Gabor Systems and Almost Periodic Functions

Gabor systems and almost periodic functions

Antonio Galbis

Strobl 2016

Joint work with Paolo Boggiatto and Carmen Fernández The concept of almost periodicity was ﬁrst studied by Harald Bohr in connection with dynamical systems and later generalized by Stepanov, Weyl, Besicovitch or von Neumann, amongst others.

Applications in harmonic analysis, group theory or differential equations. M. A. Shubin introduced scales of Sobolev spaces of almost periodic functions and recently A. Oliaro, L. Rodino and P.Wahlberg extended some of these results to almost periodic Gevrey classes.

Aim To describe the almost periodic norm of functions of one variable in terms of its coefﬁcients with respect to a Gabor system.

(Time-Frequency Analysis and Related Topics) 2 / 19 Applications in harmonic analysis, group theory or differential equations. M. A. Shubin introduced scales of Sobolev spaces of almost periodic functions and recently A. Oliaro, L. Rodino and P.Wahlberg extended some of these results to almost periodic Gevrey classes.

Aim To describe the almost periodic norm of functions of one variable in terms of its coefﬁcients with respect to a Gabor system.

The concept of almost periodicity was ﬁrst studied by Harald Bohr in connection with dynamical systems and later generalized by Stepanov, Weyl, Besicovitch or von Neumann, amongst others.

(Time-Frequency Analysis and Related Topics) 2 / 19 M. A. Shubin introduced scales of Sobolev spaces of almost periodic functions and recently A. Oliaro, L. Rodino and P.Wahlberg extended some of these results to almost periodic Gevrey classes.

Aim To describe the almost periodic norm of functions of one variable in terms of its coefﬁcients with respect to a Gabor system.

The concept of almost periodicity was ﬁrst studied by Harald Bohr in connection with dynamical systems and later generalized by Stepanov, Weyl, Besicovitch or von Neumann, amongst others.

Applications in harmonic analysis, group theory or differential equations.

(Time-Frequency Analysis and Related Topics) 2 / 19 Aim To describe the almost periodic norm of functions of one variable in terms of its coefﬁcients with respect to a Gabor system.

The concept of almost periodicity was ﬁrst studied by Harald Bohr in connection with dynamical systems and later generalized by Stepanov, Weyl, Besicovitch or von Neumann, amongst others.

(Time-Frequency Analysis and Related Topics) 2 / 19 Problem: in which sense frame-type inequalities are still possible for norm estimations in these spaces.

J.R. Partington - B. Unalmis (2001) and F. Galindo (2004) using windowed Fourier transform and wavelet transforms Y.H. Kim and A. Ron (2009) using ﬁberization techniques developed by A. Ron and Z. Shen in the context of shift-invariant systems.

Connections with frame theory

As the main spaces of almost periodic functions are non-separable they cannot admit countable frames.

(Time-Frequency Analysis and Related Topics) 3 / 19 J.R. Partington - B. Unalmis (2001) and F. Galindo (2004) using windowed Fourier transform and wavelet transforms Y.H. Kim and A. Ron (2009) using ﬁberization techniques developed by A. Ron and Z. Shen in the context of shift-invariant systems.

Connections with frame theory

As the main spaces of almost periodic functions are non-separable they cannot admit countable frames. Problem: in which sense frame-type inequalities are still possible for norm estimations in these spaces.

(Time-Frequency Analysis and Related Topics) 3 / 19 Y.H. Kim and A. Ron (2009) using ﬁberization techniques developed by A. Ron and Z. Shen in the context of shift-invariant systems.

Connections with frame theory

J.R. Partington - B. Unalmis (2001) and F. Galindo (2004) using windowed Fourier transform and wavelet transforms

(Time-Frequency Analysis and Related Topics) 3 / 19 Connections with frame theory

(Time-Frequency Analysis and Related Topics) 3 / 19 The function f(x) = cos x + cos(πx) is almost periodic but not periodic. f ∈ AP(R) if, and only if, it is continuous and can be uniformly approximated by a sequence of trigonometric polynomials

Xn ak eλk , ak ∈ C, λk ∈ R, n ∈ N, k=1

iλt where eλ(t) = e .

Almost periodic functions (Bohr) A continuous function f : R → C is said to be almost periodic (in the sense of Bohr) if for every > 0 there exists a positive number L such that every interval of length L contains a number τ with the property that

sup |f(t + τ) − f(t)| < . t

(Time-Frequency Analysis and Related Topics) 4 / 19 f ∈ AP(R) if, and only if, it is continuous and can be uniformly approximated by a sequence of trigonometric polynomials

Xn ak eλk , ak ∈ C, λk ∈ R, n ∈ N, k=1

iλt where eλ(t) = e .

sup |f(t + τ) − f(t)| < . t

The function f(x) = cos x + cos(πx) is almost periodic but not periodic.

(Time-Frequency Analysis and Related Topics) 4 / 19 iλt where eλ(t) = e .

sup |f(t + τ) − f(t)| < . t

The function f(x) = cos x + cos(πx) is almost periodic but not periodic. f ∈ AP(R) if, and only if, it is continuous and can be uniformly approximated by a sequence of trigonometric polynomials

Xn ak eλk , ak ∈ C, λk ∈ R, n ∈ N, k=1

(Time-Frequency Analysis and Related Topics) 4 / 19 Almost periodic functions (Bohr) A continuous function f : R → C is said to be almost periodic (in the sense of Bohr) if for every > 0 there exists a positive number L such that every interval of length L contains a number τ with the property that

sup |f(t + τ) − f(t)| < . t

The function f(x) = cos x + cos(πx) is almost periodic but not periodic. f ∈ AP(R) if, and only if, it is continuous and can be uniformly approximated by a sequence of trigonometric polynomials

Xn ak eλk , ak ∈ C, λk ∈ R, n ∈ N, k=1

iλt where eλ(t) = e .

(Time-Frequency Analysis and Related Topics) 4 / 19 The space AP is non-separable and non-complete with respect to the norm k · kAP. The Besicovitch space of almost periodic functions is the non separable Hilbert space

AP2(R) obtained by completion of (AP(R), k · kAP) . The exponentials (eλ)λ∈R form a non-countable orthonormal system and it turns out that for every f ∈ AP2(R), at most for countably many λ ∈ R we have

a ( ) :=bf( ) := (f e ) 0 f λ λ , λ AP2(R) , . Moreover the Parseval identity holds, that is, X kfk2 = |a (λ)|2 . AP2(R) f λ∈R

Almost periodic functions (Besicovitch) AP(R) admits an inner product deﬁned by

1 Z T (f, g)AP = lim f(t)g(t) dt. T→∞ 2T −T

Let kfkAP denote the norm induced by this inner product.

(Time-Frequency Analysis and Related Topics) 5 / 19 The Besicovitch space of almost periodic functions is the non separable Hilbert space

a ( ) :=bf( ) := (f e ) 0 f λ λ , λ AP2(R) , . Moreover the Parseval identity holds, that is, X kfk2 = |a (λ)|2 . AP2(R) f λ∈R

Almost periodic functions (Besicovitch) AP(R) admits an inner product deﬁned by

1 Z T (f, g)AP = lim f(t)g(t) dt. T→∞ 2T −T

Let kfkAP denote the norm induced by this inner product.

The space AP is non-separable and non-complete with respect to the norm k · kAP.

(Time-Frequency Analysis and Related Topics) 5 / 19 The exponentials (eλ)λ∈R form a non-countable orthonormal system and it turns out that for every f ∈ AP2(R), at most for countably many λ ∈ R we have

a ( ) :=bf( ) := (f e ) 0 f λ λ , λ AP2(R) , . Moreover the Parseval identity holds, that is, X kfk2 = |a (λ)|2 . AP2(R) f λ∈R

Almost periodic functions (Besicovitch) AP(R) admits an inner product deﬁned by

1 Z T (f, g)AP = lim f(t)g(t) dt. T→∞ 2T −T

Let kfkAP denote the norm induced by this inner product.

The space AP is non-separable and non-complete with respect to the norm k · kAP. The Besicovitch space of almost periodic functions is the non separable Hilbert space

AP2(R) obtained by completion of (AP(R), k · kAP) .

(Time-Frequency Analysis and Related Topics) 5 / 19 and it turns out that for every f ∈ AP2(R), at most for countably many λ ∈ R we have

a ( ) :=bf( ) := (f e ) 0 f λ λ , λ AP2(R) , . Moreover the Parseval identity holds, that is, X kfk2 = |a (λ)|2 . AP2(R) f λ∈R

Almost periodic functions (Besicovitch) AP(R) admits an inner product deﬁned by

1 Z T (f, g)AP = lim f(t)g(t) dt. T→∞ 2T −T

Let kfkAP denote the norm induced by this inner product.

The space AP is non-separable and non-complete with respect to the norm k · kAP. The Besicovitch space of almost periodic functions is the non separable Hilbert space

AP2(R) obtained by completion of (AP(R), k · kAP) . The exponentials (eλ)λ∈R form a non-countable orthonormal system

(Time-Frequency Analysis and Related Topics) 5 / 19 Moreover the Parseval identity holds, that is, X kfk2 = |a (λ)|2 . AP2(R) f λ∈R

Almost periodic functions (Besicovitch) AP(R) admits an inner product deﬁned by

1 Z T (f, g)AP = lim f(t)g(t) dt. T→∞ 2T −T

Let kfkAP denote the norm induced by this inner product.

The space AP is non-separable and non-complete with respect to the norm k · kAP. The Besicovitch space of almost periodic functions is the non separable Hilbert space

a ( ) :=bf( ) := (f e ) 0 f λ λ , λ AP2(R) , .

(Time-Frequency Analysis and Related Topics) 5 / 19 Almost periodic functions (Besicovitch) AP(R) admits an inner product deﬁned by

1 Z T (f, g)AP = lim f(t)g(t) dt. T→∞ 2T −T

Let kfkAP denote the norm induced by this inner product.

The space AP is non-separable and non-complete with respect to the norm k · kAP. The Besicovitch space of almost periodic functions is the non separable Hilbert space

a ( ) :=bf( ) := (f e ) 0 f λ λ , λ AP2(R) , . Moreover the Parseval identity holds, that is, X kfk2 = |a (λ)|2 . AP2(R) f λ∈R

(Time-Frequency Analysis and Related Topics) 5 / 19 2 k · k ⊂ k · k 2 ∞ ⊂ k · k Then (AP(R), ∞) S , W(L ,L ) AP2(R), AP2(R) with continuous inclusions.

AP sequences

A sequence (ak )k∈Z is said to be almost periodic if it is the restriction to the integers of a function in AP(R). This means that for every > 0 there is N ∈ N such that among any N consecutive integers there exists an integer p with the property

ak+p − ak < ∀k ∈ Z.

iλk A typical example is the sequence e˜λ(k) = e , k ∈ Z. We will always consider on AP(Z) the norm deﬁned in terms of the inner product

1 Xk (a, b)AP(Z) := lim aj bj . k→∞ 2k j=−k

Almost periodic functions (Stepanov) S2 is the closure of the trigonometric polynomials in the amalgam space W(L 2, L ∞) of those measurable functions f such that

1 Z t+1 ! 2 2 kfkW(L2,L∞) = sup |f(s)| ds < ∞. t∈R t

(Time-Frequency Analysis and Related Topics) 6 / 19 AP sequences

ak+p − ak < ∀k ∈ Z.

iλk A typical example is the sequence e˜λ(k) = e , k ∈ Z. We will always consider on AP(Z) the norm deﬁned in terms of the inner product

1 Xk (a, b)AP(Z) := lim aj bj . k→∞ 2k j=−k

Almost periodic functions (Stepanov) S2 is the closure of the trigonometric polynomials in the amalgam space W(L 2, L ∞) of those measurable functions f such that

1 Z t+1 ! 2 2 kfkW(L2,L∞) = sup |f(s)| ds < ∞. t∈R t

2 k · k ⊂ k · k 2 ∞ ⊂ k · k Then (AP(R), ∞) S , W(L ,L ) AP2(R), AP2(R) with continuous inclusions.

(Time-Frequency Analysis and Related Topics) 6 / 19 This means that for every > 0 there is N ∈ N such that among any N consecutive integers there exists an integer p with the property

ak+p − ak < ∀k ∈ Z.

iλk A typical example is the sequence e˜λ(k) = e , k ∈ Z. We will always consider on AP(Z) the norm deﬁned in terms of the inner product

1 Xk (a, b)AP(Z) := lim aj bj . k→∞ 2k j=−k

Almost periodic functions (Stepanov) S2 is the closure of the trigonometric polynomials in the amalgam space W(L 2, L ∞) of those measurable functions f such that

1 Z t+1 ! 2 2 kfkW(L2,L∞) = sup |f(s)| ds < ∞. t∈R t

2 k · k ⊂ k · k 2 ∞ ⊂ k · k Then (AP(R), ∞) S , W(L ,L ) AP2(R), AP2(R) with continuous inclusions.

AP sequences

A sequence (ak )k∈Z is said to be almost periodic if it is the restriction to the integers of a function in AP(R).

(Time-Frequency Analysis and Related Topics) 6 / 19 iλk A typical example is the sequence e˜λ(k) = e , k ∈ Z. We will always consider on AP(Z) the norm deﬁned in terms of the inner product

1 Xk (a, b)AP(Z) := lim aj bj . k→∞ 2k j=−k

Almost periodic functions (Stepanov) S2 is the closure of the trigonometric polynomials in the amalgam space W(L 2, L ∞) of those measurable functions f such that

1 Z t+1 ! 2 2 kfkW(L2,L∞) = sup |f(s)| ds < ∞. t∈R t

2 k · k ⊂ k · k 2 ∞ ⊂ k · k Then (AP(R), ∞) S , W(L ,L ) AP2(R), AP2(R) with continuous inclusions.

AP sequences

ak+p − ak < ∀k ∈ Z.

(Time-Frequency Analysis and Related Topics) 6 / 19 We will always consider on AP(Z) the norm deﬁned in terms of the inner product

1 Xk (a, b)AP(Z) := lim aj bj . k→∞ 2k j=−k

Almost periodic functions (Stepanov) S2 is the closure of the trigonometric polynomials in the amalgam space W(L 2, L ∞) of those measurable functions f such that

1 Z t+1 ! 2 2 kfkW(L2,L∞) = sup |f(s)| ds < ∞. t∈R t

2 k · k ⊂ k · k 2 ∞ ⊂ k · k Then (AP(R), ∞) S , W(L ,L ) AP2(R), AP2(R) with continuous inclusions.

AP sequences

ak+p − ak < ∀k ∈ Z.

iλk A typical example is the sequence e˜λ(k) = e , k ∈ Z.

(Time-Frequency Analysis and Related Topics) 6 / 19 Almost periodic functions (Stepanov) S2 is the closure of the trigonometric polynomials in the amalgam space W(L 2, L ∞) of those measurable functions f such that

1 Z t+1 ! 2 2 kfkW(L2,L∞) = sup |f(s)| ds < ∞. t∈R t

2 k · k ⊂ k · k 2 ∞ ⊂ k · k Then (AP(R), ∞) S , W(L ,L ) AP2(R), AP2(R) with continuous inclusions.

AP sequences

ak+p − ak < ∀k ∈ Z.

iλk A typical example is the sequence e˜λ(k) = e , k ∈ Z. We will always consider on AP(Z) the norm deﬁned in terms of the inner product

1 Xk (a, b)AP(Z) := lim aj bj . k→∞ 2k j=−k

(Time-Frequency Analysis and Related Topics) 6 / 19 where 2πibt (Ta g)(t) = g(t − a) and (Mb g)(t) = e g(t) are the translation and modulation operators.

Classical periodization lemma P ∞ ψ ∈ W 7→ k∈Z Tαk ψ ∈ L (R)

Wiener space The Wiener algebra W consists of those functions ψ ∈ L ∞(R) such that X kψkW := sup |ψ(x − k)| < ∞. ∈ k∈Z x [0,1]

The subspace of continuous functions is denoted by W0. W0 ∩ Wc0 contains the Feichtinger algebra M1.

Gabor system Throughout the presentation α, β > 0 and ψ ∈ L 2(R). We consider the family of time-frequency shifts G(ψ, α, β) = Tαk Mβ`ψ k,`∈Z

(Time-Frequency Analysis and Related Topics) 7 / 19 Classical periodization lemma P ∞ ψ ∈ W 7→ k∈Z Tαk ψ ∈ L (R)

Wiener space The Wiener algebra W consists of those functions ψ ∈ L ∞(R) such that X kψkW := sup |ψ(x − k)| < ∞. ∈ k∈Z x [0,1]

The subspace of continuous functions is denoted by W0. W0 ∩ Wc0 contains the Feichtinger algebra M1.

Gabor system Throughout the presentation α, β > 0 and ψ ∈ L 2(R). We consider the family of time-frequency shifts G(ψ, α, β) = Tαk Mβ`ψ k,`∈Z where 2πibt (Ta g)(t) = g(t − a) and (Mb g)(t) = e g(t) are the translation and modulation operators.

(Time-Frequency Analysis and Related Topics) 7 / 19 Wiener space The Wiener algebra W consists of those functions ψ ∈ L ∞(R) such that X kψkW := sup |ψ(x − k)| < ∞. ∈ k∈Z x [0,1]

The subspace of continuous functions is denoted by W0. W0 ∩ Wc0 contains the Feichtinger algebra M1.

Classical periodization lemma P ∞ ψ ∈ W 7→ k∈Z Tαk ψ ∈ L (R)

(Time-Frequency Analysis and Related Topics) 7 / 19 Wiener space The Wiener algebra W consists of those functions ψ ∈ L ∞(R) such that X kψkW := sup |ψ(x − k)| < ∞. ∈ k∈Z x [0,1]

The subspace of continuous functions is denoted by W0. W0 ∩ Wc0 contains the Feichtinger algebra M1.

Classical periodization lemma P ∞ ψ ∈ W 7→ k∈Z Tαk ψ ∈ L (R)

(Time-Frequency Analysis and Related Topics) 7 / 19 The subspace of continuous functions is denoted by W0. W0 ∩ Wc0 contains the Feichtinger algebra M1.

Classical periodization lemma P ∞ ψ ∈ W 7→ k∈Z Tαk ψ ∈ L (R)

Wiener space The Wiener algebra W consists of those functions ψ ∈ L ∞(R) such that X kψkW := sup |ψ(x − k)| < ∞. ∈ k∈Z x [0,1]

(Time-Frequency Analysis and Related Topics) 7 / 19 W0 ∩ Wc0 contains the Feichtinger algebra M1.

Classical periodization lemma P ∞ ψ ∈ W 7→ k∈Z Tαk ψ ∈ L (R)

Wiener space The Wiener algebra W consists of those functions ψ ∈ L ∞(R) such that X kψkW := sup |ψ(x − k)| < ∞. ∈ k∈Z x [0,1]

The subspace of continuous functions is denoted by W0.

(Time-Frequency Analysis and Related Topics) 7 / 19 Gabor system Throughout the presentation α, β > 0 and ψ ∈ L 2(R). We consider the family of time-frequency shifts G(ψ, α, β) = Tαk Mβ`ψ k,`∈Z where 2πibt (Ta g)(t) = g(t − a) and (Mb g)(t) = e g(t) are the translation and modulation operators.

Classical periodization lemma P ∞ ψ ∈ W 7→ k∈Z Tαk ψ ∈ L (R)

Wiener space The Wiener algebra W consists of those functions ψ ∈ L ∞(R) such that X kψkW := sup |ψ(x − k)| < ∞. ∈ k∈Z x [0,1]

The subspace of continuous functions is denoted by W0. W0 ∩ Wc0 contains the Feichtinger algebra M1.

(Time-Frequency Analysis and Related Topics) 7 / 19 Moreover, its µ-Fourier coefﬁcient is 1 c = bψ(µ)ha, e˜ i µ α αµ AP(Z) and the map X k · k → k · k 7→ AP(Z), AP(Z) AP(R), AP2(R) , a ak Tαk ψ, k∈Z is continuous.

Proposition

Let α > 0 and ψ ∈ W0 be given. For every sequence a = (ak )k∈Z ∈ AP(Z) the function X g = ak Tαk ψ k∈Z is almost periodic in the sense of Bohr.

(Time-Frequency Analysis and Related Topics) 8 / 19 and the map X k · k → k · k 7→ AP(Z), AP(Z) AP(R), AP2(R) , a ak Tαk ψ, k∈Z is continuous.

Proposition

Let α > 0 and ψ ∈ W0 be given. For every sequence a = (ak )k∈Z ∈ AP(Z) the function X g = ak Tαk ψ k∈Z is almost periodic in the sense of Bohr. Moreover, its µ-Fourier coefﬁcient is 1 c = bψ(µ)ha, e˜ i µ α αµ AP(Z)

(Time-Frequency Analysis and Related Topics) 8 / 19 Proposition

(Time-Frequency Analysis and Related Topics) 8 / 19 For every k ∈ Z,

sup Tkαψ ⊂ [(−q + k)α, (q + k)α]. Consequently, for any N > q,     Z Nα X Z Nα  X    −iµt   −iµt  ak Tkαψ (t)e dt =  ak Tkαψ (t)e dt −   −   Nα k Nα |k|6N−q

Z ∞ X −iµt = ak (Tkαψ) (t)e dt −∞ |k|6N−q

X −iµkα = bψ(µ) ak e . |k|6N−q

Since   1 Z Nα X   −iµt cµ = lim  ak Tkαψ (t)e dt, N→∞ 2Nα   −Nα k the conclusion follows.

Proof (Fourier coefﬁcients): Assume ψ is compactly supported, let say on [−qα, qα], q ∈ N.

(Time-Frequency Analysis and Related Topics) 9 / 19 Consequently, for any N > q,     Z Nα X Z Nα  X    −iµt   −iµt  ak Tkαψ (t)e dt =  ak Tkαψ (t)e dt −   −   Nα k Nα |k|6N−q

Z ∞ X −iµt = ak (Tkαψ) (t)e dt −∞ |k|6N−q

X −iµkα = bψ(µ) ak e . |k|6N−q

Since   1 Z Nα X   −iµt cµ = lim  ak Tkαψ (t)e dt, N→∞ 2Nα   −Nα k the conclusion follows.

Proof (Fourier coefﬁcients): Assume ψ is compactly supported, let say on [−qα, qα], q ∈ N. For every k ∈ Z,

sup Tkαψ ⊂ [(−q + k)α, (q + k)α].

(Time-Frequency Analysis and Related Topics) 9 / 19   Z Nα  X    −iµt =  ak Tkαψ (t)e dt −   Nα |k|6N−q

Z ∞ X −iµt = ak (Tkαψ) (t)e dt −∞ |k|6N−q

X −iµkα = bψ(µ) ak e . |k|6N−q

Since   1 Z Nα X   −iµt cµ = lim  ak Tkαψ (t)e dt, N→∞ 2Nα   −Nα k the conclusion follows.

Proof (Fourier coefﬁcients): Assume ψ is compactly supported, let say on [−qα, qα], q ∈ N. For every k ∈ Z,

sup Tkαψ ⊂ [(−q + k)α, (q + k)α]. Consequently, for any N > q,   Z Nα X   −iµt  ak Tkαψ (t)e dt −Nα k

(Time-Frequency Analysis and Related Topics) 9 / 19 Z ∞ X −iµt = ak (Tkαψ) (t)e dt −∞ |k|6N−q

X −iµkα = bψ(µ) ak e . |k|6N−q

Since   1 Z Nα X   −iµt cµ = lim  ak Tkαψ (t)e dt, N→∞ 2Nα   −Nα k the conclusion follows.

Proof (Fourier coefﬁcients): Assume ψ is compactly supported, let say on [−qα, qα], q ∈ N. For every k ∈ Z,

(Time-Frequency Analysis and Related Topics) 9 / 19 X −iµkα = bψ(µ) ak e . |k|6N−q

Since   1 Z Nα X   −iµt cµ = lim  ak Tkαψ (t)e dt, N→∞ 2Nα   −Nα k the conclusion follows.

Proof (Fourier coefﬁcients): Assume ψ is compactly supported, let say on [−qα, qα], q ∈ N. For every k ∈ Z,

Z ∞ X −iµt = ak (Tkαψ) (t)e dt −∞ |k|6N−q

(Time-Frequency Analysis and Related Topics) 9 / 19 Since   1 Z Nα X   −iµt cµ = lim  ak Tkαψ (t)e dt, N→∞ 2Nα   −Nα k the conclusion follows.

Proof (Fourier coefﬁcients): Assume ψ is compactly supported, let say on [−qα, qα], q ∈ N. For every k ∈ Z,

Z ∞ X −iµt = ak (Tkαψ) (t)e dt −∞ |k|6N−q

X −iµkα = bψ(µ) ak e . |k|6N−q

(Time-Frequency Analysis and Related Topics) 9 / 19 Proof (Fourier coefﬁcients): Assume ψ is compactly supported, let say on [−qα, qα], q ∈ N. For every k ∈ Z,

Z ∞ X −iµt = ak (Tkαψ) (t)e dt −∞ |k|6N−q

X −iµkα = bψ(µ) ak e . |k|6N−q

Since   1 Z Nα X   −iµt cµ = lim  ak Tkαψ (t)e dt, N→∞ 2Nα   −Nα k the conclusion follows.

(Time-Frequency Analysis and Related Topics) 9 / 19 Proof:

The case ψ ∈ Wc0 is well-known and follows from the deﬁnition.

2π Let us now assume ψ ∈ W0, take α = γ and consider, for each λ ∈ R, the almost periodic function X iλk gλ = e Tαk ψ. k∈Z

The non-zero Fourier coefﬁcients of gλ are precisely 1 bψ(λ + γk), k ∈ Z. α Hence X 2 2 iλk bψ(λ + γk) 6 Ckgλk 6 Ck e k . AP k∈Z AP k Since the almost periodic norm of this sequence is independent of λ, the conclusion follows.

Corollary

Let ψ ∈ W0 ∪ Wc0 and γ > 0 be given. Then

X 2 sup bψ(λ + γk) < ∞. λ k∈Z

(Time-Frequency Analysis and Related Topics) 10 / 19 2π Let us now assume ψ ∈ W0, take α = γ and consider, for each λ ∈ R, the almost periodic function X iλk gλ = e Tαk ψ. k∈Z

Corollary

Let ψ ∈ W0 ∪ Wc0 and γ > 0 be given. Then

X 2 sup bψ(λ + γk) < ∞. λ k∈Z

Proof:

The case ψ ∈ Wc0 is well-known and follows from the deﬁnition.

(Time-Frequency Analysis and Related Topics) 10 / 19 The non-zero Fourier coefﬁcients of gλ are precisely 1 bψ(λ + γk), k ∈ Z. α Hence X 2 2 iλk bψ(λ + γk) 6 Ckgλk 6 Ck e k . AP k∈Z AP k Since the almost periodic norm of this sequence is independent of λ, the conclusion follows.

Corollary

Let ψ ∈ W0 ∪ Wc0 and γ > 0 be given. Then

X 2 sup bψ(λ + γk) < ∞. λ k∈Z

Proof:

The case ψ ∈ Wc0 is well-known and follows from the deﬁnition.

2π Let us now assume ψ ∈ W0, take α = γ and consider, for each λ ∈ R, the almost periodic function X iλk gλ = e Tαk ψ. k∈Z

(Time-Frequency Analysis and Related Topics) 10 / 19 Hence X 2 2 iλk bψ(λ + γk) 6 Ckgλk 6 Ck e k . AP k∈Z AP k Since the almost periodic norm of this sequence is independent of λ, the conclusion follows.

Corollary

Let ψ ∈ W0 ∪ Wc0 and γ > 0 be given. Then

X 2 sup bψ(λ + γk) < ∞. λ k∈Z

Proof:

The case ψ ∈ Wc0 is well-known and follows from the deﬁnition.

2π Let us now assume ψ ∈ W0, take α = γ and consider, for each λ ∈ R, the almost periodic function X iλk gλ = e Tαk ψ. k∈Z

The non-zero Fourier coefﬁcients of gλ are precisely 1 bψ(λ + γk), k ∈ Z. α

(Time-Frequency Analysis and Related Topics) 10 / 19 Since the almost periodic norm of this sequence is independent of λ, the conclusion follows.

Corollary

Let ψ ∈ W0 ∪ Wc0 and γ > 0 be given. Then

X 2 sup bψ(λ + γk) < ∞. λ k∈Z

Proof:

The case ψ ∈ Wc0 is well-known and follows from the deﬁnition.

2π Let us now assume ψ ∈ W0, take α = γ and consider, for each λ ∈ R, the almost periodic function X iλk gλ = e Tαk ψ. k∈Z

The non-zero Fourier coefﬁcients of gλ are precisely 1 bψ(λ + γk), k ∈ Z. α Hence X 2 2 iλk bψ(λ + γk) 6 Ckgλk 6 Ck e k . AP k∈Z AP k

(Time-Frequency Analysis and Related Topics) 10 / 19 Corollary

Let ψ ∈ W0 ∪ Wc0 and γ > 0 be given. Then

X 2 sup bψ(λ + γk) < ∞. λ k∈Z

Proof:

The case ψ ∈ Wc0 is well-known and follows from the deﬁnition.

2π Let us now assume ψ ∈ W0, take α = γ and consider, for each λ ∈ R, the almost periodic function X iλk gλ = e Tαk ψ. k∈Z

(Time-Frequency Analysis and Related Topics) 10 / 19 Only translations Given ψ ∈ L 1(R), the map S : AP(R), k · kAP(R) → AP(Z), k · kAP(Z) , f → (hf, Tkαψi)k ,

is continuous if and only if X k sup |bψ(λ + 2π )|2 < ∞. λ α k In particular, this is the case when ψ ∈ W or ψ ∈ L 1(R) ∩ Wb .

Analysis map

We will denote by h·, ·i the usual duality involving L p spaces.

(Time-Frequency Analysis and Related Topics) 11 / 19 In particular, this is the case when ψ ∈ W or ψ ∈ L 1(R) ∩ Wb .

Analysis map

We will denote by h·, ·i the usual duality involving L p spaces.

Only translations Given ψ ∈ L 1(R), the map S : AP(R), k · kAP(R) → AP(Z), k · kAP(Z) , f → (hf, Tkαψi)k , is continuous if and only if X k sup |bψ(λ + 2π )|2 < ∞. λ α k

(Time-Frequency Analysis and Related Topics) 11 / 19 Analysis map

We will denote by h·, ·i the usual duality involving L p spaces.

Only translations Given ψ ∈ L 1(R), the map S : AP(R), k · kAP(R) → AP(Z), k · kAP(Z) , f → (hf, Tkαψi)k , is continuous if and only if X k sup |bψ(λ + 2π )|2 < ∞. λ α k In particular, this is the case when ψ ∈ W or ψ ∈ L 1(R) ∩ Wb .

(Time-Frequency Analysis and Related Topics) 11 / 19 Moreover there is a constant C such that X kh i k2 k k2 f, Tαk Mβ`ψ k∈Z AP(Z) 6 C f AP(R) `∈Z

for every f ∈ AP(R). This means that the vector-valued map S : AP(R) → `2(AP(Z)) given by

2 S(f) := hf, Tαk Mβ`ψi ∈ ` (AP(Z)) k∈Z `∈Z is continuous.

`2(AP(Z)) consists of all vector sequences a = a` where each a` ∈ AP(Z) and `∈Z X 2 ` 2 kak := ka kAP(Z) < ∞. `∈Z

Analysis map

Bessel type estimate

(Time-Frequency Analysis and Related Topics) 12 / 19 This means that the vector-valued map S : AP(R) → `2(AP(Z)) given by

2 S(f) := hf, Tαk Mβ`ψi ∈ ` (AP(Z)) k∈Z `∈Z is continuous.

`2(AP(Z)) consists of all vector sequences a = a` where each a` ∈ AP(Z) and `∈Z X 2 ` 2 kak := ka kAP(Z) < ∞. `∈Z

Analysis map

Bessel type estimate

1 Let ψ ∈ W0 or ψ ∈ L (R) ∩ Wc0. Then, for every almost periodic function f ∈ AP(R) and every ` ∈ Z, the sequence hf, Tαk Mβ`ψi , k∈Z obtained after taking the inner product of f with a sequence of translations of a ﬁxed modulation of ψ, is almost periodic. Moreover there is a constant C such that X kh i k2 k k2 f, Tαk Mβ`ψ k∈Z AP(Z) 6 C f AP(R) `∈Z for every f ∈ AP(R).

(Time-Frequency Analysis and Related Topics) 12 / 19 `2(AP(Z)) consists of all vector sequences a = a` where each a` ∈ AP(Z) and `∈Z X 2 ` 2 kak := ka kAP(Z) < ∞. `∈Z

Analysis map

Bessel type estimate

2 S(f) := hf, Tαk Mβ`ψi ∈ ` (AP(Z)) k∈Z `∈Z is continuous.

(Time-Frequency Analysis and Related Topics) 12 / 19 Analysis map

Bessel type estimate

2 S(f) := hf, Tαk Mβ`ψi ∈ ` (AP(Z)) k∈Z `∈Z is continuous.

`2(AP(Z)) consists of all vector sequences a = a` where each a` ∈ AP(Z) and `∈Z X 2 ` 2 kak := ka kAP(Z) < ∞. `∈Z (Time-Frequency Analysis and Related Topics) 12 / 19 We ﬁrst observe that ( ) 2π D ` E λ ∈ [0, ): a , e˜λα , 0 for some ` ∈ Z α n o∞ is a countable set, let say λj . j=1

D ` E On the other hand, if a , e˜λα , 0 for some ` ∈ Z then

2π λ = λ + p j α for some j ∈ N and p ∈ Z.

1 Proof (the case ψ ∈ L (R) and bψ ∈ W0): We consider the formal adjoint of S, which is the map T : `2(AP(Z)) → AP (R), deﬁned for 2 a = a` by `∈Z X D E (Ta e ) = ( − ) · a` e˜ , λ AP2(R) bψ λ `β , λα . `∈Z

(Time-Frequency Analysis and Related Topics) 13 / 19 D ` E On the other hand, if a , e˜λα , 0 for some ` ∈ Z then

2π λ = λ + p j α for some j ∈ N and p ∈ Z.

1 Proof (the case ψ ∈ L (R) and bψ ∈ W0): We consider the formal adjoint of S, which is the map T : `2(AP(Z)) → AP (R), deﬁned for 2 a = a` by `∈Z X D E (Ta e ) = ( − ) · a` e˜ , λ AP2(R) bψ λ `β , λα . `∈Z We ﬁrst observe that ( ) 2π D ` E λ ∈ [0, ): a , e˜λα , 0 for some ` ∈ Z α n o∞ is a countable set, let say λj . j=1

(Time-Frequency Analysis and Related Topics) 13 / 19 1 Proof (the case ψ ∈ L (R) and bψ ∈ W0): We consider the formal adjoint of S, which is the map T : `2(AP(Z)) → AP (R), deﬁned for 2 a = a` by `∈Z X D E (Ta e ) = ( − ) · a` e˜ , λ AP2(R) bψ λ `β , λα . `∈Z We ﬁrst observe that ( ) 2π D ` E λ ∈ [0, ): a , e˜λα , 0 for some ` ∈ Z α n o∞ is a countable set, let say λj . j=1

D ` E On the other hand, if a , e˜λα , 0 for some ` ∈ Z then

2π λ = λ + p j α for some j ∈ N and p ∈ Z.

(Time-Frequency Analysis and Related Topics) 13 / 19 We can apply Schur lemma in order to conclude that, for every j, the matrix bj deﬁnes `,p `,p a bounded operator on `2(Z) and the norm of that operator is bounded by a constant C independent of j. Finally 2 X X X X 2 j ` (Ta, eλ)AP (R) = b a , e˜λ α 2 `,p j AP(Z) λ j p `

X X 2 2 ` 6 C a , e˜λ α j AP(Z) j ` X 2 ` 2 2 2 = C ka kAP(Z) = C kak < ∞. `

continuation

We put 2π bj = bψ(λ + p − `β). `,p j α

(Time-Frequency Analysis and Related Topics) 14 / 19 Finally 2 X X X X 2 j ` (Ta, eλ)AP (R) = b a , e˜λ α 2 `,p j AP(Z) λ j p `

X X 2 2 ` 6 C a , e˜λ α j AP(Z) j ` X 2 ` 2 2 2 = C ka kAP(Z) = C kak < ∞. `

continuation

We put 2π bj = bψ(λ + p − `β). `,p j α We can apply Schur lemma in order to conclude that, for every j, the matrix bj deﬁnes `,p `,p a bounded operator on `2(Z) and the norm of that operator is bounded by a constant C independent of j.

(Time-Frequency Analysis and Related Topics) 14 / 19 continuation

X X 2 2 ` 6 C a , e˜λ α j AP(Z) j ` X 2 ` 2 2 2 = C ka kAP(Z) = C kak < ∞. `

(Time-Frequency Analysis and Related Topics) 14 / 19 Consequently

∗ 2 T : AP2(R) → ` (AP2(Z))

is a continuous map. The proof is complete after showing that

∗ D E 2 T (eλ) = eλ, TkαM`βψ ∈ ` (AP(Z)) (1) k ` for every λ ∈ R.

Continuation

This shows that (Ta, eλ) are the Fourier coefﬁcients of an element Ta ∈ AP2(R) and AP2(R) λ the map T is well-deﬁned and continuous.

(Time-Frequency Analysis and Related Topics) 15 / 19 The proof is complete after showing that

∗ D E 2 T (eλ) = eλ, TkαM`βψ ∈ ` (AP(Z)) (1) k ` for every λ ∈ R.

Continuation

This shows that (Ta, eλ) are the Fourier coefﬁcients of an element Ta ∈ AP2(R) and AP2(R) λ the map T is well-deﬁned and continuous. Consequently

∗ 2 T : AP2(R) → ` (AP2(Z)) is a continuous map.

(Time-Frequency Analysis and Related Topics) 15 / 19 Continuation

This shows that (Ta, eλ) are the Fourier coefﬁcients of an element Ta ∈ AP2(R) and AP2(R) λ the map T is well-deﬁned and continuous. Consequently

∗ 2 T : AP2(R) → ` (AP2(Z)) is a continuous map. The proof is complete after showing that

∗ D E 2 T (eλ) = eλ, TkαM`βψ ∈ ` (AP(Z)) (1) k ` for every λ ∈ R.

(Time-Frequency Analysis and Related Topics) 15 / 19 2π (a) There is a system hλ,` , λ ∈ [0, α ), ` ∈ Z of continuous almost periodic functions such that there exist A, B > 0 such that

X X 2 Akfk2 f, h Bkfk2 . (2) AP2(R) 6 λ,` AP2(R) 6 AP2(R) `∈Z 2π λ∈[0, α )

for every f ∈ AP2(R).

(b) In the case ψ ∈ W0 we have

2 X D E 2 2 Akfk 6 k f, Tαk Mβ`ψ k 6 Bkfk AP2(R) k AP(Z) AP2(R) `

for every f ∈ S2.

Almost periodic frame

Theorem

1 Let ψ ∈ W0 or ψ ∈ L ∩ Wc0 and assume that G(ψ, α, β) is a Gabor frame. Then

(Time-Frequency Analysis and Related Topics) 16 / 19 (b) In the case ψ ∈ W0 we have

2 X D E 2 2 Akfk 6 k f, Tαk Mβ`ψ k 6 Bkfk AP2(R) k AP(Z) AP2(R) `

for every f ∈ S2.

Almost periodic frame

Theorem

1 Let ψ ∈ W0 or ψ ∈ L ∩ Wc0 and assume that G(ψ, α, β) is a Gabor frame. Then

2π (a) There is a system hλ,` , λ ∈ [0, α ), ` ∈ Z of continuous almost periodic functions such that there exist A, B > 0 such that

X X 2 Akfk2 f, h Bkfk2 . (2) AP2(R) 6 λ,` AP2(R) 6 AP2(R) `∈Z 2π λ∈[0, α ) for every f ∈ AP2(R).

(Time-Frequency Analysis and Related Topics) 16 / 19 Almost periodic frame

Theorem

1 Let ψ ∈ W0 or ψ ∈ L ∩ Wc0 and assume that G(ψ, α, β) is a Gabor frame. Then

2π (a) There is a system hλ,` , λ ∈ [0, α ), ` ∈ Z of continuous almost periodic functions such that there exist A, B > 0 such that

X X 2 Akfk2 f, h Bkfk2 . (2) AP2(R) 6 λ,` AP2(R) 6 AP2(R) `∈Z 2π λ∈[0, α ) for every f ∈ AP2(R).

(b) In the case ψ ∈ W0 we have

2 X D E 2 2 Akfk 6 k f, Tαk Mβ`ψ k 6 Bkfk AP2(R) k AP(Z) AP2(R) ` for every f ∈ S2.

(Time-Frequency Analysis and Related Topics) 16 / 19 One of the keys in the proof of the previous theorem consists in obtaining the identity D E k f, T M ψ k = k f, h k (3) αk β` λ,` AP2(R) 2 2π k AP(Z) λ ` ([0, α )) for every f ∈ AP(R) and ` ∈ Z in the case ψ ∈ W. Although the family of Gabor frames is stable under Fourier transformation the proof of 1 the theorem for the case ψ ∈ L (R) and bψ ∈ W0 cannot be deduced from the case ψ ∈ W0 or vice versa.

The functions hλ,` form a generalized frame in the sense of Kaiser or Antoine, Gazeau, Han, Gabardo.

(Time-Frequency Analysis and Related Topics) 17 / 19 Although the family of Gabor frames is stable under Fourier transformation the proof of 1 the theorem for the case ψ ∈ L (R) and bψ ∈ W0 cannot be deduced from the case ψ ∈ W0 or vice versa.

The functions hλ,` form a generalized frame in the sense of Kaiser or Antoine, Gazeau, Han, Gabardo. One of the keys in the proof of the previous theorem consists in obtaining the identity D E k f, T M ψ k = k f, h k (3) αk β` λ,` AP2(R) 2 2π k AP(Z) λ ` ([0, α )) for every f ∈ AP(R) and ` ∈ Z in the case ψ ∈ W.

(Time-Frequency Analysis and Related Topics) 17 / 19 The functions hλ,` form a generalized frame in the sense of Kaiser or Antoine, Gazeau, Han, Gabardo. One of the keys in the proof of the previous theorem consists in obtaining the identity D E k f, T M ψ k = k f, h k (3) αk β` λ,` AP2(R) 2 2π k AP(Z) λ ` ([0, α )) for every f ∈ AP(R) and ` ∈ Z in the case ψ ∈ W. Although the family of Gabor frames is stable under Fourier transformation the proof of 1 the theorem for the case ψ ∈ L (R) and bψ ∈ W0 cannot be deduced from the case ψ ∈ W0 or vice versa.

(Time-Frequency Analysis and Related Topics) 17 / 19 1 The case ψ ∈ L (R) ∩ Wc0 The functions are constructed from their Fourier coefﬁcients.

2π For each λ ∈ [0, α ) consider  0 if α(µ − λ) < 2πZ  cλ,` =  µ   [ 2π M`βψ(λ + α p) if α(µ − λ) = 2πp

λ,` There exists hλ,` ∈ AP2(R) whose Fourier coefﬁcients are precisely cµ . Our hypothesis on ψ implies that the sequence of Fourier coefﬁcients is in `1 and consequently hλ,` is a continuous almost periodic function, that is, hλ,` ∈ AP(R).

Construction of the functions hλ,`

The case ψ ∈ W0

X iλkα hλ,` = e TkαM`βψ. k∈Z

(Time-Frequency Analysis and Related Topics) 18 / 19 2π For each λ ∈ [0, α ) consider  0 if α(µ − λ) < 2πZ  cλ,` =  µ   [ 2π M`βψ(λ + α p) if α(µ − λ) = 2πp

Construction of the functions hλ,`

The case ψ ∈ W0

X iλkα hλ,` = e TkαM`βψ. k∈Z

1 The case ψ ∈ L (R) ∩ Wc0 The functions are constructed from their Fourier coefﬁcients.

(Time-Frequency Analysis and Related Topics) 18 / 19 λ,` There exists hλ,` ∈ AP2(R) whose Fourier coefﬁcients are precisely cµ . Our hypothesis on ψ implies that the sequence of Fourier coefﬁcients is in `1 and consequently hλ,` is a continuous almost periodic function, that is, hλ,` ∈ AP(R).

Construction of the functions hλ,`

The case ψ ∈ W0

X iλkα hλ,` = e TkαM`βψ. k∈Z

1 The case ψ ∈ L (R) ∩ Wc0 The functions are constructed from their Fourier coefﬁcients.

2π For each λ ∈ [0, α ) consider  0 if α(µ − λ) < 2πZ  cλ,` =  µ   [ 2π M`βψ(λ + α p) if α(µ − λ) = 2πp

(Time-Frequency Analysis and Related Topics) 18 / 19 Our hypothesis on ψ implies that the sequence of Fourier coefﬁcients is in `1 and consequently hλ,` is a continuous almost periodic function, that is, hλ,` ∈ AP(R).

Construction of the functions hλ,`

The case ψ ∈ W0

X iλkα hλ,` = e TkαM`βψ. k∈Z

1 The case ψ ∈ L (R) ∩ Wc0 The functions are constructed from their Fourier coefﬁcients.

2π For each λ ∈ [0, α ) consider  0 if α(µ − λ) < 2πZ  cλ,` =  µ   [ 2π M`βψ(λ + α p) if α(µ − λ) = 2πp

λ,` There exists hλ,` ∈ AP2(R) whose Fourier coefﬁcients are precisely cµ .

(Time-Frequency Analysis and Related Topics) 18 / 19 Construction of the functions hλ,`

The case ψ ∈ W0

X iλkα hλ,` = e TkαM`βψ. k∈Z

1 The case ψ ∈ L (R) ∩ Wc0 The functions are constructed from their Fourier coefﬁcients.

2π For each λ ∈ [0, α ) consider  0 if α(µ − λ) < 2πZ  cλ,` =  µ   [ 2π M`βψ(λ + α p) if α(µ − λ) = 2πp

(Time-Frequency Analysis and Related Topics) 18 / 19 Example

2π 2π Assume Λ = {λr : r ∈ N} ⊂ [0, α ) is a countable set and M = Λ + α Z. Then hλr ,` : r ∈ N, ` ∈ Z

is a Hilbert frame in AP2(M).

Separable subspaces

Deﬁnition

For any countable subset M ⊂ R we denote by AP2(M) the closed subspace of AP2(R) consisting of those f ∈ AP2(R) whose spectrum is contained in M.

(Time-Frequency Analysis and Related Topics) 19 / 19 Separable subspaces

Deﬁnition

For any countable subset M ⊂ R we denote by AP2(M) the closed subspace of AP2(R) consisting of those f ∈ AP2(R) whose spectrum is contained in M.

Example

2π 2π Assume Λ = {λr : r ∈ N} ⊂ [0, α ) is a countable set and M = Λ + α Z. Then hλr ,` : r ∈ N, ` ∈ Z is a Hilbert frame in AP2(M).

(Time-Frequency Analysis and Related Topics) 19 / 19