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Multi-dimensional sampling approaches are often faced with certain intricacies as we need to deal with matrices and the non-commutativity of matrices precludes straightforward extensions of the usual one-dimensional results. In this article, we reformulate the Papoulis' sampling theorem for the reconstruction of higher-dimensional signals that are band-limited in the sense of free metaplectic transformation, which encapsulates the classical Fourier transform as well as the ensuing generalized phase-space transforms. The key idea is to invoke the arbitrary lattice sampling of the higher-dimensional Euclidean space based on general separable or non-separable matrices. Besides, a newly formulated convolution operation satisfying the fundamental properties of commutativity and associativity is employed for the construction of a set of filters allowing data acquisition on a per-channel basis, which is an intrinsic feature of the Papoulis' sampling theorem. Towards the culmination, it is demonstrated that the proposed sampling procedure is also of substantial importance in the context of fan beam tomography. [ABSTRACT FROM AUTHOR]

We study the recovery of square-integrable signals from the absolute values of their wavelet transforms, also called wavelet phase retrieval. We present a new uniqueness result for wavelet phase retrieval. To be precise, we show that any wavelet with finitely many vanishing moments allows for the unique recovery of real-valued bandlimited signals up to global sign. Additionally, we present the first uniqueness result for sampled wavelet phase retrieval in which the underlying wavelets are allowed to be complex-valued and we present a uniqueness result for phase retrieval from sampled Cauchy wavelet transform measurements. [ABSTRACT FROM AUTHOR]

Stochastic subspace identification (SSI) and its variants are widely used in conventional structural health monitoring (SHM) owing to their high computation robustness and estimation accuracy. Conventional SHM is based on a Nyquist (high-rate and uniform) sampling dictated by Shannon–Nyquist sampling theorem. As data quantity of SHM is increasing with duration and equipment at an ever-growing rate and new applications of SHM with strict constricts on sampling are emerging one by one, sub-Nyquist sampling, also known as compressive sampling has increasingly been used in SHM in recent years. Therefore, despite the success of SSI method and its algorithmic variants, a significant drawback lies in their handling of sub-Nyquist data. To overcome this drawback, we set out to extend the applicable scenarios of SSI from Nyquist (high-rate/uniform) sampling to sub-Nyquist (low-rate/nonuniform) sampling. In this paper, we propose a compressive stochastic subspace identification (CSSI) which can identify modal parameters from sub-Nyquist/compressed data. Specifically, CSSI involves two aspects: covariance matrix reconstruction and system parameter identification. We find the complete covariance matrix in classic SSI can also be estimated/reconstructed by sub-Nyquist samples. More importantly, this reconstruction can be physically guaranteed through special sampling patterns without sparsity prior. Based on the above findings, we develop a compressed covariance sensing-based covariance matrix reconstruction technique and derive the optimal sub-Nyquist sampling pattern such that the reconstruction is hassle-free and efficient. Then, we perform modal parameter identification on the basis of the reconstructed covariance matrix. Finally, the effectiveness of CSSI is validated by simulations and experiments. In short, CSSI, as a new extension of SSI, can extract modal parameters at lower sampling rates than what is prescribed by the classic SSI. The implication of this is the possibility that CSSI for operational modal analysis will require fewer measurements thus it would reduce the storage requirement and the bandwidth for transmitting data. • We propose a compressive stochastic subspace identification (CSSI) algorithm. • We develop a universal and efficient covariance reconstruction technique for CSSI. • We derive the optimal sampling pattern such that CSSI works in practice. • We validate the effectiveness of CSSI with both simulations and experiments. • CSSI is expected to reduce the cost of operational modal analysis. [ABSTRACT FROM AUTHOR]

We derive sufficient conditions for sampling with derivatives in shift-invariant spaces generated by a periodic exponential B-spline. The sufficient conditions are expressed with a new notion of measuring the gap between consecutive samples. These conditions are near optimal, and, in particular, they imply the existence of sampling sets with lower Beurling density arbitrarily close to the necessary density. [ABSTRACT FROM AUTHOR]

Many areas of science and engineering encounter data defined on spherical manifolds. Modelling and analysis of spherical data often necessitates spherical harmonic transforms, at high degrees, and increasingly requires efficient computation of gradients for machine learning or other differentiable programming tasks. We develop novel algorithmic structures for accelerated and differentiable computation of generalised Fourier transforms on the sphere S 2 and rotation group SO (3) , i.e. spherical harmonic and Wigner transforms, respectively. We present a recursive algorithm for the calculation of Wigner d -functions that is both stable to high harmonic degrees and extremely parallelisable. By tightly coupling this with separable spherical transforms, we obtain algorithms that exhibit an extremely parallelisable structure that is well-suited for the high throughput computing of modern hardware accelerators (e.g. GPUs). We also develop a hybrid automatic and manual differentiation approach so that gradients can be computed efficiently. Our algorithms are implemented within the JAX differentiable programming framework in the S2FFT ▪ software code. Numerous samplings of the sphere are supported, including equiangular and HEALPix sampling. Computational errors are at the order of machine precision for spherical samplings that admit a sampling theorem. When benchmarked against alternative C codes we observe up to a 400-fold acceleration. Furthermore, when distributing over multiple GPUs we achieve very close to optimal linear scaling with increasing number of GPUs due to the highly parallelised and balanced nature of our algorithms. Provided access to sufficiently many GPUs our transforms thus exhibit an unprecedented effective linear time complexity. • Differentiable and accelerated spherical harmonic and Wigner transforms. • Novel Wigner d-function recursion which is stable and scalable to high resolution. • Highly efficient gradient propagation scheme for harmonic analysis. • Open sourced and professionally developed JAX software package. [ABSTRACT FROM AUTHOR]

The Existential Theory of the Reals (ETR) consists of existentially quantified Boolean formulas over equalities and inequalities of polynomial functions of real variables. In this paper we propose and study the approximate existential theory of the reals (ϵ -ETR) in which the constraints are only satisfied approximately. We first show that when the domain of the variables is the reals then ϵ -ETR = ETR under polynomial time reductions, and then study the constrained ϵ -ETR problem where groups of variables are constrained to lie in bounded convex sets. Our main result is a sampling theorem that discretizes the domain in a grid-like manner whose density depends on various properties of the ETR formula. A consequence of our theorem is that we obtain a (quasi-)polynomial time approximation scheme ((Q)PTAS) for a fragment of constrained ϵ -ETR. We use this theorem to create several new PTAS and QPTAS for problems from a variety of fields. [ABSTRACT FROM AUTHOR]

We introduce a new concept of variable bandwidth that is based on the frequency truncation of Wilson expansions. For this model we derive sampling theorems, a complete reconstruction of f from its samples, and necessary density conditions for sampling. Numerical simulations support the interpretation of this model of variable bandwidth. In particular, chirps, as they arise in the description of gravitational waves, can be modeled in a space of variable bandwidth. [ABSTRACT FROM AUTHOR]

We consider the problem of phase retrieval from magnitudes of short-time Fourier transform (STFT) measurements. It is well-known that signals are uniquely determined (up to global phase) by their STFT magnitude when the underlying window has an ambiguity function that is nowhere vanishing. It is less clear, however, what can be said in terms of unique phase-retrievability when the ambiguity function of the underlying window vanishes on some of the time-frequency plane. In this note, we demonstrate that by considering signals in Paley–Wiener spaces, it is possible to prove new uniqueness results for STFT phase retrieval. Among those, we establish a first uniqueness theorem for STFT phase retrieval from magnitude-only samples for real-valued signals. [ABSTRACT FROM AUTHOR]

We consider elliptic second order partial differential operators with Lipschitz continuous leading order coefficients on finite cubes and the whole Euclidean space. We prove quantitative sampling and equidistribution theorems for eigenfunctions. The estimates are scale-free, in the sense that for a sequence of growing cubes we obtain uniform estimates. These results are applied to prove lifting of eigenvalues as well as the infimum of the essential spectrum, and an uncertainty relation (aka spectral inequality) for short energy interval spectral projectors. Several applications including random operators are discussed. In the proof we have to overcome several challenges posed by the variable coefficients of the leading term. [ABSTRACT FROM AUTHOR]

We provide the foundations of a Hilbert space theory for the short-time Fourier transform (STFT) where the flat tori T N 2 = R 2 / (Z × N Z) = [ 0 , 1 ] × [ 0 , N ] act as phase spaces. We work on an N -dimensional subspace S N of distributions periodic in time and frequency in the dual S 0 ′ (R) of the Feichtinger algebra S 0 (R) and equip it with an inner product. To construct the Hilbert space S N we apply a suitable double periodization operator to S 0 (R). On S N , the STFT is applied as the usual STFT defined on S 0 ′ (R). This STFT is a continuous extension of the finite discrete Gabor transform from the lattice onto the entire flat torus. As such, sampling theorems on flat tori lead to Gabor frames in finite dimensions. For Gaussian windows, one is lead to spaces of analytic functions and the construction allows to prove a necessary and sufficient Nyquist rate type result, which is the analogue, for Gabor frames in finite dimensions, of a well known result of Lyubarskii and Seip-Wallstén for Gabor frames with Gaussian windows and which, for N odd, produces an explicit full spark Gabor frame. The compactness of the phase space, the finite dimension of the signal spaces and our sampling theorem offer practical advantages in some applications. We illustrate this by discussing a problem of current research interest: recovering signals from the zeros of their noisy spectrograms. [ABSTRACT FROM AUTHOR]

We prove a partial fraction decomposition of a quotient of two functions E α (i t x) and I α (i t) which are defined in terms of the Bessel functions J α and J α + 1 of the first kind. This expansion leads naturally to the introduction of an orthonormal system with respect to the measure | x | 2 α + 1 d x 2 α + 1 Γ (α + 1) in [ − 1 , 1 ] , which we call the Fourier–Dunkl system of the second kind. Euler–Dunkl polynomials E n , α (x) of degree n are defined by considering E α (t x) ∕ I α (t) as a generating function. It is shown that the sum ∑ m = 1 ∞ 1 ∕ j m , α 2 k , where j m , α are the positive zeros of J α , is equal (up to an explicit factor) to E 2 k − 1 , α (1). For α = 1 ∕ 2 this leads to classical results of Euler since the function E 1 ∕ 2 (x) is the exponential function and E n , 1 ∕ 2 (x) are (essentially) the Euler polynomials. In the second part of the paper a sampling theorem of Whittaker–Shannon–Kotel'nikov type is established which is strongly related to the above-mentioned partial decomposition and which holds for all functions in the Payley–Wiener space defined by the Dunkl transform in [ − 1 , 1 ]. [ABSTRACT FROM AUTHOR]

Fast reconstruction of a bandlimited function from its finite oversampling data has been a fundamental problem in sampling theory. As the number of sample data increases to infinity, exponentially-decaying reconstruction errors can be achieved by many methods in the literature. In fact, it is generally conjectured that when the optimal method is used, the dominant term in the error of reconstructing a function bandlimited to [ − δ , δ ] (δ < π) from its data sampled at the integer points on [ − n , n ] is exp (− λ (π − δ) n). By far, the best estimate for the constant λ among regularization methods is 1 ∕ 2 and is achieved by the highly efficient Gaussian regularized Whittaker–Kotelnikov–Shannon sampling series. We prove in this paper that the exponential constant 1 ∕ 2 is optimal for this method. Moreover, the optimal variance of the Gaussian regularizer is provided. [ABSTRACT FROM AUTHOR]

The Special Affine Fourier Transformation or the SAFT generalizes a number of well known unitary transformations as well as signal processing and optics related mathematical operations. Shift-invariant spaces also play an important role in sampling theory, multiresolution analysis, and many other areas of signal and image processing. Shannon's sampling theorem, which is at the heart of modern digital communications, is a special case of sampling in shift-invariant spaces. Furthermore, it is well known that the Poisson summation formula is equivalent to the sampling theorem and that the Zak transform is closely connected to the sampling theorem and the Poisson summation formula. These results have been known to hold in the Fourier transform domain for decades and were recently shown to hold in the Fractional Fourier transform domain by A. Bhandari and A. Zayed. The main goal of this article is to show that these results also hold true in the SAFT domain. We provide a short, self-contained proof of Shannon's theorem for functions bandlimited in the SAFT domain and then show that sampling in the SAFT domain is equivalent to orthogonal projection of functions onto a subspace of bandlimited basis associated with the SAFT domain. This interpretation of sampling leads to least-squares optimal sampling theorem. Furthermore, we show that this approximation procedure is linked with convolution and semi-discrete convolution operators that are associated with the SAFT domain. We conclude the article with an application of fractional delay filtering of SAFT bandlimited functions. [ABSTRACT FROM AUTHOR]

A novel power spectrum blind sampling (PSBS) approach is proposed supporting low-power wireless sensor networks for Operational Modal Analysis (OMA) applications. The developed approach relies on sensors, employing deterministic non-uniform in time multi-coset sampling to acquire structural response acceleration signals at sub-Nyquist sampling rates. These signals are treated as realizations of stationary random processes without making any assumption about the average signal frequency content and spectral support. The acquired compressed measurements are transmitted to a central server and collectively processed via a PSBS technique, herein extended to the multi-sensor case, to estimate the power spectral density matrix of an underlying spatially correlated stationary response acceleration random process directly from the compressed measurements. Structural modal properties are then extracted through standard frequency domain decomposition (FDD). The efficacy of the proposed approach to resolve closely-spaced modes is numerically tested for various data compression levels using noisy response acceleration signals of a white-noise excited finite element model of a space truss as well as field-recorded acceleration time-histories of an instrumented bridge under operational loading. It is shown that accurate mode shapes based on the modal assurance criterion can be obtained from as low as 89% less measurements compared to conventional non-compressive FDD at Nyquist sampling rate. Further, significant gains in energy consumption and 3 to 5 times battery life extension are estimated for wireless sensors operating on multi-coset sampling at different data compression levels. It is, therefore, concluded that the proposed PSBS approach could provide long-term structural health monitoring systems with low-maintenance cost once wireless sensors with multi-coset sampling capabilities become commercially available. [ABSTRACT FROM AUTHOR]

• The enriched finite elements of local harmonic functions are used for modeling discontinuous or high-frequency wave propagation. • The elastic wave propagation in an impacting rod is studied with both dynamic modeling and experimental test. • The sampling theorem and quantitative criterion for spatial and temporal discretization with the enriched elements are proposed through dispersion analysis. • The numerical results based on the Hertzian contact model are in good agreement with the experimental observations, especially at wave interfaces. The wave propagation in an impacting rod is instant and nearly discontinuous, which remains a challenging problem for both dynamic modeling and experimental test. This paper presents an Enriched Finite Element Method (Enriched FEM) with controllable dispersion error to accurately and efficiently model the discontinuous wave propagation. The Enriched FEM approximates the displacement solutions based on a linear interpolation enriched with a series of harmonic functions. The paper focuses on the dispersion error of the enriched finite element from spatial-temporal discretization and the sampling theorem of the enriched finite element with different cutoff numbers, and then gives a simple quantitative criterion on the interdependent relationship between the size of mesh and time step so as to reduce the numerical dispersion error. In addition, the paper presents the experimental study on the wave propagation of an elastic rod colliding with a rigid wall. The experimental results explicitly show the incidence and reflection of the stress wave in the rod in the contact duration. They also indicate that the amplitude of the strain almost does not decay with the wave propagation along the rod. The numerical results based on the Hertzian contact model are in good agreement with the experimental observations, especially at wave interfaces. [ABSTRACT FROM AUTHOR]

Periodic nonuniform sampling (PNS) is widely used in sub-Nyquist sampling schemes of multitone signals. However, what sampling pattern of PNS enables the signal to recover from the sub-Nyquist samples? Although this problem has been discussed in different schemes, the corresponding answers vary greatly. In this paper, we introduce the recurrent delay times to describe a PNS and provide a mathematical definition of signal recoverability from the perspective of delay. On this basis, we review the well-known Shannon–Nyquist sampling theorem and suggest a delay sampling theorem (DST) that a multitone signal bandlimited to [ 0 , f max ] can be perfectly reconstructed from its periodic nonuniform samples when its minimum combined delay time τ min is not more than the Nyquist interval 1 / 2 f max . To explain and demonstrate the DST, subspace theory-based algorithms and active aliasing and de-aliasing algorithm (AADA) are proposed and used to recover the spectrum or waveform of multitone signals. The high consistency between practical and theoretical results evidently verifies the correctness of DST. In terms of sampling and recovery of multitone signal, DST extends the Shannon–Nyquist sampling theorem and naturally unifies several existing sampling strategies from the perspective of delay. DST provides a new guideline for the sub-Nyquist sampling of multitone signals. According to the DST criterion, we can optimize the sampling patterns to reduce cosets and improve the supremum frequency such that the reconstructed spectrum is alias-free. By generalizing the sampling theorem from the view of delay, DST opens many avenues for the development of multitone signal processing. [ABSTRACT FROM AUTHOR]

The classical assumption in sampling and spline theories is that the input signal is square-integrable, which prevents us from applying such techniques to signals that do not decay or even grow at infinity. In this paper, we develop a sampling theory for multidimensional non-decaying signals living in weighted L p spaces. The sampling and reconstruction of an analog signal can be done by a projection onto a shift-invariant subspace generated by an interpolating kernel. We show that, if this kernel and its biorthogonal counterpart are elements of appropriate hybrid-norm spaces, then both the sampling and the reconstruction are stable. This is an extension of earlier results by Aldroubi and Gröchenig. The extension is required because it allows us to develop the theory for the ideal sampling of non-decaying signals in weighted Sobolev spaces. When the d -dimensional signal and its d / p + ε derivatives, for arbitrarily small ε > 0 , grow no faster than a polynomial in the L p sense, the sampling operator is shown to be bounded even without a sampling kernel. As a consequence, the signal can also be interpolated from its samples with a nicely behaved interpolating kernel. [ABSTRACT FROM AUTHOR]

Real world Structural Health Monitoring (SHM) systems consist of sensors in the scale of hundreds, each sensor generating extremely large amounts of data, often arousing the issue of the cost associated with data transfer and storage. Sensor energy is a major component included in this cost factor, especially in Wireless Sensor Networks (WSN). Data compression is one of the techniques that is being explored to mitigate the effects of these issues. In contrast to traditional data compression techniques, Compressive Sensing (CS) – a very recent development – introduces the means of accurately reproducing a signal by acquiring much less number of samples than that defined by Nyquist's theorem. CS achieves this task by exploiting the sparsity of the signal. By the reduced amount of data samples, CS may help reduce the energy consumption and storage costs associated with SHM systems. This paper investigates CS based data acquisition in SHM, in particular, the implications of CS on damage detection and localization. CS is implemented in a simulation environment to compress structural response data from a Reinforced Concrete (RC) structure. Promising results were obtained from the compressed data reconstruction process as well as the subsequent damage identification process using the reconstructed data. A reconstruction accuracy of 99% could be achieved at a Compression Ratio (CR) of 2.48 using the experimental data. Further analysis using the reconstructed signals provided accurate damage detection and localization results using two damage detection algorithms, showing that CS has not compromised the crucial information on structural damages during the compression process. [ABSTRACT FROM AUTHOR]

The article starts with generalizations of some classical results and new truncation error upper bounds in the sampling theorem for bandlimited stochastic processes. Then, it investigates Lp ( [0, T]) and uniform approximations of φ -sub-Gaussian random processes by finite time sampling sums. Explicit truncation error upper bounds are established. Some specifications of the general results for which the assumptions can be easily verified are given. Direct analytical methods are employed to obtain the results. [ABSTRACT FROM AUTHOR]

• Blade tip timing (BTT) is used to track the natural frequency shift of the rotor blades for crack propagation monitoring. • Block-accelerated orthogonal least-squares can reconstruct the undersampled BTT signal in the time-frequency domain. • A frequency prediction method is provided to avoid comparing among different blades for crack propagation monitoring. • BTT and strain gauges are used to monitor the crack propagation simultaneously to verify the reliability of BTT. Blade tip timing (BTT) as a non-contact measurement technique can identify the blade vibration parameters. However, the signal recorded by BTT usually does not satisfy the Nyquist sampling theorem. To monitor blade crack propagation, it is important to accurately reconstruct the BTT signal to track the natural frequency shift of the rotor blade. In this paper, a sparse reconstruction method named block-accelerated orthogonal least-squares (Block-AOLS) is proposed to reconstruct the undersampled BTT signal in the time-frequency domain. Block-AOLS not only inherits the advantage of the high accuracy of AOLS but also makes use of the characteristic of block sparsity to reduce the interference components in the BTT signal. Furthermore, Block-AOLS is used to reconstruct the undersampled BTT signal to track the natural frequencies of cracked blades. The advantage of Block-AOLS is illustrated through simulation by comparing with the classical orthogonal matching pursuit. In the spin testing, the vibration of a blisk with eight blades is measured by both BTT and strain gauges, where two symmetrical blades without artificial cracks suddenly broke. The process of crack propagation can be inferred according to the shift of natural frequencies between the perfect and cracked blades. Moreover, a vibration frequency prediction method is provided to avoid comparing the natural frequencies among different blades for crack propagation monitoring, which is suitable to monitor the mistuning blades. To verify the reliability of the crack propagation monitoring using BTT, the strain signal with interval sample is analyzed by synchroextracting transform. Compared the natural frequency of the perfect blade with that of the cracked blade, the time when the natural frequency of the cracked blade shift is the same as that of the BTT result. [ABSTRACT FROM AUTHOR]

This paper deals with predicting the occurrence of self-excited vibrations during drilling. Previous work postulates that these are due to the coupling between the distributed drill string system and the regenerative effect in the bit-rock interaction law. We use a previously developed distributed model and the linearized bit-rock interaction law to derive a graphical condition for stability based on the Nyquist stability criterion. [ABSTRACT FROM AUTHOR]

We analyze two standard methods to compute the diffusion coefficient of a tracer particle in a medium from molecular dynamics (MD) simulation, the velocity autocorrelation function (VACF) method, and the mean-squared displacement (MSD) method. We show that they are equivalent in the sense that they provide the same mean values with the same level of statistical errors. We obtain analytic expressions for the level of the statistical errors present in the time-dependent diffusion coefficient as well as the VACF and the MSD. Under the assumption that the velocity of the tracer particle is a Gaussian process, all results are expressed in terms of the VACF. Hence, the standard errors of all relevant quantities are computable once the VACF is obtained from MD simulation. By using analytic models described by the Langevin equations driven by Gaussian white noise and Poissonian white shot noise, we verify our theoretical error estimates and discuss the non-Gaussianity effect in the error estimates when the Gaussian process approximation does not hold exactly. For validation, we perform MD simulations for the self-diffusion of a Lennard-Jones fluid and the diffusion of a large and massive colloid particle suspended in the fluid. Our theoretical framework is also applicable to mesoscopic simulations, e.g., Langevin dynamics and dissipative particle dynamics. [ABSTRACT FROM AUTHOR]

We prove the existence of sampling sets and interpolation sets near the critical density, in Paley Wiener spaces of a locally compact abelian (LCA) group G . This solves a problem left by Gröchenig, Kutyniok, and Seip (2008) [7] . To achieve this result, we prove the existence of universal Riesz bases of characters for L 2 ( Ω ) , provided that the relatively compact subset Ω of the dual group G ˆ satisfies a multi-tiling condition. This last result generalizes Fuglede's theorem, and extends to LCA groups setting recent constructions of Riesz bases of exponentials in bounded sets of R d . [ABSTRACT FROM AUTHOR]

Abstract: It is shown that three versions of the sampling theorem of signal analysis are equivalent in the sense that each can be proved as a corollary of one of the others. The theorems in question are the sampling theorem for functions belonging to the Bernstein space , the sampling theorem for functions in , , and the approximate sampling theorem for non-bandlimited function. One essential difference to an earlier paper of two of the authors is the avoidance of the deep Paley–Wiener theorem of Fourier analysis. [Copyright &y& Elsevier]

• A stereo-camera 3D-DIC system based on two low frame rate cameras and a multimedia projector is presented. • A down-sampling approach is used to overcome the Nyquist-Shannon theorem limitation. • A post-processing algorithm is proposed to acquire bandpass high-frequency vibration signal with low-speed equipment. • The constraint of pure sinusoidal excitation is removed. • A specific function to control the shaker is proposed, having a constant amplitude in the frequency band. A low-speed stereo-camera DIC setup was used in this paper to measure the down-sampled bandpass vibration signal of a plate in a given frequency range, which is much higher than the available frame rate. Down-sampled vibration measurements are well known in the literature. However, they are always subject to the strong hypothesis of having a single frequency component in the excitation source (e.g., pure sinusoidal excitation). In this scenario, several approaches can be found in the literature to reconstruct the actual response from the down-sampled data. In this paper, a data postprocessing algorithm is newly introduced to properly reconstruct the target response in the case of a frequency band excitation, thus relaxing the single frequency excitation constraint and allowing to explore a given frequency range with a single measurement. Additionally, a custom excitation signal is presented in this paper to achieve a constant intensity in the studied frequency range. The proposed approach is presented and experimentally validated by measuring a cantilever plate vibration in the kHz range using a 100 fps acquisition. A single acquisition of 98 frames allowed to describe the deformed shapes at 49 different frequency values in the range 1110–1160 Hz, highlighting two resonance peaks. The comparison with the results of multiple conventional single-frequency tests and LDV measurements confirmed the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

Abstract: Operators are studied for a class of band-limited functions φ and a wide class of tempered distributions . Convergence of to f as in the -norm is proved under a very mild assumption on φ, , and the rate of convergence is equal to the order of Strang–Fix condition for φ. To study convergence in , , we assume that there exists such that a.e. on , a.e. on for all . For appropriate band-limited or compactly supported functions , the estimate , where denotes the r-th modulus of continuity, is obtained for arbitrary . For tempered distributions , we proved that tends to f in the -norm, , with an arbitrary large approximation order. In particular, for some class of differential operators L, we consider such that . The corresponding wavelet frame-type expansions are found. [Copyright &y& Elsevier]

Abstract: We consider the problem of reconstructing a wide sense stationary band-limited process from its local averages taken either at the Nyquist rate or above. As a result we obtain a sufficient condition under which average sampling expansions hold in mean square and for almost all sample functions. Truncation and aliasing errors of the expansion are also discussed. [Copyright &y& Elsevier]

Abstract: Motivated by [B. Han, Z. Shen, Dual wavelet frames and Riesz bases in Sobolev spaces, Constr. Approx. 29 (2009) 369–406], we establish a sampling theorem in , , using a special pair of dual frames. The converging rate of the corresponding sampling series is investigated, and then sampling approximation to a signal in Sobolev space is established. If a signal to be reconstructed satisfies a mild condition in Fourier transform, then its sampling series converges exponentially fast. As an application, the sampling approximation is used to modify interpolating error, which arises when using an interpolating refinable function to reconstruct , such that can be arbitrarily approximated by its samples. [Copyright &y& Elsevier]

Abstract: In this paper we analyze the structure of bandlimited BMO-functions. Using a recently found equation for the calculation of the Hilbert transform of bounded bandlimited functions, we derive a decomposition result for bandlimited BMO-functions, which is similar to the well-known Fefferman–Stein decomposition. Based on this decomposition we characterize the range of the Hilbert transform. Moreover, we present interesting applications of this result. We characterize the peak value behavior of bandlimited BMO-functions, show that the derivative of bandlimited BMO-functions is bounded, and prove a sampling theorem for bandlimited BMO-functions. [Copyright &y& Elsevier]

Abstract: We obtain a q-linear analogue of Gegenbauerʼs expansion of the plane wave. It is expanded in terms of the little q-Gegenbauer polynomials and the third Jackson q-Bessel function. The result is obtained by using a method based on bilinear biorthogonal expansions. [Copyright &y& Elsevier]

Abstract: Sampling theory is concerned with the problem of reconstructing a signal in a Hilbert space from a given a collection of sampled values of . If a certain decomposition of the Hilbert space is possible (in terms of the sampling and reconstruction subspaces) then a consistent reconstruction can be obtained. In this paper we treat the case in which such a decomposition cannot be found. For this situation, we study the quasi-consistent reconstructions which are an extension of the consistent reconstructions. We relate the previous concepts to generalized inverses. We also present some new results and problems regarding consistent sampling. [Copyright &y& Elsevier]

Abstract: We introduce a sampling theory for cyclic lattice invariant spaces in the context of locally compact Abelian groups. The key element of the theory is a new periodization condition which, for a lattice in a group G, characterizes the cyclic invariant subspaces which allow sampling. When is contained in a larger lattice , we characterize the collection of -invariant subspaces of which are also -invariant. We use this to show that, if V is cyclic with respect to both and , then V admits sampling with respect to if and only if V admits sampling with respect to . [Copyright &y& Elsevier]

Abstract: We study some problems related to the effect of bounded, additive sample noise in the bandlimited interpolation given by the Whittaker–Shannon–Kotelnikov (WSK) sampling formula. We establish a generalized form of the WSK series that allows us to consider the bandlimited interpolation of any bounded sequence at the zeros of a sine-type function. The main result of the paper is that if the samples in this series consist of independent, uniformly distributed random variables, then the resulting bandlimited interpolation almost surely has a bounded global average. In this context, we also explore the related notion of a bandlimited function with bounded mean oscillation. We prove some properties of such functions, and in particular, we show that they are either bounded or have unbounded samples at any positive sampling rate. We also discuss a few concrete examples of functions that demonstrate these properties. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the error analysis of the derivative of the classical sampling theorem of bandlimited functions. We consider truncation, amplitude, and time-jitter errors. Both pointwise and uniform estimates are given. We derive analogues of the results of Piper (1975), Brown (1969), Jagerman (1966) and Li (1998) in a generalized manner. The amplitude and time-jitter errors are studied in the view of the works of Butzer (1983) and Butzer et al. (1988), provided that the bandlimited function satisfies some decay properties. [Copyright &y& Elsevier]

We present the sampling theorem with sampling functions of general form for entire functions satisfying one of the growth conditions 1+&z.sfnc;y&z.sfnc;f(z)&les;A1+&z.sfnc;z&z.sfnc;N1expτ&z.sfnc;x&z.sfnc;+σ&z.sfnc;y&z.sfnc;,1+&z.sfnc;x&z.sfnc;f(z)&les;A1+&z.sfnc;z&z.sfnc;N1expτ&z.sfnc;x&z.sfnc;+σ&z.sfnc;y&z.sfnc; for some A>0, τ,σ&ges;0, N1∈N∪{0} and any z=x+iy∈C. It will be shown that many well-known sampling theorems included in SIAM J. Math. Anal. 19 (1988) 1198–1203 and Inform. Control 8 (1965) 143–158 can be interpreted as special cases of this sampling theorem. As examples, we provide sampling representations for entire functions which are bounded, of polynomial growth, or of exponential growth on R. We also provide sampling representations involving derivatives of entire functions and nonuniform sampling representations. Taking the set of sampling points in which a finite number of points are arbitrarily distributed, we obtain a sampling representation. [Copyright &y& Elsevier]

• A stereo-camera 3D-DIC system for full-field vibration measurements is presented. • The optical setup is based on two low frame rate cameras and a multimedia projector. • A down-sampling approach is used to overcome the Nyquist-Shannon theorem limitation. • Fringe projection is introduced to solve the stereo matching problem. • An innovative time-domain image filtering is proposed to enhance measurement results. • 3D vibration amplitudes lower than 10 μm at 6.5 kHz have been measured with a 178 fps camera. The digital image correlation (DIC) was used in this paper to obtain full-field measurements of a target vibrating at a frequency higher than the maximum cameras' frame rate. The down-sampling technique was implemented to compensate for the cameras' moderate frame rate, thus getting an accurate displacement acquisition even at 6.5 kHz. Two innovative methods to support the DIC application were introduced. The use of fringe projection (or structured light), initially applied on the sample at rest, reduced the effort and time required for the stereo matching task's solution and improved this setting's accuracy and reliability. Additionally, a new time-domain image filtering was proposed and tested to improve the quality of the obtained DIC maps. In combination with the down-sampling, the effect of this filtering technique was tested in this work at (approx.) 2500 and 6500 Hz by measuring the response of a bladed disk to sinusoidal excitation. Evidence of improved results was observed for both frequencies for amplitudes in the range of 10 µm. [ABSTRACT FROM AUTHOR]

• Improve compressive sensing used for crack segmentation in practical applications. • Replace sparsity constraint in compressive sensing by deep generative models. • Reduce size of training data with imperfect generative model capturing cracks well. • Achieve high segmentation accuracy and robustness to noise with the proposed method. In a structural health monitoring (SHM) system that uses digital cameras to monitor cracks of structural surfaces, techniques for reliable and effective data compression are essential to ensure a stable and energy-efficient crack images transmission in wireless devices, e.g., drones and robots with high definition cameras installed. Compressive sensing (CS) is a signal processing technique that allows accurate recovery of a signal from a sampling rate much smaller than the limitation of the Nyquist sampling theorem. Different from the popular approach of simultaneously training encoder and decoder using neural network models, the CS theory ensures a high probability of accurate signal reconstruction based on random measurements that is shorter than the length of the original signal under a sparsity constraint. Such method is particularly useful when measurements are expensive, such as wireless sensing of civil structures, because its hardware implementation allows down sampling of signals during the sensing process. Hence, CS methods can achieve significant energy saving for the sensing devices. However, the strong assumption of the signals being highly sparse in an invertible space is relatively hard to guarantee for many real images, such as image of cracks. In this paper, we present a new approach of CS that replaces the sparsity regularization with a generative model that is able to effectively capture a low dimension representation of targeted images. We develop a recovery framework for automatic crack segmentation of compressed crack images based on this new CS method. We demonstrate the remarkable performance of our method that takes advantage of the strong capability of generative models to capture the necessary features required in the crack segmentation task even the backgrounds of the generated images are not well reconstructed. The superior performance of our recovery framework is illustrated by comparisons to three existing CS algorithms. Furthermore, we show that our framework is potentially extensible to other common problems in automatic crack segmentation, such as defect recovery from motion blurring and occlusion. [ABSTRACT FROM AUTHOR]