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In this work we connect machine learning techniques, in particular kernel machine learning, to inverse source and scattering problems. We show the proposed kernel machine learning has demonstrated generalization capability and has a rigorous mathematical foundation. The proposed learning is based on the Mercer kernel, the reproducing kernel Hilbert space, the kernel trick, as well as the mathematical theory of inverse source and scattering theory, and the restricted Fourier integral operator. The kernel machine learns a multilayer neural network which outputs an \epsilon -neighborhood average of the unknown or its nonlinear transformation. We then apply the general architecture to the multifrequency inverse source problem for a fixed observation direction and the Born inverse medium scattering problem. We establish a mathematically justified kernel machine indicator with demonstrated capability in both shape identification and parameter identification, under very general assumptions on the physical unknowns. More importantly, stability estimates are established in the case of both noiseless and noisy measurement data. Of central importance is the interplay between a restricted Fourier integral operator and a corresponding Sturm--Liouville differential operator. Several numerical examples are presented to demonstrate the capability of the proposed kernel machine learning. [ABSTRACT FROM AUTHOR]

This study explores efficient methods for computing eigenvalues and function values associated with Chebyshev-type prolate spheroidal wave functions (CPSWFs). Applying the expansion of the factor e i c x y and the inherent properties of Chebyshev polynomials, we present an exact and stable numerical approximation for the exact eigenvalues of the integral operator to CPSWFs. Additionally, we illustrate the efficiency of employing fast Fourier transform and barycentric interpolation techniques for computing CPSWF values and related quantities, which are essential for various numerical applications based on these functions. The analysis is supported by numerical examples, providing validation for the accuracy and reliability of our proposed approach. [ABSTRACT FROM AUTHOR]

Near-field acoustic holography (NAH) based on compressing sensing (CS) theory enables accurate reconstruction of sound fields using a limited number of sampling points. However, the successful implementation of this technique depends on two crucial factors: (1) the appropriate selection or construction of the spatial basis and (2) an effective sparse regularization process. To enhance reconstruction performance for elongated sound sources, this paper proposes a novel sound field reconstruction method that combines prolate spheroidal wave functions (PSWFs) with the orthogonal matching pursuit (OMP) algorithm. In this method, PSWFs serve as a sparse spatial basis for representing the radiated sound field. The sparse coefficients are determined by the OMP algorithm in a linear subspace composed of basic functions that best match the residual error. The OMP algorithm effectively identifies significant components before potentially selecting incorrect ones by setting an appropriate stopping rule. Numerical simulations are conducted using a line-array source model. The results show that the proposed method can accurately reconstruct the sound pressures of the elongated source model using a relatively small number of samplings. In addition, the proposed method exhibits robustness across a wide frequency range, diverse array configurations and various sampling numbers. The experimental results further validate the feasibility and reliability of the proposed method. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish an extension of the prolate spheroidal wave functions which is the eigenfunction of the finite Bessel type of linear canonical transform (LCT) so‐called finite canonical Fourier–Bessel transform. We study the concentration problem, uncertainty principles about the essential supports, and signal recovery related to the canonical Fourier–Bessel transform on the space of band‐limited functions. [ABSTRACT FROM AUTHOR]

We introduce an efficient scheme for the construction of quadrature rules for bandlimited functions. While the scheme is predominantly based on well-known facts about prolate spheroidal wave functions of order zero, it has the asymptotic CPU time estimate O (n log ⁡ n) to construct an n -point quadrature rule. Moreover, the size of the " n log ⁡ n " term in the CPU time estimate is small, so for all practical purposes the CPU time cost is proportional to n. The performance of the algorithm is illustrated by several numerical examples. [ABSTRACT FROM AUTHOR]

This paper proposes a method to correct severe, but known, probe positioning errors affecting a phaseless Planar Wide-Mesh Scanning (PWMS) Near-Field to Far-Field (NFFF) transformation technique and shows its numerical assessment. The involved technique exploits a scanning strategy by applying the non-redundant sampling representation of the squared amplitude of the electromagnetic field and by considering the source as surrounded by an oblate spheroid. In this way, the acquired NF samples are drastically reduced as compared to the classical phaseless NFFF transformation techniques. For the considered technique, the phaseless problem is addressed as a quadratic inverse one by exploiting the squared amplitude of the data gathered on two separate scanning planes and by devising a proper representation of the searched for unknowns. By assuming the planar positioning errors as known, the squared amplitude of the NF samples, at the nominal sampling points of the non-redundant representation, are recovered, from the positioning errors corrupted ones, by a Singular Value Decomposition based approach. After the compensation error step, an optimal sampling interpolation algorithm allows one to get, from the so obtained correctly positioned PWMS squared amplitude NF data, those on the classical plane-rectangular grid necessary to apply the phase retrieval procedure. The effectiveness of the proposed compensation approach is here shown by numerical results.

Contiguous submatrices of the Fourier matrix are known to be ill-conditioned. In a recent paper in SIAM review A. Barnett has provided new bounds on the rate of ill-conditioning (Barnett in SIAM Rev 64:105–131, 2022). In this paper we focus on the corresponding singular value decomposition. The singular vectors can be computed from the so-called periodic discrete prolate spheroidal sequences, named in analogy to spheroidal wave functions which are associated with the continuous Fourier transform. Their numerical computation is hampered by the clustering of singular values. We collect and expand known results on the stable numerical computation of the singular value decomposition of Fourier submatrices. The prolate sequences are eigenvectors of a tridiagonal matrix whose spectrum is free of clusters and this enables their computation. We collect these observations in a simple and convenient algorithm. The corresponding singular values can be accurately computed as well, except when they are small. Even then, small singular values can be computed in high-precision arithmetic with modest computational effort, even for large and extremely ill-conditioned submatrices. We illustrate the computations and point out a few applications in which Fourier submatrices arise. [ABSTRACT FROM AUTHOR]

Gaussian upper and lower bounds and Hölder continuity are established for the heat kernel associated to the prolate spheroidal wave functions (PSWFs) of order zero. These results are obtained by application of a general perturbation principle using the fact that the PSWF operator is a perturbation of the Legendre operator. Consequently, the Gaussian bounds and Hölder inequality for the PSWF heat kernel follow from the ones in the Legendre case. As an application of the general perturbation principle, we also establish Gaussian bounds for the heat kernels associated to generalized univariate PSWFs and PSWFs on the unit ball in R d . Further, we develop the related to the PSWFs of order zero smooth functional calculus, which provides the necessary groundwork in developing the theory of Besov and Triebel–Lizorkin spaces associated to the PSWFs. One of our main results on Besov and Triebel–Lizorkin spaces associated to the PSWFs asserts that they are the same as the Besov and Triebel–Lizorkin spaces generated by the Legendre operator. [ABSTRACT FROM AUTHOR]

Time and band limiting operators are expressed as functions of the confluent Heun operator arising in the spheroidal wave equation. Explicit formulas are obtained when the bandwidth parameter is either small or large and results on the complete Fourier transform are recovered. [ABSTRACT FROM AUTHOR]

This study explores efficient methods for computing eigenvalues and function values associated with Chebyshev-type prolate spheroidal wave functions (CPSWFs). Applying the expansion of the factor eicxy and the inherent properties of Chebyshev polynomials, we present an exact and stable numerical approximation for the exact eigenvalues of the integral operator to CPSWFs. Additionally, we illustrate the efficiency of employing fast Fourier transform and barycentric interpolation techniques for computing CPSWF values and related quantities, which are essential for various numerical applications based on these functions. The analysis is supported by numerical examples, providing validation for the accuracy and reliability of our proposed approach.

Near-field acoustic holography (NAH) based on compressing sensing (CS) theory enables accurate reconstruction of sound fields using a limited number of sampling points. However, the successful implementation of this technique depends on two crucial factors: (1) the appropriate selection or construction of the spatial basis and (2) an effective sparse regularization process. To enhance reconstruction performance for elongated sound sources, this paper proposes a novel sound field reconstruction method that combines prolate spheroidal wave functions (PSWFs) with the orthogonal matching pursuit (OMP) algorithm. In this method, PSWFs serve as a sparse spatial basis for representing the radiated sound field. The sparse coefficients are determined by the OMP algorithm in a linear subspace composed of basic functions that best match the residual error. The OMP algorithm effectively identifies significant components before potentially selecting incorrect ones by setting an appropriate stopping rule. Numerical simulations are conducted using a line-array source model. The results show that the proposed method can accurately reconstruct the sound pressures of the elongated source model using a relatively small number of samplings. In addition, the proposed method exhibits robustness across a wide frequency range, diverse array configurations and various sampling numbers. The experimental results further validate the feasibility and reliability of the proposed method.

Faster than Nyquist signaling increases the spectral efficiency of pulse amplitude modulation by accepting inter-symbol interference, where an equalizer is needed at the receiver. Since the complexity of an optimal equalizer increases exponentially with the number of the interfering symbols, practical truncated equalizers assume shorter memory. The power of the resulting residual interference depends on the transmit filter and limits the performance of truncated equalizers. In this letter, we use numerical optimizations and the prolate spheroidal wave functions to find optimal time-limited pulses that achieve minimum residual interference. Compared to root raised cosine pulses, the new pulses decrease the residual interference by an order of magnitude, for example, a decrease by 32 dB is achieved for an equalizer that considers four interfering symbols at 57% faster transmissions. As a proof of concept, for the 57% faster transmissions of binary symbols, we showed that using the new pulse with a 4-state equalizer has better bit error rate performance compared to using a root raised cosine pulse with a 128-state equalizer. [ABSTRACT FROM AUTHOR]

We give new formulas for finding a compactly supported function v on R d , d ≥ 1 , from its Fourier transform F v given within the ball B r. For the one-dimensional case, these formulas are based on the theory of prolate spheroidal wave functions (PSWF's). In multidimensions, well-known results of the Radon transform theory reduce the problem to the one-dimensional case. Related results on stability and convergence rates are also given. [ABSTRACT FROM AUTHOR]

For a pair of sets T , Ω ⊂ R the time-frequency localization operator is defined as S T , Ω = P T F − 1 P Ω F P T , where F is the Fourier transform and P T , P Ω are projection operators onto T and Ω, respectively. We show that in the case when both T and Ω are intervals, the eigenvalues of S T , Ω satisfy λ n (T , Ω) ≥ 1 − δ | T | | Ω | if n ≤ (1 − ε) | T | | Ω | , where ε > 0 is arbitrary and δ = δ (ε) < 1 , provided that | T | | Ω | > c ε. This improves the result of Bonami, Jaming and Karoui, who proved it for ε ≥ 0.42. The proof is based on the properties of the Bargmann transform. [ABSTRACT FROM AUTHOR]

Prolate spheroidal wave function (PSWF) signals are the best energy-gathering signals in the time-frequency domain,here is an exploratory application of PSWF signals to continuous phase modulation technology.Firstly,the process of generating CPM modulated signals based on PSWF signals was described.Secondly,a calculation method for the power spectral density of CPM modulated signals based on PSWF using the autocorrelation function method was given.Finally,combined with numerical calculation and analysis,the power spectral density and signal occupied bandwidth characteristics of minimum frequency shift keying,sinusoidal frequency shift keying,Gaussian filtered minimum shift keying and CPM modulation signal based on Gaussian function were compared.Numerical calculation results show that the PSWF signal can obtain CPM modulation signal with better spectral performance and energy concentration,compared with rectangular pulse,raised cosine pulse,Gaussian pulse and Gauss-like pulse as the baseband frequency modulation signal of CPM.

A novel frequency domain multi-carrier modulation (MCM-FD) scheme is proposed based on prolate spheroidal wave functions (PSWFs) for reducing the high complexity in the conventional time domain PSWFs multi-carrier modulation (MCM-PSWFs-TD) schemes. By constructing the relationship between discrete representation and exponential function representation of MCM-PSWFs-TD signals, it can be observed that the orthogonality and parity symmetry of the waveforms of PSWFs in the frequency domain are the same as that in the time domain. Thus, the PSWFs signal can be divided into two groups based on their parity symmetry, while these two groups of PSWFs signals can be processed simultaneously. Based on this concept, signal waveforms with only half spectrum range are invoked in the process of information loading and signal detection for reducing the number of sampling points participated in the signal operation. Compared to the MCM-PSWFs-TD scheme, the proposed MCM-PSWFs-FD scheme is capable of significantly reducing the computational complexity without severely degrading the system performance, such as spectral efficiency (SE), bit error rate performance, signal energy concentration and peak-to-average power ratio (PAPR). Furthermore, the cyclic-prefix orthogonal frequency division multiplexing (CP-OFDM), OFDM with weighted overlap and add (WOLA-OFDM), filter OFDM (F-OFDM), universal filtered multi-carrier (UFMC), as well as the filter bank multi-carrier with offset quadrature amplitude modulation (FBMC-OQAM) are also demonstrated as benchmarks. Simulation results are provided for illustrating that the proposed MCM-PSWFs-FD scheme is capable of striking a favorable tradeoff between the computational complexity and the system performance (i.e. SE, out-of-band energy leakage, adjacent frequency band interference, and PAPR), while the signal waveform design of the MCM-PSWFs-FD scheme is also more concise and flexible than the benchmarks.

The discrete prolate spheroidal sequences (DPSSs) are a set of orthonormal sequences in ℓ 2 (Z) which are strictly bandlimited to a frequency band [ − W , W ] and maximally concentrated in a time interval { 0 , ... , N − 1 }. The timelimited DPSSs (sometimes referred to as the Slepian basis) are an orthonormal set of vectors in C N whose discrete time Fourier transform (DTFT) is maximally concentrated in a frequency band [ − W , W ]. Due to these properties, DPSSs have a wide variety of signal processing applications. The DPSSs are the eigensequences of a timelimit-then-bandlimit operator and the Slepian basis vectors are the eigenvectors of the so-called prolate matrix. The eigenvalues in both cases are the same, and they exhibit a particular clustering behavior – slightly fewer than 2 N W eigenvalues are very close to 1, slightly fewer than N − 2 N W eigenvalues are very close to 0, and very few eigenvalues are not near 1 or 0. This eigenvalue behavior is critical in many of the applications in which DPSSs are used. There are many asymptotic characterizations of the number of eigenvalues not near 0 or 1. In contrast, there are very few non-asymptotic results, and these don't fully characterize the clustering behavior of the DPSS eigenvalues. In this work, we establish two novel non-asymptotic bounds on the number of DPSS eigenvalues between ϵ and 1 − ϵ. Also, we obtain bounds detailing how close the first ≈ 2 N W eigenvalues are to 1 and how close the last ≈ N − 2 N W eigenvalues are to 0. Furthermore, we extend these results to the eigenvalues of the prolate spheroidal wave functions (PSWFs), which are the continuous-time version of the DPSSs. Finally, we present numerical experiments demonstrating the quality of our non-asymptotic bounds on the number of DPSS eigenvalues between ϵ and 1 − ϵ. [ABSTRACT FROM AUTHOR]

In this paper, we provide further illustrations of prolate interpolation and pseudospectral differentiation based on the barycentric perspectives. The convergence rates of the barycentric prolate interpolation and pseudospectral differentiation are derived. Furthermore, we propose the new preconditioner, which leads to the well-conditioned prolate collocation scheme. Numerical examples are included to show the high accuracy of the new method. We apply this approach to solve the second-order boundary value problem and Helmholtz problem. [ABSTRACT FROM AUTHOR]

Copyright of Systems Engineering & Electronics is the property of Journal of Systems Engineering & Electronics Editorial Department and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Predicting the wave-induced response in the near-future is of importance to ensure safety of ships. To achieve this target, a possible method for deterministic and conditional prediction of future responses utilizing measured data from the most recent past has been developed. Herein, accurate derivation of the autocorrelation function (ACF) is required. In this study, a new approach for deriving ACFs from measurements is proposed by introducing the Prolate Spheroidal Wave Functions (PSWF). PSWF can be used in two ways: fitting the measured response itself or fitting the sample ACF from the measurements. The paper contains various numerical demonstrations, using a stationary heave motion time series of a containership, and the effectiveness of the present approach is demonstrated by comparing with both a non-parametric and a parametric spectrum estimation method; in this case, Fast Fourier Transformation (FFT) and an Auto-Regressive (AR) model, respectively. The present PSWF-based approach leads to two important properties: (1) a smoothed ACF from the measurements, including an expression of the memory time, (2) a high frequency resolution in power spectrum densities (PSDs). Finally, the paper demonstrates that a fitting of the ACF using PSWF can be applied for deterministic motion predictions ahead of current time. [ABSTRACT FROM AUTHOR]

Prolate spheroidal wave functions are an orthogonal family of bandlimited functions on R that have the highest concentration within a specific time interval. They are also identified as the eigenfunctions of a time-frequency limiting operator (TFLO), and the associated eigenvalues belong to the interval [ 0 , 1 ]. Previous work has studied the asymptotic distribution and clustering behavior of the TFLO eigenvalues. In this paper, we extend these results to multiple dimensions. We prove estimates on the eigenvalues of a spatio-spectral limiting operator (SSLO) on L 2 (R d) , which is an alternating product of projection operators associated to given spatial and frequency domains in R d. If one of the domains is a hypercube, and the other domain is convex body satisfying a symmetry condition, we derive quantitative bounds on the distribution of the SSLO eigenvalues in the interval [ 0 , 1 ]. To prove our results, we design an orthonormal system of wave packets in L 2 (R d) that are highly concentrated in the spatial and frequency domains. We show that these wave packets are "approximate eigenfunctions" of a spatio-spectral limiting operator. To construct the wave packets, we use a variant of the Coifman-Meyer local sine basis for L 2 [ 0 , 1 ] , and we lift the basis to higher dimensions using a tensor product. [ABSTRACT FROM AUTHOR]

Abstract The paper studies covert communication over a continuous-time Gaussian channel. The covertness condition requires that the channel output must statistically resemble pure noise. When the additive Gaussian noise is “white” over the bandwidth of interest, a formal coding theorem is proven, extending earlier results on covert Gaussian communication in discrete time. This coding theorem is applied to study scenarios where the input bandwidth can be infinite and where positive or even infinite per-second rates may be achievable.

We prove that under very mild conditions for any interpolation formula f (x) = ∑ λ ∈ Λ f (λ) a λ (x) + ∑ μ ∈ M f ^ (μ) b μ (x) we have a lower bound for the counting functions n Λ (R 1) + n M (R 2) ≥ 4 R 1 R 2 - C log 2 (4 R 1 R 2) which very closely matches recent interpolation formulas of Radchenko and Viazovska and of Bondarenko, Radchenko and Seip. [ABSTRACT FROM AUTHOR]

Copyright of Telecommunications Science is the property of Beijing Xintong Media Co., Ltd. and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The method presented and studied in [1-2] for solving self-adjoint multiparameter spectral problems for weakly coupled systems of ordinary differential equations is based on marching with respect to a parameter introduced into the problem. Although the method is formally applicable to systems of ordinary differential equations with singularities, its direct use for the numerical solution of the problem indicated in this paper's title is limited. A modification of the method is proposed that applies to the computation of various, including high-frequency, acoustic oscillations in both nearly spherical and strongly prolate spheroids. [ABSTRACT FROM AUTHOR]

• A grid-free compressive beamforming method is proposed based on prolate spheroidal wave functions. • It is compatible with arbitrary linear microphone arrays. • It can identify sources accurately, even using only a single snapshot. • The increase in snapshots helps the method perform better. Grid-free compressive beamforming is a new attractive method for acoustic source identification based on compressive sensing theory, which extracts the source information from the microphone array measurements without discretizing the target source region. In the framework of linear microphone array measurements, many studies have been carried out around this method. They have strict requirements on the distribution of microphones. Specifically, only the uniform arrays with equally spaced microphones and sparse arrays consisting of a selection of microphones from uniform arrays are available. In this paper, based on the prolate spheroidal wave functions, we propose a grid-free compressive beamforming method compatible with arbitrary linear microphone arrays. Some examples are conducted to demonstrate the correctness and superiority of the proposed method. Plenty of Monte Carlo simulations are performed to reveal the effects of source coherence, source separation, noise, and number of snapshots. This study helps to promote the popularization and application of grid-free compressive beamforming method and offers the possibility to improve the acoustic source identification performance by optimizing microphone distribution. [ABSTRACT FROM AUTHOR]

The paper studies covert communication over a continuous-time Gaussian channel. The covertness condition requires that the channel output must statistically resemble pure noise. When the additive Gaussian noise is "white" over the bandwidth of interest, a formal coding theorem is proven, extending earlier results on covert Gaussian communication in discrete time. This coding theorem is applied to study scenarios where the input bandwidth can be infinite and where positive or even infinite per-second rates may be achievable. [ABSTRACT FROM AUTHOR]

Abstract Prolate spheroidal wave functions of order zero (PSWFs) are widely used in scientific computation. There are few results about the error bounds of the prolate interpolation and differentiation. In this paper, based on the Cauchy's residue theorem and asymptotics of PSWFs, the convergence rates are derived. To get stable approximation, the first barycentric formula is applied. These theoretical results and high accuracy are illustrated by numerical examples. [ABSTRACT FROM AUTHOR]

We present an approximation scheme for functions in three dimensions, that requires only their samples on the Cartesian grid, under the assumption that the functions are sufficiently concentrated in both space and frequency. The scheme is based on expanding the given function in the basis of generalized prolate spheroidal wavefunctions, with the expansion coefficients given by weighted dot products between the samples of the function and the samples of the basis functions. As numerical implementations require all expansions to be finite, we present a truncation rule for the expansions. Finally, we derive a bound on the overall approximation error in terms of the assumed space/frequency concentration. [ABSTRACT FROM AUTHOR]

We consider a communication system whereby T-seconds time-limited codewords are transmitted over a W-Hz band-limited additive white Gaussian noise channel. In the asymptotic regime as WT→∞, it is known that the maximal achievable rates with such a scheme converge to Shannon’s capacity with the presence of 2WT degrees of freedom. In this work we study the degrees of freedom and the achievable information rates for finite values of WT. We use prolate spheroidal wave functions to obtain an information lossless equivalent discrete formulation and then we apply Polyanskiy’s results on coding in the finite block-length regime. We derive upper and lower bounds on the achievable rates and the corresponding degrees of freedom and we numerically evaluate them for sample values of 2WT. The bounds are asymptotically tight and numerical computations show the gap between them decreases as 2WT increases. Additionally, the possible decrease from 2WT in the available degrees of freedom is upper-bounded by a logarithmic function of 2WT.

For fixed real numbers c>0,α>-12, the finite Hankel transform operator, denoted by Hcα is given by the integral operator defined on L2(0,1) with kernel Kα(x,y)=cxyJα(cxy). To the operator Hcα, we associate a positive, self-adjoint compact integral operator Qcα=cHcαHcα. Note that the integral operators Hcα and Qcα commute with a Sturm-Liouville differential operator Dcα. In this paper, we first give some useful estimates and bounds of the eigenfunctions φn,c(α) of Hcα or Qcα. These estimates and bounds are obtained by using some special techniques from the theory of Sturm-Liouville operators, that we apply to the differential operator Dcα. If (μn,α(c))n and λn,α(c)=c|μn,α(c)|2 denote the infinite and countable sequence of the eigenvalues of the operators Hc(α) and Qcα, arranged in the decreasing order of their magnitude, then we show an unexpected result that for a given integer n≥0,λn,α(c) is decreasing with respect to the parameter α. As a consequence, we show that for α≥12, the λn,α(c) and the μn,α(c) have a super-exponential decay rate. Also, we give a lower decay rate of these eigenvalues. As it will be seen, the previous results are essential tools for the analysis of a spectral approximation scheme based on the eigenfunctions of the finite Hankel transform operator. Some numerical examples will be provided to illustrate the results of this work. [ABSTRACT FROM AUTHOR]

For two real numbers we study some spectral properties of the weighted finite bilateral Laplace transform operator, defined over the space by In particular, we use a technique based on the Min-Max theorem to prove that the sequence of the eigenvalues of this operator has a super-exponential decay rate to zero. Moreover, we give a lower bound with a magnitude of order for the largest eigenvalue of the operator Also, we give some local estimates and bounds of the eigenfunctions of Moreover, we show that these eigenfunctions are good candidates for the spectral approximation of a function that can be written as a weighted finite Laplace transform of an other function. Finally, we give some numerical examples that illustrate the different results of this work. In particular, we provide an example that illustrate the Laplace-based spectral method, for the inversion of the finite Laplace transform. [ABSTRACT FROM AUTHOR]

In this paper, we show that the generalized prolate spheroidal wave functions (GPSWFs), called sometimes Slepian’s functions, can be defined as the most concentrated functions among functions in some reproducing kernel Hilbert spaces (RKHS for short). As a consequence, we use them the GPSWFs to approximate the sets of K-bandlimited functions that are essentially time limited to an interval (a, b). [ABSTRACT FROM AUTHOR]

Slepian functions provide a solution to the optimization problem of joint time-frequency localization. Here, this concept is extended by using a generalized optimization criterion that favors energy concentration in one interval while penalizing energy in another interval, leading to the “augmented” Slepian functions. Mathematical foundations together with examples are presented in order to illustrate the most interesting properties that these generalized Slepian functions show. Also the relevance of this novel energy-concentration criterion is discussed along with some of its applications. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce a new methodology for the fault diagnosis of induction machines working in the transient regime, when time-frequency analysis tools are used. The proposed method relies on the use of the optimized Slepian window for performing the short time Fourier transform (STFT) of the stator current signal. It is shown that for a given sequence length of finite duration, the Slepian window has the maximum concentration of energy, greater than can be reached with a gated Gaussian window, which is usually used as the analysis window. In this paper, the use and optimization of the Slepian window for fault diagnosis of induction machines is theoretically introduced and experimentally validated through the test of a 3.15-MW induction motor with broken bars during the start-up transient. The theoretical analysis and the experimental results show that the use of the Slepian window can highlight the fault components in the current's spectrogram with a significant reduction of the required computational resources. [ABSTRACT FROM AUTHOR]

We introduce an approximation scheme for almost bandlimited functions which are sufficiently concentrated in a disk, based on their equally spaced samples on a Cartesian grid. The scheme is based on expanding the function into a series of two-dimensional prolate spheroidal wavefunctions, and estimating the expansion coefficients using the available samples. We prove that the approximate expansion coefficients have particularly simple formulas, in the form of a dot product of the available samples with samples of the basis functions. We also derive error bounds for the error incurred by approximating the expansion coefficients as well as by truncating the expansion. In particular, we derive a bound on the approximation error in terms of the assumed space/frequency concentration, and provide a simple truncation rule to control the length of the expansion and the resulting approximation error. [ABSTRACT FROM AUTHOR]

As modern scientific image datasets typically consist of a large number of images of high resolution, devising methods for their accurate and efficient processing is a central research task. In this paper, we consider the problem of obtaining the steerable principal components of a dataset, a procedure termed \steerable PCA" (steerable principal component analysis). The output of the procedure is the set of orthonormal basis functions which best approximate the images in the dataset and all of their planar rotations. To derive such basis functions, we first expand the images in an appropriate basis, for which the steerable PCA reduces to the eigen-decomposition of a block-diagonal matrix. If we assume that the images are well localized in space and frequency, then such an appropriate basis is the prolate spheroidal wave functions (PSWFs). We derive a fast method for computing the PSWFs expansion coefficients from the images' equally spaced samples, via a specialized quadrature integration scheme, and show that the number of required quadrature nodes is similar to the number of pixels in each image. We then establish that our PSWF-based steerable PCA is both faster and more accurate then existing methods, and more importantly, provides us with rigorous error bounds on the entire procedure. [ABSTRACT FROM AUTHOR]

Recently, there is a growing interest in the spectral approximation by the Prolate Spheroidal Wave Functions (PSWFs) ψ n , c , c > 0 . This is due to the promising new contributions of these functions in various classical as well as emerging applications from Signal Processing, Geophysics, Numerical Analysis, etc. The PSWFs form a basis with remarkable properties not only for the space of band-limited functions with bandwidth c , but also for the Sobolev space H s ( [ − 1 , 1 ] ) . The quality of the spectral approximation and the choice of the parameter c when approximating a function in H s ( [ − 1 , 1 ] ) by its truncated PSWFs series expansion, are the main issues. By considering a function f ∈ H s ( [ − 1 , 1 ] ) as the restriction to [ − 1 , 1 ] of an almost time-limited and band-limited function, we try to give satisfactory answers to these two issues. Also, we illustrate the different results of this work by some numerical examples. [ABSTRACT FROM AUTHOR]

For fixed c , the Prolate Spheroidal Wave Functions (PSWFs) ψ n , c form a basis with remarkable properties for the space of band-limited functions with bandwidth c . They have been largely studied and used after the seminal work of D. Slepian, H. Landau and H. Pollack. Many of the PSWFs applications rely heavily on the behavior and the decay rate of the eigenvalues ( λ n ( c ) ) n ≥ 0 of the time and frequency limiting operator, which we denote by Q c . Hence, the issue of the accurate estimation of the spectrum of this operator has attracted a considerable interest, both in numerical and theoretical studies. In this work, we give an explicit integral approximation formula for these eigenvalues. This approximation holds true starting from the plunge region where the spectrum of Q c starts to have a fast decay. As a consequence of our explicit approximation formula, we give a precise description of the super-exponential decay rate of the λ n ( c ) . Also, we mention that the described approximation scheme provides us with fairly accurate approximations of the λ n ( c ) with low computational load, even for very large values of the parameters c and n . Finally, we provide the reader with some numerical examples that illustrate the different results of this work. [ABSTRACT FROM AUTHOR]

This paper presents a new approach for estimating encountered wave elevation sequences by use of measured ship responses, where wind-waves and swell may come from different directions, i.e. bi-directional waves. The main assumption of the approach, making use of Prolate Spheroidal Wave Functions (PSWF), is that the wave field is represented by multi-directional irregular waves. Thus, combining available measured responses, the phases and amplitudes of the multi-directional irregular waves are derived as the solution, by which the wave profiles can be estimated. Numerical investigations using artificially generated response measurements (sway, heave, pitch, vertical bending moment) of a bulk carrier in uni-directional and bi-directional long-crested as well as short-crested sea states are made. It is shown that the present approach can accurately estimate wave elevation sequences in such sea states. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce a new methodology for the fault diagnosis of induction machines working in the transient regime, when time-frequency analysis tools are used. The proposed method relies on the use of the optimized Slepian window for performing the short time Fourier transform (STFT) of the stator current signal. It is shown that for a given sequence length of finite duration, the Slepian window has the maximum concentration of energy, greater than can be reached with a gated Gaussian window, which is usually used as the analysis window. In this paper, the use and optimization of the Slepian window for fault diagnosis of induction machines is theoretically introduced and experimentally validated through the test of a 3.15-MW induction motor with broken bars during the start-up transient. The theoretical analysis and the experimental results show that the use of the Slepian window can highlight the fault components in the current’s spectrogram with a significant reduction of the required computational resources.

The aim of this paper is to investigate the quality of approximation of almost time- and almost band-limited functions by its expansion in two classical orthogonal polynomials bases: the Hermite basis and the ultraspherical polynomials bases (which include Legendre and Chebyshev bases as particular cases). As a corollary, this allows us to obtain the quality of approximation in the L 2 -Sobolev space by these orthogonal polynomials bases. Also, we obtain the rate of convergence of the Legendre series expansion of the prolate spheroidal wave functions. [ABSTRACT FROM AUTHOR]