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A planar large sieve and sparsity of time-frequency representations

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A planar large sieve and sparsity of time-frequency representations arxiv: v [math.fa] 27 Feb 207 Luís Daniel Abreu Acoustics Research Institute, Wohllebengasse 2-4, Vienna A-040, Austria. Abstract
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A planar large sieve and sparsity of time-frequency representations arxiv: v [math.fa] 27 Feb 207 Luís Daniel Abreu Acoustics Research Institute, Wohllebengasse 2-4, Vienna A-040, Austria. Abstract With the aim of measuring the sparsity of a real signal, Donoho and Logan introduced the concept of maximum Nyquist density, and used it to extend Bombieri s principle of the large sieve to bandlimited functions. This led to several recovery algorithms based on the minimization of the L -norm. In this paper we introduce the concept of planar maximum Nyquist density, which measures the sparsity of the time-frequency distribution of a function. We obtain a planar large sieve principle which applies to time-frequency representations with a gaussian window, or equivalently, to Fock spaces, allowing for perfect recovery of the short-fourier transform STFT) of functions in the modulation space M also known as Feichtinger s algebra S 0 ) corrupted by sparse noise and for approximation of missing STFT data in M, by L -minimization. I. INTRODUTION With the aim of measuring the sparsity of a real signal, Donoho and Logan introduced the concept of maximum Nyquist density, defined in [2] as ρt,w) := W sup T [t,t+/w] t R W T, where T R and W is the band-size in the space of band-limited functions } B W := {f L : ˆfω) = 0, ω πw. If the set T is sparse in terms of low Lebesgue measure small concentration in any interval of length /W ), then Michael Speckbacher Acoustics Research Institute, Wohllebengasse 2-4, Vienna A-040, Austria. ρt,w) can be considerably small compared to the natural Nyquist density W T. We will write P A f = χ A f for the multiplication by the indicator function of A. In [2, Theorem 7], Donoho and Logan proved that, if f B W and δ 2 W, the inequality πw/2 P T f sup T [t,t+θ] t R sinπwθ/2) f ) holds. In particular, if δt) denotes the norm of the projection operator P T and θ = /W, then δt) π 2ρT,W). Note that the inequality ) falls within the realm of quantitative uncertainty principles [], [20], [2], [23], [36], which paved the way to the modern theory of compressed sensing see [9, Section.6] or [0], [25]). Donoho and Logan s interest in such inequalities, in particular in obtaining good constants depending on the sparsity of the set T, was motivated by signal recovery problems []. As an application, they derived the following results, which allows to perfectly reconstruct a bandlimited signal corrupted by sparse noise using L - norm minimization. orollary: [2, orolllary ]) Suppose that g = b+n is observed, b B W, n L, and that the unknown support T of the noise n satisfies ρt,w) /π. 2) Then the solution of the minimization problem βg) = arg min b BW g b is unique and recovers the signal b perfectly βg) = b). This extends the so-called Logan s phenomenon [28], see also the discussion in [, Section 6.2]). The following recovery result for missing data extends the results from [] from the L 2 to the L setting. We will give a proof in the STFT context in orollary 2. orollary: Suppose that we observe h = P T cb + n), where b B W, n ε and ρt,w) 2/π. Then, any solution of the minimization problem satisfies σh) := arg min b BW P T c b h) b σh) 4ε 2 π ρt,w). Let ϕt) = 2 /4 e πt2 be the normalized gaussian. The short-time Fourier transform STFT) is defined as follows V ϕ fx,ω) = ft)ϕt x)e 2πiωt dt. 3) Moreover, define the modulation spaces R M p := { f S R) : V ϕ f p }, p. Modulation spaces are ubiquitous in time-frequency analysis [6], [26]. They were introduced in [4]. It is the purpose of this paper to obtain a planar version of ) and apply it to recovery problems for the short-time Fourier transform of functions in M using L -minimization. The space M is also known as Feichtinger s algebra S 0 and it can be identified with the Bargmann-Fock space F ) of entire functions. As a planar analogue of ρt, W), we introduce the following concept. Let R 2. The planar maximum Nyquist density ρ,r) is defined as ρ,r) := sup z R 2 z +D /R ), 4) where D /R R 2 is the disc of radius /R centered in the origin. If the set is sparse in the sense of Lesbegue measure small concentration in any disc of radius /R) then ρ, R) can be considerably smaller than the natural Nyquist density see [3], [3], [7], [22], [3], [33] for natural Nyquist densities in the context of Fock and modulation spaces and [8] for a survey on the current state of the art of the topic). Our main result is the following. Theorem : onsider R 2 and let f M, then, for every 0 R , it holds Set By Theorem, P V ϕ f) ρ,r) e π/r2 V ϕ f. 5) δ ) := sup f M P V ϕ f) V ϕ f. δ ) e π/r2 ) ρ,r). 6) Moreover, if δ ) 2 then every f M satisfies P V ϕ f) P cv ϕ f). ombined with the argument in [, Section 6.2], this implies the following result, which allows to perfectly reconstruct the STFT of a signal in M corrupted by sparse noise, using L -norm minimization. orollary : Suppose that G = V ϕ f +N is observed, where f M, N L R 2 ) and that the unknown support of N satisfies ρ,r) 2 e π/r2 ), 7) for some R 0. Then δ ) 2 and the solution of the minimization problem βg) = arg min g M G V ϕ g is unique and recovers the signal f perfectly βg) = f). One can also derive an analogue for the recovery of missing data. orollary 2: Let f M and suppose that one observes H = P cv ϕ f+n), where N ε and that the domain of missing data satisfies ρ,r) e π/r2 ), 8) for some R 0. Then any solution of satisfies σh) = arg min h M P ch V ϕ h) V ϕ f σh) Proof: First, observe that Hence, 2ε e π/r2 ) e π/r2 ρ,r). P ch V ϕ σh)) n ε. Vϕ f σh) = P cv ϕ f σh) + P V ϕ b σg) P c Vϕ f H + P c H Vϕ σh) +δ ) Vϕ f σh) 2ε+δ ) Vϕ f σh), which concludes the proof using 6) and 8). There are other approaches to the recovery of sparse time-frequency representations which concentrate on the set-up of finite sparse time-frequency representations [29], [30]. Another consequence of Theorem is the following refined L uncertainty principle for the STFT see [26, Proposition 3.3.] and [7], [9], [32] for other uncertainty principles for the STFT). orollary 3: Suppose thatf M satisfies V ϕ f = and that R 2 and ε 0 are such that ε V ϕ fx,ω) dxdω, then ) ρ,r) ε inf. R 0 e π/r2 In particular, orollary 3 shows that the mass of the STFT of a function cannot be concentrated on sets that are locally small over the whole time-frequency plane. Our arguments to prove Theorem are an adaptation of Selberg s argument for the large sieve see [6], [8], [24]), along the lines of [2]. The analysis reveals that, at least in the continuous case, dealing with joint time-frequency representations leads to considerable simplifications, due to the existence of local reproducing formulas [33]. This is not surprising since, as observed earlier by Daubechies [3] and Seip [33], the study of joint time-frequency restriction operators with a gaussian window tends to be simplified. In particular, the functions best concentrated in a disc have a simple explicit formula when written in the phase space. This is in contrast with the classical time and band-limiting problem which has been studied in detail by Landau [27] see also [4] for an alternative approach). II. MODULATION AND FOK SPAES We will follow notations and definitions from [26]. The Bargmann transform on is defined by Bfz) = 2 /4 ft)e 2πt z πt2 πz 2 /2 dt. 9) R Writing z = x+iω, a simple calculation shows that e iπxω V ϕ fx, ω) = Bfz)e π z 2 /2. 0) Let F p ) be the space of entire functions equipped with the norm F p L p := if p and Fz) p e πp z 2 /2 dz, F L := sup Fz) e π z 2 /2, z if p =. The Bargmann transform is a unitary operator from L 2 R) to F 2 ) and extends to a bijective operator from M p to F p ), for p see [35], or [] for a proof that extends to polyanalytic Fock spaces). As in [34], we define the translation operator T w on F p ) as follows: T w Fz) := e πw z π w 2 /2 Fz w). It acts isometrically on every F p ), p. The corresponding convolution is F Gz) := Fw)Gz w)e πz w w 2) dw = Fw)T w Gz)e π w 2 /2 dw. ) III. PROOF SKETH OF MAIN RESULTS A. oncentration estimates We say that a function G L ) is concentrated on Ω if I P Ω )G L = 0. Our main results will follow from the following statement which corresponds to ). We will give a full proof and more general results in [5]. Proposition : Suppose that there exists G L ) which is concentrated on Ω, such that F F G, F ) F ) is bounded and boundedly invertible. Then F dµ G L Λµ,Ω) νg) F L, ) Φ L where νg) := sup Φ F) and Φ G L ) Λµ,Ω) := sup e π z 2 /2 dµz). w w+ω Proof sketch: For F F ), there exists F F ) unique such that F = F G. Hence, replacing F by F G and using, one after another, I P Ω )G L = 0, Fubini s theorem, and Hölder s inequality p = ) yields Fz) dµz) G L Λµ,Ω) F L. The observation that statement. B. Proof of Theorem Define F L F G L = thus implies our dµz) := χ z)e π z 2 /2 dz, with some subset of nonzero measure. onsequently, F L,µ = P F L and setting Ω = D /R yields Λµ,D /R ) = sup z +D /R ) = ρ,r). z Let F be entire and R 0, then for any z, the following local reproducing formula holds [34]: Fz) = e π/r2 ) F χ D/R )z). 2) Now, let R 0, choosing G = G R := χ D/R yields that convolution with G R gives a bounded and invertible operator on F ). Then G R L =, ) Φ L νg R ) = sup =, Φ F Φ G R L e π/r2 and Proposition yields P F L F L ρ,r) e π/r2. This proves the result for F F ). Since the Bargmann transform extends to a bijective operator from M to F ), there exists f M such that Fz) = Bfz) = e iπxω+π z 2 /2 V ϕ fx, ω). This completes the proof. AKNOWLEDGEMENT L.D. Abreu and M. Speckbacker were supported by the Austrian Science Foundation FWF) START-project FLAME Frames and Linear Operators for Acoustical Modeling and Parameter Estimation, Y 55-N3) REFERENES [] L. D. Abreu, K. Gröchenig. Banach Gabor frames with Hermite functions: polyanalytic spaces from the Heisenberg group. Appl. Anal., 9: , 202. [2] L. D. Abreu, M. Dörfler. An inverse problem for localization operators. Inverse Problems, 28: 500, 202. [3] L. D. Abreu, K. Gröchenig, J. L. Romero. On accumulated spectrograms. Trans. Amer. Math. Soc., 368: , 206. [4] L. D. Abreu, J. M. Pereira. Measures of localization and quantitative Nyquist densities. Appl. omput. Harmon. Anal., 38 3): , 205 [5] L. D. Abreu, M. Speckbacher. In preparation. [6]. Aubel, H. Bölcskei. 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