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This work is concerned with the interpolation of a function f when using a low number of interpolation points, as required by the finite element method for solving PDEs numerically. The function f is assumed to have a jump or a steep derivative, and our goal is to minimize the oscillations produced by the Gibbs phenomenon while preserving the approximation properties for smoother functions. This is achieved by interpolating the transform f ^ = g ∘ f using Lagrange polynomials, where g is a rational transformation chosen by minimizing a suitable functional depending on the values of f . The mapping g is monotonic and constructed to possess boundary layers that remove the Gibbs phenomenon. No previous knowledge of the location of the jump is required. The extension to functions of several variables is straightforward, of which we provide several examples. Finally, we show how the interpolation fits the finite element method and compare it with known strategies. [ABSTRACT FROM AUTHOR]

In this study, we proposed a family of m -point quaternary approximating subdivision schemes, characterized by an explicit formula involving three parameters. One of these parameters served as a shape control parameter, allowing for flexible curve design, while the other two parameters identify different members of the family and determined the smoothness of the resulting limit curves. We conducted a thorough analysis of the proposed schemes, covering their smoothness properties, polynomial generation, and reproduction capabilities. Additionally, we examined the behavior of the Gibbs phenomenon within the family both theoretically and graphically, highlighting the advantages of the proposed schemes in eliminating undesirable oscillations. A comparative study with existing subdivision schemes demonstrated the effectiveness and versatility of our approach. The results indicated that the proposed family offered enhanced smoothness and control, making it suitable for a wide range of applications in computer graphics and geometric modeling. [ABSTRACT FROM AUTHOR]

In this study, we proposed a family of m-point quaternary approximating subdivision schemes, characterized by an explicit formula involving three parameters. One of these parameters served as a shape control parameter, allowing for flexible curve design, while the other two parameters identify different members of the family and determined the smoothness of the resulting limit curves. We conducted a thorough analysis of the proposed schemes, covering their smoothness properties, polynomial generation, and reproduction capabilities. Additionally, we examined the behavior of the Gibbs phenomenon within the family both theoretically and graphically, highlighting the advantages of the proposed schemes in eliminating undesirable oscillations. A comparative study with existing subdivision schemes demonstrated the effectiveness and versatility of our approach. The results indicated that the proposed family offered enhanced smoothness and control, making it suitable for a wide range of applications in computer graphics and geometric modeling.

Copyright of Journal of South China University of Technology (Natural Science Edition) is the property of South China University of Technology 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.)

In a photoinjector electron source, the initial transverse electron bunch properties are determined by the spatial properties of the laser beam on the photocathode. Spatial shaping of the laser is commonly achieved by relay imaging an illuminated circular mask onto the photocathode. However, the Gibbs phenomenon shows that recreating the sharp edge and discontinuity of the cut profile at the mask on the cathode is not possible with an optical relay of finite aperture. Furthermore, the practical injection of the laser into the photoinjector results in the beam passing through small or asymmetrically positioned apertures. This work uses wavefront propagation to show how the transport apertures cause ripple structures to appear in the transverse laser profile even when effectively the full laser power is transmitted. The impact of these structures on the propagated electron bunch has also been studied with electron bunches of high and low charge density. With high charge density, the ripples in the initial charge distribution rapidly wash-out through space charge effects. However, for bunches with low charge density, the ripples can persist through the bunch transport. Although statistical properties of the electron bunch in the cases studied are not greatly affected, there is the potential for the distorted electron bunch to negatively impact machine performance. Therefore, these effects should be considered in the design phase of accelerators using photoinjectors.

Approximating non-smooth functions presents a significant challenge due to the emergence of unwanted oscillations near discontinuities, commonly known as Gibbs’ phenomena. Traditional methods like finite Fourier or Chebyshev representations only achieve convergence on the order of O(1). A promising avenue in addressing this issue lies in nonlinear and essentially non-oscillatory approximation techniques, such as rational or Padé approximation. A recent and notable endeavor to mitigate Gibbs’ oscillations is through singular Padé-Chebyshev approximation. However, a drawback of this approach is the requirement to specify the discontinuity location within the algorithm, which is often unknown in practical applications. To tackle this obstacle, we propose a localized Padé-Chebyshev approximation method. Fortunately, our efforts yield success; the proposed localized variant effectively captures jump locations in non-smooth functions while maintaining an essentially non-oscillatory character. Furthermore, we employ Padé-Chebyshev approximation within a finite volume framework to address scalar hyperbolic conservation laws. Remarkably, the resulting rational numerical scheme demonstrates stability regardless of wave propagation direction. Consequently, we introduce a central rational numerical scheme for scalar hyperbolic conservation laws, offering robust and accurate computation of solutions. [ABSTRACT FROM AUTHOR]

In a photoinjector electron source, the initial transverse electron bunch properties are determined by the spatial properties of the laser beam on the photocathode. Spatial shaping of the laser is commonly achieved by relay imaging an illuminated circular mask onto the photocathode. However, the Gibbs phenomenon shows that recreating the sharp edge and discontinuity of the cut profile at the mask on the cathode is not possible with an optical relay of finite aperture. Furthermore, the practical injection of the laser into the photoinjector results in the beam passing through small or asymmetrically positioned apertures. This work uses wavefront propagation to show how the transport apertures cause ripple structures to appear in the transverse laser profile even when effectively the full laser power is transmitted. The impact of these structures on the propagated electron bunch has also been studied with electron bunches of high and low charge density. With high charge density, the ripples in the initial charge distribution rapidly wash‐out through space charge effects. However, for bunches with low charge density, the ripples can persist through the bunch transport. Although statistical properties of the electron bunch in the cases studied are not greatly affected, there is the potential for the distorted electron bunch to negatively impact machine performance. Therefore, these effects should be considered in the design phase of accelerators using photoinjectors. [ABSTRACT FROM AUTHOR]

Piecewise affine or linear approximation has garnered significant attention as a technique for approximating piecewise-smooth functions. In this study, we propose a novel approach: piecewise non-linear approximation based on rational approximation, aimed at approximating non-smooth functions. We introduce a method termed piecewise Padé Chebyshev (PiPC) tailored for approximating univariate piecewise smooth functions. Our investigation focuses on assessing the effectiveness of PiPC in mitigating the Gibbs phenomenon during the approximation of piecewise smooth functions. Additionally, we provide error estimates and convergence results of PiPC for non-smooth functions. Notably, our technique excels in capturing singularities, if present, within the function with minimal Gibbs oscillations, without necessitating the explicit specification of singularity locations. To the best of our knowledge, prior research has not explored the use of piecewise non-linear approximation for approximating non-smooth functions. Finally, we validate the efficacy of our methods through numerical experiments, employing PiPC to reconstruct a non-trivial non-smooth function, thus demonstrating its capability to significantly alleviate the Gibbs phenomenon.

Many attempts have been made in the past to regain the spectral accuracy of the spectral methods, which is lost drastically due to the presence of discontinuity. In this article, an attempt has been made to show that mollification using Legendre and Chebyshev polynomial based kernels improves the convergence rate of the Fourier spectral method. Numerical illustrations are provided with examples involving one or more discontinuities and compared with the existing Dirichlet kernel mollifier. Dependence of the efficiency of the polynomial mollifiers on the parameter P$$ P $$ is analogous to that in the Dirichlet mollifier, which is detailed by analyzing the numerical solution. Further, they are extended to linear scalar conservation law problems. [ABSTRACT FROM AUTHOR]

Linear approximation methods suffer from Gibbs oscillations when approximating functions with jumps. Essentially non oscillatory subcell-resolution (ENO-SR) is a local technique avoiding oscillations and with a full order of accuracy, but a loss of regularity of the approximant appears. The goal of this paper is to introduce a new approach having both properties of full accuracy and regularity. In order to obtain it, we propose a three-stage algorithm: first, the data is smoothed by subtracting an appropriate non-smooth data sequence; then a chosen high order linear approximation operator is applied to the smoothed data and finally, an approximation with the proper jump or corner (jump in the first order derivative) discontinuity structure is reinstated by correcting the smooth approximation with the non-smooth element used in the first stage. This new procedure can be applied as subdivision scheme to design curves and surfaces both in point-value and in cell-average contexts. Using the proposed algorithm, we are able to construct approximations with high precision, with high piecewise regularity, and without smearing nor oscillations in the presence of discontinuities. These are desired properties in real applications as computer aided design or car design, among others. [ABSTRACT FROM AUTHOR]

One of the major challenges in finite element methods is the mitigation of spurious oscillations near sharp layers and discontinuities known as the Gibbs phenomenon. In this paper, we propose a set of functionals to identify spurious oscillations in best approximation problems in finite element spaces. Subsequently, we adopt these functionals in the formulation of constraints in an effort to eliminate the Gibbs phenomenon. By enforcing these constraints in best approximation problems, we can entirely eliminate over- and undershoot in one-dimensional continuous approximations, and significantly suppress them in one- and higher-dimensional discontinuous approximations. [ABSTRACT FROM AUTHOR]

This paper studies and analyzes the approximation of one-dimensional smooth closed-form functions with compact support using a mixed Fourier series (i.e., a combination of partial Fourier series and other forms of partial series). To explore the potential of this approach, we discuss and revise its application in signal processing, especially because it allows us to control the decreasing rate of Fourier coefficients and avoids the Gibbs phenomenon. Therefore, this method improves the signal processing performance in a wide range of scenarios, such as function approximation, interpolation, increased convergence with quasi-spectral accuracy using the time domain or the frequency domain, numerical integration, and solutions of inverse problems such as ordinary differential equations. Moreover, the paper provides comprehensive examples of one-dimensional problems to showcase the advantages of this approach. [ABSTRACT FROM AUTHOR]

Discretization-based spectral approximation methods provide spectrally accurate reconstruction of an analytic function. The expansion of non-smooth functions is contaminated by high frequency non-diminishing oscillations near discontinuity points, and this behaviour is named as Gibbs phenomenon. This problem can be well resolved by well-chosen post-processing technique, and one possible choice is spectral filtering. In this paper, a comparison scenario of adaptive spectral filtering for resolution of Gibbs phenomenon is presented. Several spectral filter functions are compared using Chebyshev collocation and Legendre collocation spectral methods, in terms of pointwise and L 2 normed-wise convergence analysis of computed filtered approximations. [ABSTRACT FROM AUTHOR]

We consider the problem of computing the Lebesgue volume of compact basic semi-algebraic sets. In full generality, it can be approximated as closely as desired by a converging hierarchy of upper bounds obtained by applying the Moment-SOS (sums of squares) methodology to a certain infinite-dimensional linear program (LP). At each step one solves a semidefinite relaxation of the LP which involves pseudo-moments up to a certain degree. Its dual computes a polynomial of same degree which approximates from above the discontinuous indicator function of the set, hence with a typical Gibbs phenomenon which results in a slow convergence of the associated numerical scheme. Drastic improvements have been observed by introducing in the initial LP additional linear moment constraints obtained from a certain application of Stokes' theorem for integration on the set. However and so far there was no rationale to explain this behavior. We provide a refined version of this extended LP formulation. When the set is the smooth super-level set of a single polynomial, we show that the dual of this refined LP has an optimal solution which is a continuous function. Therefore in this dual one now approximates a continuous function by a polynomial, hence with no Gibbs phenomenon, which explains and improves the already observed drastic acceleration of the convergence of the hierarchy. Interestingly, the technique of proof involves recent results on Poisson's partial differential equation (PDE). [ABSTRACT FROM AUTHOR]

Context. The Gibbs phenomenon introduces significant distortions for most popular 2D graphics standards because they use a finite sum of harmonics when image processing by expansion of the signal into a two-dimensional Fourier series is used in order to reduce the size of the graphical file. Thus, the reduction of this phenomenon is a very important problem. Objective. The aim of the current work is the application of two-dimensional Padé-type approximations with the aim of elimination of the Gibbs phenomenon in image processing and reduction of the size of the resulting image file. Method. We use the two-dimensional Padé-type approximants method which we have developed earlier to reduce the Gibbs phenomenon for the harmonic two-dimensional Fourier series. A definition of a Padé-type functional is proposed. For this purpose, we use the generalized two-dimensional Padé approximation proposed by Chisholm when the range of the frequency values on the integer grid is selected according to the Vavilov method. The proposed scheme makes it possible to determine a set of series coefficients necessary and sufficient for construction of a Padé-type approximation with a given structure of the numerator and denominator. We consider some examples of Padé approximants application to simple discontinuous template functions for both formulaic and discrete representation. Results. The study gives us an opportunity to make some conclusions about practical usage of the Padé-type approximation and about its advantages. They demonstrate effective elimination of distortions inherent to Gibbs phenomena for the Padé-type approximant. It is well seen that Padé-type approximant is significantly more visually appropriate than Fourier one. Application of the Padétype approximation also leads to sufficient decrease of approximants’ parameter number without the loss of precision. Conclusions. The applicability of the technique and the possibility of its application to improve the accuracy of calculations are demonstrated. The study gives us an opportunity to make conclusions about the advantages of the Padé-type approximation practical usage. [ABSTRACT FROM AUTHOR]

The problem of nonparametric estimation of a univariate density with rth continuous derivative on compact support is addressed ( r ≥ 2). If the density function has compact support and is non-zero at either boundary, regular kernel estimator will be completely biased at such boundary. Although several correction methods were proposed to improve the bias at the boundary to h 2 in the last decades, this paper initiates a way to further improve bias to higher order ( h r ) for interior area of density function support, while remaining the order of bias h 2 at boundary. We will first review flat-top kernel estimator and flat-top series estimator, then propose the Transformed Flat-top Series estimator. The theoretical analysis is supplemented with simulation results as well as real data applications. [ABSTRACT FROM AUTHOR]

The Gibbs phenomenon describes oscillations of small or large amplitudes that occur, when a signal with steep gradients or noise components is approximated. Such interruptions can degrade the quality of desired signal. Reduction of such oscillations is an essential task to extract vital information from the desired signal. This paper, therefore, presents the Chebyshev spectral method (CSM) that is combined with two novel concepts to reduce the influence of oscillatory structures. The first notion uses a thresholding approach to estimate true expansion coefficients in a noisy environment, while the second concept introduces a new smoothing function. The basic framework of the proposed concept is to introduce an additional threshold procedure into pre-existing Chebyshev collocation method to handle the fluctuations of noise interferences. Moreover, the CSM is the global-behaviour approximation based on the points of an entire domain, which allows for high-order convergence to be recovered. The method is implemented for sharp gradient-contained function and to a signal that has been distorted by noise. Through computational experiments, efficiency of the proposed method is verified graphically and numerically. Signal-to-noise ratio of 37.4810 dB is achieved with corresponding mean square error about 1.79e−04. The percentage root-mean-square difference (PRD) and maximum error are obtained as 1.3402% and 0.0399, respectively. [ABSTRACT FROM AUTHOR]

This paper studies and analyzes the approximation of one-dimensional smooth closed-form functions with compact support using a mixed Fourier series (i.e., a combination of partial Fourier series and other forms of partial series). To explore the potential of this approach, we discuss and revise its application in signal processing, especially because it allows us to control the decreasing rate of Fourier coefficients and avoids the Gibbs phenomenon. Therefore, this method improves the signal processing performance in a wide range of scenarios, such as function approximation, interpolation, increased convergence with quasi-spectral accuracy using the time domain or the frequency domain, numerical integration, and solutions of inverse problems such as ordinary differential equations. Moreover, the paper provides comprehensive examples of one-dimensional problems to showcase the advantages of this approach.

In this paper, we want to clarify the Gibbs phenomenon when continuous and discontinuous finite elements are used to approximate discontinuous or nearly discontinuous PDE solutions from the approximation point of view. For a simple step function, we explicitly compute its continuous and discontinuous piecewise constant or linear projections on discontinuity matched or non-matched meshes. For the simple discontinuity-aligned mesh case, piecewise discontinuous approximations are always good. For the general non-matched case, we explain that the piecewise discontinuous constant approximation combined with adaptive mesh refinements is a good choice to achieve accuracy without overshoots. For discontinuous piecewise linear approximations, non-trivial overshoots will be observed unless the mesh is matched with discontinuity. For continuous piecewise linear approximations, the computation is based on a "far-away assumption", and non-trivial overshoots will always be observed under regular meshes. We calculate the explicit overshoot values for several typical cases. Numerical tests are conducted for a singularly-perturbed reaction-diffusion equation and linear hyperbolic equations to verify our findings in the paper. Also, we discuss the L 1 -minimization-based methods and do not recommend such methods due to their similar behavior to L 2 -based methods and more complicated implementations. [ABSTRACT FROM AUTHOR]

In this article, a nonlinear binary three-point non-interpolatory subdivision scheme is presented. It is based on a nonlinear perturbation of the three-point subdivision scheme: A new three point approximating C 2 subdivision scheme, where the convergence and the stability of this linear subdivision scheme are analyzed. It is possible to prove that this scheme does not present Gibbs oscillations in the limit functions obtained. The numerical experiments show that the linear scheme is stable even in the presence of jump discontinuities. Even though, close to jump discontinuities, the accuracy is loosed. This order reduction is equivalent to the introduction of some diffusion. Diffusion is a good property for subdivision schemes when the discontinuities are numerical, i.e., they appear when discretizing a continuous function close to high gradients. On the other hand, if the initial control points come from the discretization of a piecewise continuous function, it can be interesting that the subdivision scheme produces a piecewise continuous limit function. For instance, in the approximation of conservation laws, real discontinuities appear as shocks in the solution. The nonlinear modification introduced in this work allows to attain this objective. As far as we know, this is the first subdivision scheme that appears in the literature with these properties. [ABSTRACT FROM AUTHOR]

This paper is devoted to the construction and analysis of some new families of n-point ternary subdivision schemes. Some members of the families were adapted to the presence of discontinuities converging to limit functions without Gibbs oscillations. We present a numerical comparison where we check the theoretical properties. [ABSTRACT FROM AUTHOR]

Given a set of Lagrange type data in two dimensions and assuming that the points of the data set are associated with the graph of an explicit function with discontinuities, one wants to determine the points at which the function exhibits discontinuities. This is a concrete problem that appears in the area of approximation of curves with discontinuities and is present in different scientific areas. To do this, it is necessary to recognize discontinuity placement at the function. This need to characterize the placement of discontinuity points is fundamental for the development of mathematical models that take into account the discontinuities of functions. In this work, a new methodology is proposed to determine the points where the discontinuities of a function occur using an approximation space constructed from continuous trigonometric functions. The approach used to locate the discontinuity points of the function is based on the Gibbs phenomenon which is related to the oscillations found at the points of discontinuity when the discontinuous function is represented by a continuous function. Results will be presented and show the numerical process to approximate the placements of discontinuity points is successful.

This article continues a series of publications under the same name. It performs further improvement of the method for restoring discontinuous functions of two variables using projections to improve the accuracy of approximation without the Gibbs phenomenon for the case, where the discontinuity lines are the boundaries of squares nested into each other. We consider the case of the discontinuity lines having angular points, where the derivative along the normal is undefined. A discontinuous spline is constructed so that the difference between the function being approximated and this spline is a differentiable function. This function is restored using finite Fourier sums whose Fourier coefficients are found using projections. Analysis of the computing results confirmed the theoretical statement of the study. [ABSTRACT FROM AUTHOR]

Copyright of Cybernetics & Systems Analysis / Kibernetiki i Sistemnyj Analiz is the property of V.M. Glushkov Institute of Cybernetics of NAS of Ukraine 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.)

In this work, we propose and substantiate the expediency of using the Gibbs phenomenon. This phenomenon occurs at the edge of the function gap, for the analysis of which. The creation of a method of constructing such multidimensional approximations to the approximation and determination of the set of coefficients of the series necessary to construct an approximation of the given structure is considered. The conditions of convergence of approximants of approximate multidimensional functions constructed according to the developed method are determined. It is shown that in the calculations of two-dimensional approximations, the use of the Gibbs phenomenon makes it possible to significantly reduce the quality of image processing for most popular graphic standards, since they use a finite sum of harmonics. Image compression plays a very important role in many technological applications, such as document and medical imaging, television video conferencing, etc. Image compression methods are often divided into two categories: lossless and lossy. Lossless methods are usually chosen for applications where it is important to preserve very fine image details. These include medical and space research, remote sensing. On the other hand, lossy methods are used in situations where a significant degree of compression is required. This applies, for example, to digital photography, where, as a rule, the loss of some image details is not critical. Image compression methods are often divided into two categories: lossless and lossy. Lossless methods are usually chosen for applications where it is important to preserve very fine image details. These include medical and space research, remote sensing. On the other hand, lossy methods are used in situations where a significant degree of compression is required. This applies, for example, to digital photography, where, as a rule, the loss of some image details is not critical. When choosing a compression algorithm, it is important to understand its positive and negative sides. When choosing algorithms with the loss of part of the data, it is necessary to clearly understand under what conditions the image will lose the quality of human perception. This is due to the fact that blurring occurs at the borders of a sharp change in contrast and leads to the appearance of false optical shadows and poor-quality analysis when processing the results of radiological and sonar research. A two-dimensional Padé-type approximation method was developed, which reduces the Gibbs phenomenon for a harmonic two-dimensional Fourier series. This approach makes it possible to develop a method application scheme, an image processing algorithm and its effectiveness. Thanks to this, it becomes possible to optimize calculations and build a transformation algorithm. [ABSTRACT FROM AUTHOR]

Recent developments in the context of minimum residual finite element methods are paving the way for designing quasi-optimal discretization methods in non-standard function spaces, such as q-type Sobolev spaces. For q → 1, these methods have demonstrated huge potential in avoiding the notorious Gibbs phenomena, i.e., the occurrence of spurious non-physical oscillations near thin layers and jump discontinuities. In this work we provide theoretical results that explain some of these numerical observations. In particular, we investigate the Gibbs phenomena for q-best approximations of discontinuities in finite element spaces with 1 ≤ q < ∞. We prove sufficient conditions on meshes in one and two dimensions such that over- and undershoots vanish in the limit q → 1. Moreover, we include examples of meshes such that Gibbs phenomena remain present even for q = 1 and demonstrate that our results can be used to design meshes so as to eliminate the Gibbs phenomenon. [ABSTRACT FROM AUTHOR]

In this paper, we collect the basic theory and the most important applications of a novel technique that has shown to be suitable for scattered data interpolation, quadrature, bio-imaging reconstruction. The method relies on polynomial mapped bases allowing, for instance, to incorporate data or function discontinuities in a suitable mapping function. The new technique substantially mitigates the Runge's and Gibbs effects. [ABSTRACT FROM AUTHOR]

The mapped bases or Fake Nodes Approach (FNA), introduced in De Marchi et al. (J Comput Appl Math 364:112347, 2020c), allows to change the set of nodes without the need of resampling the function. Such scheme has been successfully applied for mitigating the Runge's phenomenon, using the S-Runge map, or the Gibbs phenomenon, with the S-Gibbs map. However, the original S-Gibbs suffers of a subtle instability when the interpolant is constructed at equidistant nodes, due to the Runge'sphenomenon. Here, we propose a novel approach, termed Gibbs–Runge-Avoiding Stable Polynomial Approximation (GRASPA), where both Runge's and Gibbs phenomena are mitigated simultaneously. After providing a theoretical analysis of the Lebesgue constant associated with the mapped nodes, we test the new approach by performing various numerical experiments which confirm the theoretical findings. [ABSTRACT FROM AUTHOR]

The paper deals with a special filtered approximation method, which originates interpolation polynomials at Chebyshev zeros by using de la Vallée Poussin filters. In order to get an optimal approximation in spaces of locally continuous functions equipped with weighted uniform norms, the related Lebesgue constants have to be uniformly bounded. In previous works this has already been proved under different sufficient conditions. Here, we complete the study by stating also the necessary conditions to get it. Several numerical experiments are also given to test the theoretical results and make comparisons to Lagrange interpolation at the same nodes. [ABSTRACT FROM AUTHOR]

Abstract When a Fourier series is used to approximate a function with a jump discontinuity, the Gibbs phenomenon always exists. This similar phenomenon exists for wavelets expansions. Based on the Gibbs phenomenon of a Fourier series and wavelet expansions of a function with a jump discontinuity, in this paper, we consider that a Gibbs phenomenon occurs for the p-ary subdivision schemes. Similar to the method of (Appl. Math. Lett. 76:157–163, 2018), we generalize the results about the stationary binary subdivision schemes in (Appl. Math. Lett. 76:157–163, 2018) to the case of p-ary subdivision schemes. By considering the masks of subdivision schemes, we obtain a sufficient condition to determine whether there exists a Gibbs phenomenon for p-ary subdivision schemes in the limit function close to the discontinuous point. This condition consists of the positivity of the partial sums of the values of the masks. By applying this condition, we can avoid the Gibbs phenomenon for p-ary subdivision schemes near discontinuity points. Finally, some examples in classical subdivision schemes are given to illustrate the results in this paper.

In this article, we provide the first systematic numerical study on, via the popular Fourier-cosine (COS) method, finite-time Gerber--Shiu functions with the risk process being driven by a generic L\'evy subordinator. These functions play a major role in modern actuarial science, and there are still many open problems left behind such as the one here of looking for a universal effective numerical scheme for them. By extending the celebrated Ballot Theorem to the continuous setting, we first derive an explicit integral expression for these functions, with an arbitrary penalty, in terms of their infinite-time counterpart. As is common in actuarial or financial practice, an advanced knowledge of the characteristic function of the driving L\'evy process facilitates the applicants of the Fourier-cosine method to this integral expression. Under some mild and practically feasible assumptions, a comprehensive and rigorous (yet demanding) error analysis is provided; indeed, up to an arbitrarily chosen error tolerance level, the numerical scheme is linear in computational complexity, which can even reach the theoretically fastest possible rate of 3; all of these are the most effective records of the contemporary state of the art in actuarial science. Finally, the effectiveness of our approximation method is illustrated through different representative numerical experiments, some of which, such as those driven by Gamma and Generalized Stable Processes, are even achieved for the first time in the literature, due to the limitations of most common existing approaches; we shall discuss this more in this article. [ABSTRACT FROM AUTHOR]

Inverse problems are of importance in many fields of science and engineering. Electromagnetic inversion deals with estimating information contained in electromagnetic measurements. The inversion scheme needs to be designed properly to compensate for Gibbs oscillations effects in the solution, and hence give better validation for the estimated quantities. In this paper an inversion methodology based on simulated annealing is presented that has the ability to extract information about electrical conductivity and dielectric permittivity of a vertically stratified medium using the scattered electric field. Furthermore, Gibbs phenomenon and its oscillation effect on the inversion solution have been studied, and an efficient approach has been developed to render more accurate estimations. Results of implementing the proposed approach and its resolution compared with the original methodology are presented. [ABSTRACT FROM AUTHOR]

This paper attempts to explain some methodological issues regarding EEG signal analysis which might lead to misinterpretation and therefore to unsubstantiated conclusions. The so called “split-alpha,” a “new phenomenon” in EEG spectral analysis described lately in few papers is such a case. We have shown that spectrum feature presented as a “split alpha” can be the result of applying improper means of analysis of the spectrum of the EEG signal that did not take into account the significant properties of the applied Fast Fourier Transform (FFT) method. Analysis of the shortcomings of the FFT method applied to EEG signal such as limited duration of analyzed signal, dependence of frequency resolution on time window duration, influence of window duration and shape, overlapping and spectral leakage was performed. Our analyses of EEG data as well as simulations indicate that double alpha spectra called as “split alpha” can appear, as spurious peaks, for short signal window when the EEG signal being studied shows multiple frequencies and frequency bands. These peaks have no relation to any frequencies of the signal and are an effect of spectrum leakage. Our paper is intended to explain the reasons underlying a spectrum pattern called as a “split alpha” and give some practical indications for using spectral analysis of EEG signal that might be useful for readers and allow to avoid EEG spectrum misinterpretation in further studies and publications as well as in clinical practice.

This study develops a new response filtering approach for recovering dynamic mechanical stresses under impact loading. For structural safety, it is important to consider the propagation of transient mechanical stresses inside structures under impact loads. Commonly, mechanical stress waves can be obtained by solving Newton's second law using explicit or implicit finite element procedures. Regardless of the numerical approach, large discrepancies called the Gibb's phenomenon are observed between the numerical solution and the analytical solution. To reduce these discrepancies and enhance the accuracy of the numerical solution, this study develops a response filtering method (RFM). The RFM averages the transient responses within split time domains. By solving several benchmark problems and analyzing the stresses in the frequency domain, it was possible to verify that the RFM can provide an improved solution that converges toward the analytical solution. A mathematical theory is also presented to correlate the relationship between the filtering length and the frequency components of the filtered stress values. [ABSTRACT FROM AUTHOR]

This paper attempts to explain some methodological issues regarding EEG signal analysis which might lead to misinterpretation and therefore to unsubstantiated conclusions. The so called " split-alpha ," a "new phenomenon" in EEG spectral analysis described lately in few papers is such a case. We have shown that spectrum feature presented as a " split alpha " can be the result of applying improper means of analysis of the spectrum of the EEG signal that did not take into account the significant properties of the applied Fast Fourier Transform (FFT) method. Analysis of the shortcomings of the FFT method applied to EEG signal such as limited duration of analyzed signal, dependence of frequency resolution on time window duration, influence of window duration and shape, overlapping and spectral leakage was performed. Our analyses of EEG data as well as simulations indicate that double alpha spectra called as " split alpha " can appear, as spurious peaks, for short signal window when the EEG signal being studied shows multiple frequencies and frequency bands. These peaks have no relation to any frequencies of the signal and are an effect of spectrum leakage. Our paper is intended to explain the reasons underlying a spectrum pattern called as a " split alpha " and give some practical indications for using spectral analysis of EEG signal that might be useful for readers and allow to avoid EEG spectrum misinterpretation in further studies and publications as well as in clinical practice. [ABSTRACT FROM AUTHOR]

In this paper, the properties of a new family of nonlinear dyadic subdivision schemes are presented and studied depending on the conditions imposed to the mean used to rewrite the linear scheme upon which the new scheme is based. The convergence, stability, and order of approximation of the schemes of the family are analyzed in general. Also, the elimination of the Gibbs oscillations close to discontinuities is proved in particular cases. It is proved that these schemes converge towards limit functions that are Hölder continuous with exponent larger than 1. The results are illustrated with several examples. [ABSTRACT FROM AUTHOR]

In this article we consider the numerical approximation of the convection-diffusion-reaction equation. One of the main challenges of designing a numerical method for this problem is that boundary layers occurring in the convection-dominated case can lead to non-physical oscillations in the numerical approximation, often referred to as Gibbs phenomena. The idea of this article is to consider the approximation problem as a residual minimization in dual norms in L q -type Sobolev spaces, with 1 < q < ∞. We then apply a non-standard, non-linear Petrov–Galerkin discretization, that is applicable to reflexive Banach spaces such that the space itself and its dual are strictly convex. Similar to discontinuous Petrov–Galerkin methods, this method is based on minimizing the residual in a dual norm. Replacing the intractable dual norm by a suitable discrete dual norm gives rise to a non-linear inexact mixed method. This generalizes the Petrov–Galerkin framework developed in the context of discontinuous Petrov–Galerkin methods to more general Banach spaces. For the convection–diffusion–reaction equation, this yields a generalization of a similar approach from the L 2 -setting to the L q -setting. A key advantage of considering a more general Banach space setting is that, in certain cases, the oscillations in the numerical approximation vanish as q tends to 1, as we will demonstrate using a few simple numerical examples. [ABSTRACT FROM AUTHOR]

This paper is devoted to the construction and analysis of some new non-linear subdivision and multiresolution schemes using the Lehmer means. Our main objective is to attain adaption close to discontinuities. We present theoretical, numerical results and applications for different schemes. The main theoretical result is related to the four-point interpolatory scheme, that we write as a perturbation of a linear scheme. Our aim is to establish a one-step contraction property that allows to prove the stability of the new scheme. Indeed with a one-step contraction property for the scheme of differences, it is possible to prove the stability of the 2D algorithm constructed using a tensor product approach. In this article, we also consider the associated three points cell-average scheme, that we will use to present some results for image compression, and a non-interpolatory scheme, that we will use to introduce an application to subdivision curves in 2D. These applications show that the use of the Lehmer mean is suitable for the design of subdivision schemes for the generation of curves and for image processing. [ABSTRACT FROM AUTHOR]

In this paper we address the problem of approximating functions with discontinuities via kernel-based methods. The main result is the construction of discontinuous kernel-based basis functions. The linear spaces spanned by these discontinuous kernels lead to a very flexible tool which sensibly or completely reduces the well-known Gibbs phenomenon in reconstructing functions with jumps. For the new basis we provide error bounds and numerical results that support our claims. The method is also effectively tested for approximating satellite images. [ABSTRACT FROM AUTHOR]