1 + |ξ|)s.
Scalar product: 〈f, g〉M := 〈Vgfwm, Vghwm〉L2 .
What would Φ be so that f(y) = 〈f, φy〉M?
Roza Aceska http://nuhag.eu
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
Variable bandwidth reproducing kernel
〈f, φy〉M =
〈Vgfwm, Vg(φy)wm〉L2 = 〈e−2πixξT Vgf , Vg(Tyφ)w2m〉L2
(use Vgf(x, ξ) = e−2πixξVgf(ξ,−x) = e−2πixξT Vgf(x, ξ))
= 〈f , V ∗g T −1(Vg(φy)e2πixξw2
m
)〉L2 .
By uniqueness,
V ∗g T −1(Vg(φy)e2πixξw2
m
)= e2πiy·,
i.e.
Φ(·, y) = φy(·) = V ∗g
(e−2πixξw−2
s (x, ξ)T Vg(e2πiy·)(x, ξ)).
(13)
Roza Aceska http://nuhag.eu
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
Variable bandwidth reproducing kernel
〈f, φy〉M =
〈Vgfwm, Vg(φy)wm〉L2 = 〈e−2πixξT Vgf , Vg(Tyφ)w2m〉L2
(use Vgf(x, ξ) = e−2πixξVgf(ξ,−x) = e−2πixξT Vgf(x, ξ))
= 〈f , V ∗g T −1(Vg(φy)e2πixξw2
m
)〉L2 .
By uniqueness,
V ∗g T −1(Vg(φy)e2πixξw2
m
)= e2πiy·,
i.e.
Φ(·, y) = φy(·) = V ∗g
(e−2πixξw−2
s (x, ξ)T Vg(e2πiy·)(x, ξ)).
(13)
Roza Aceska http://nuhag.eu
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
Variable bandwidth reproducing kernel
〈f, φy〉M =
〈Vgfwm, Vg(φy)wm〉L2 = 〈e−2πixξT Vgf , Vg(Tyφ)w2m〉L2
(use Vgf(x, ξ) = e−2πixξVgf(ξ,−x) = e−2πixξT Vgf(x, ξ))
= 〈f , V ∗g T −1(Vg(φy)e2πixξw2
m
)〉L2 .
By uniqueness,
V ∗g T −1(Vg(φy)e2πixξw2
m
)= e2πiy·,
i.e.
Φ(·, y) = φy(·) = V ∗g
(e−2πixξw−2
s (x, ξ)T Vg(e2πiy·)(x, ξ)).
(13)
Roza Aceska http://nuhag.eu
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
Variable bandwidth reproducing kernel
〈f, φy〉M =
〈Vgfwm, Vg(φy)wm〉L2 = 〈e−2πixξT Vgf , Vg(Tyφ)w2m〉L2
(use Vgf(x, ξ) = e−2πixξVgf(ξ,−x) = e−2πixξT Vgf(x, ξ))
= 〈f , V ∗g T −1(Vg(φy)e2πixξw2
m
)〉L2 .
By uniqueness,
V ∗g T −1(Vg(φy)e2πixξw2
m
)= e2πiy·,
i.e.
Φ(·, y) = φy(·) = V ∗g
(e−2πixξw−2
s (x, ξ)T Vg(e2πiy·)(x, ξ)).
(13)
Roza Aceska http://nuhag.eu
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
Variable bandwidth reproducing kernel
〈f, φy〉M =
〈Vgfwm, Vg(φy)wm〉L2 = 〈e−2πixξT Vgf , Vg(Tyφ)w2m〉L2
(use Vgf(x, ξ) = e−2πixξVgf(ξ,−x) = e−2πixξT Vgf(x, ξ))
= 〈f , V ∗g T −1(Vg(φy)e2πixξw2
m
)〉L2 .
By uniqueness,
V ∗g T −1(Vg(φy)e2πixξw2
m
)= e2πiy·,
i.e.
Φ(·, y) = φy(·) = V ∗g
(e−2πixξw−2
s (x, ξ)T Vg(e2πiy·)(x, ξ)).
(13)
Roza Aceska http://nuhag.eu
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
Summary
Weighted modulation spaceBandwidth shiftDilation invarianceSumming band-limited functionsReproducing kernelFuture work
Roza Aceska http://nuhag.eu
Outline
Intro
Weightedmodulationspaces
Properties
Bandwidthshift
Summingbandlim-itedfunctions
what wecan dowith band-limitedfunctions
Reprodu-cingkernel
Summary
Bibliography
Grochenig, K.; 2001: Foundations of Time-FrequencyAnalysisFeichtinger, H.G.; 1981: On a new Segal algebraFeichtinger, H.,Werther, T.; 2002:Robustness ofminimal norm interpolation in Sobolev algebras.http://homepage.univie.ac.at/roza.aceska/papers.html
Roza Aceska http://nuhag.eu