Non-periodic Functions Research Articles

Review In Fourier series

It is known that Fourier series are one of linear transformations and have many uses in mathematics, physics, engineering…etc. It is used to influence on periodic functions that repeats in a certain period to transform it into a sum of sine and cosine functions also used for non-periodic functions, as we will notice, and it has many other uses. In this article, I study, with examples, the important uses of Fourier series in physics, engineering, and mathematics, with proofs and examples where it necessary. One of results that we have reached in this article is that use Fourier series to facilitated a solution of differential equations in particular partial differential equations by the sinusoidal part of it, also obtaining the best approximation element for a particular function by means of its mathematical mean which called Feger operators. Moreover, there are more than one auxiliary conclusion for, example if a function has two different periods, then the sum of these periods, its subtraction, and its multiplying by constant, will be a new period for this function. Here, I refer to another conclusion that, Fourier series itself do not give the best approximation element for a particular function.

Uniform Convergence of Fourier Series for Hölder Continuous Functions and Functions of Bounded Variation

This article discusses two significant results about the uniform convergence of Fourier series. One is the uniform convergence of the Fourier series for Hölder-α(α>0) continuous functions, while the other is the uniform convergence of the Fourier series for functions that can be expressed as the sum of finitely many continuous monotonic functions. The two results are particularly useful. Because Hölder continuity condition ensures a specific regularity that facilitates the analysis and provides a robust framework for proving uniform convergence and many practical functions can be decomposed into monotonic components, making the theorem broadly applicable. By providing detailed proofs and applications, this study demonstrates the significant implications of uniform convergence in both theoretical and practical contexts. Moreover, this work paves the way for future research in higher-dimensional Fourier analysis, non-periodic functions, and adaptive Fourier techniques, underscoring the ongoing relevance and versatility of Fourier series in modern mathematics. Several well-known results emerge as corollaries of these findings, highlighting the robustness and applicability of these theorems in Fourier analysis.

Functional description of fiber orientation in paperboard based on orientation tensors resulting from μ-CT scans

Modern μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\upmu$$\\end{document}-CT scans offer non-destructive three-dimensional microstructure analysis of various materials. Despite small density contrasts, this technology also accurately captures the complex network structure of paper and paperboard. It provides detailed insight into the fiber orientation distribution in paper, which was previously unattainable with existing measurement methods. The image analysis results are applicable for numerical fiber network models, and simulations based on these models can significantly improve our understanding of the microstructural influence on macrostructural paper properties, such as strength, stiffness, and curl tendency. This work presents a new method for investigating the microstructural fiber orientation in paperboard. Orientation tensors obtained from μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\upmu$$\\end{document}-CT scan image processing are extracted and evaluated. Based on the direction of the orientation tensor’s first eigenvector, the fiber orientation distribution is determined and then approximated by periodic and non-periodic probability density functions (PDFs). In this study, 40 commercial paperboard samples, each consisting of three plies, were analyzed. From a given set of commonly used PDFs, the best fitting ones have been identified based on their original location in the paper roll and on their ply. It was found that the von Mises PDF provided the most accurate representation of fiber orientation distribution in the middle ply, while the Elliptical PDF was the most suitable for the outer plies. Moreover, a more pronounced fiber orientation anisotropy was observed at the edges of the paper roll than in its center. The best PDFs and their function parameters are provided to allow for direct usage in numerical microstructure models of paperboard, thus enhancing their representativeness.

Variable transformations in combination with wavelets and ANOVA for high-dimensional approximation

We use hyperbolic wavelet regression for the fast reconstruction of high-dimensional functions having only low-dimensional variable interactions. Compactly supported periodic Chui-Wang wavelets are used for the tensorized hyperbolic wavelet basis on the torus. With a variable transformation, we are able to transform the approximation rates and fast algorithms from the torus to other domains. We perform and analyze scattered data approximation for smooth but arbitrary density functions by using a least squares method. The corresponding system matrix is sparse due to the compact support of the wavelets, which leads to a significant acceleration of the matrix vector multiplication. For non-periodic functions, we propose a new extension method. A proper choice of the extension parameter together with the piecewise polynomial Chui-Wang wavelets extends the functions appropriately. In every case, we are able to bound the approximation error with high probability. Additionally, if the function has a low effective dimension (i.e., only interactions of a few variables), we qualitatively determine the variable interactions and omit ANOVA terms with low variance in a second step in order to decrease the approximation error. This allows us to suggest an adapted model for the approximation. Numerical results show the efficiency of the proposed method.

New estimates related with the best polynomial approximation

In some old results, we find estimates the best approximation \(E_{n,p}(f)\) of a periodic function satisfying \(f^{(r)}\in\mathbb{L}^p_{2\pi}\) in terms of the norm of \(f^{(r)}\) (Favard inequality). In this work, we look for a similar result under the weaker assumption \(f^{(r)}\in \mathbb{L}^q_{2\pi}\), with \(1<q<p<\infty\). We will present inequalities of the form \(E_{n,p}(f)\leq C(n)\Vert D^{(r)}f\Vert_q\), where \(D^{(r)}\) is a differential operator. We also study the same problem in spaces of non-periodic functions with a Jacobi weight.

Deep learning soliton dynamics and complex potentials recognition for 1D and 2D [formula omitted]-symmetric saturable nonlinear Schrödinger equations

In this paper, we firstly extend the physics-informed neural networks (PINNs) to learn data-driven stationary and non-stationary solitons of 1D and 2D saturable nonlinear Schrödinger equations (SNLSEs) with two fundamental PT-symmetric Scarf-II and periodic potentials in optical fibers. Secondly, the data-driven inverse problems are studied for PT-symmetric potential functions discovery rather than just potential parameters in the 1D and 2D SNLSEs. Particularly, we propose a modified PINNs (mPINNs) scheme to identify directly the PT potential functions of the 1D and 2D SNLSEs by the solution data. And the inverse problems about 1D and 2D PT-symmetric potentials depending on propagation distance z are also investigated using mPINNs method. We also identify the potential functions by the PINNs applied to the stationary equation of the SNLSE. Furthermore, two network structures are compared under different parameter conditions such that the predicted PT potentials can achieve the similar high accuracy. These results illustrate that the established deep neural networks can be successfully used in 1D and 2D SNLSEs with high accuracies. Moreover, some main factors affecting neural networks performance are discussed in 1D and 2D PT Scarf-II and periodic potentials, including activation functions, structures of the networks, and sizes of the training data. In particular, twelve different nonlinear activation functions are in detail analyzed containing the periodic and non-periodic functions such that it is concluded that selecting activation functions according to the form of solution and equation usually can achieve better effect.

The Use of Trigonometric Series for the Study of Isotropic Beam Deflection

The average deformed fiber is a continuous and smooth function of the fourth order. The deflection and rotation of beams can be determined by various methods available in the literature. Thus, in this paper, the expression of the average deformed fiber is defined in advance, then it is considered an infinite sum of sinusoidal functions that are successively evaluated using sinusoidal trigonometric series if it is considered a periodic function, or using Fourier series if it is considered a non-periodic function. The method is examined by solving several beam problems. The results indicate that the method can be used with confidence for solving any bending beam problem.

Compartmental Unpredictable Functions

There is a huge family of recurrent functions, which starts with equilibria and ends with Poisson stable functions. They are fundamental in theoretical and application senses, and they admit a famous history. Recently, we have added the unpredictable functions to the family. The research has been performed in several papers and books. Obviously, theoretical and application merits of functions increase if one provides rigorously approved efficient methods of construction of concrete examples, as well as their numerical simulations. In the present study, we met the challenges for unpredictability by considering functions of two variables on diagonals. Algorithms have been created, and they are both deterministic and random. Characteristics are introduced to evaluate contributions of periodic and unpredictable components to the dynamics, and they are clearly illustrated in graphs of the functions. Definitions of non-periodic compartmental functions are provided as suggestions for the research in the future.

Non-periodic solutions of the Goła̧b–Schinzel type functional equation

We determine the solutions of the Goła̧b-Schinzel type functional equation in the class of non-periodic functions. Applying this result we give a positive answer to the problem raised by E. Jabłońska (Aequationes Math 87:125–133, 2014).

An exact solution to the Fourier Transform of band-limited periodic functions with nonequispaced data and application to non-periodic functions

The need to Fourier transform data sets with irregular sampling is shared by various domains of science. This is the case for example in astronomy or seismology. Iterative methods have been developed that allow to reach approximate solutions. Here an exact solution to the problem for band-limited periodic signals is presented. The exact spectrum can be deduced from the spectrum of the non-equispaced data through the inversion of a Toeplitz matrix. The result applies to data of any dimension. This method also provides an excellent approximation for non-periodic band-limit signals. The method allows to reach very high dynamic ranges (1013 with double-float precision) which depend on the regularity of the samples.

Fourier Continuation Discontinuous Galerkin Methods for Linear Hyperbolic Problems

Fourier continuation (FC) is an approach used to create periodic extensions of non-periodic functions to obtain highly-accurate Fourier expansions. These methods have been used in partial differential equation (PDE)-solvers and have demonstrated high-order convergence and spectrally accurate dispersion relations in numerical experiments. Discontinuous Galerkin (DG) methods are increasingly used for solving PDEs and, as all Galerkin formulations, come with a strong framework for proving the stability and the convergence. Here we propose the use of FC in forming a new basis for the DG framework.

On some quasi-periodic approximations

Trigonometric approximation or interpolation of a non-smooth function on a finite interval has poor convergence properties. This is especially true for discontinuous functions. The case of infinitely differentiable but non-periodic functions with discontinuous periodic extensions onto the real axis has attracted interest from many researchers. In a series of works, we discussed an approach based on quasi-periodic trigonometric basis functions whose periods are slightly bigger than the length of the approximation interval. We proved validness of the approach for trigonometric interpolations. In this paper, we apply those ideas to classical Fourier expansions.

Visualizing Solutions of the One-Dimensional Schrödinger Equation Using a Finite Difference Method

FINDIF is a Windows application that numerically solves the one-dimensional (1D) Schrödinger equation and displays the eigenstates, eigenvalues, and probability density of the system. FINDIF accepts both nonperiodic and periodic 1D potential energy functions as input and uses the finite difference method to evaluate the energy of the quantum system. This Technology Report illustrates the use of FINDIF with applications, such as the classic 1D particle-in-a-box, the particle-in-a-box with internal barrier, the modified Kronig–Penney model of a linear array of rectangular wells, the harmonic oscillator with visualization of the eigenstates and tunneling effect, the anharmonic Morse potential of the Ar dimer, and the periodic torsional potential for internal rotation of ethane. Students can explore other quantum chemical examples by considering both realistic and fictitious model potential energy functions, making outcome predictions before running FINDIF calculations, visualizing the results afterward, and then comparing their predictions with the results they observe. Such exercises assist students as they develop insights into the behavior and properties of quantum mechanical systems.

A novel approach to radially global gyrokinetic simulation using the flux-tube code stella

A novel approach to global gyrokinetic simulation is implemented in the flux-tube code stella. This is done by using a subsidiary expansion of the gyrokinetic equation in the perpendicular scale length of the turbulence, originally derived by Parra and Barnes (2015) [23], which allows the use of Fourier basis functions while enabling the effect of radial profile variation to be included in a perturbative way. Radial variation of the magnetic geometry is included by utilizing a global extension of the Grad-Shafranov equation and the Miller equilibrium equations which is obtained through Taylor expansion. Radial boundary conditions that employ multiple flux-tube simulations are also developed, serving as a more physically motivated replacement to the conventional Dirichlet radial boundary conditions that are used in global simulation. It is shown that these new boundary conditions eliminate much of the numerical artefacts generated near the radial boundary when expressing a non-periodic function using a spectral basis. We then benchmark the new approach both linearly and non-linearly using a number of standard test cases.

Analysis on Cattaneo-Christov heat flux in three-phase oscillatory flow of non-Newtonian fluid through porous zone bounded by hybrid nanofluids

The primary aspect of this study is a theoretical analysis of heat transfer enhancement in the unsteady three-phase oscillatory flow of Casson fluid bounded by the layers of nanofluid and hybrid nanofluids flowing adjacent to the walls of a long-infinite horizontal composite channel. Stoke’s model is employed to define Casson fluid, and one phase model is implemented for nanofluid. Moreover, Cattaneo–Christov heat flux model is also considered. This model amends Fourier’s law of heat transport using thermal relaxation time due to which heat passes through the channel within a finite limit. Analytical solutions are attained through a regular perturbation technique by employing periodic and non-periodic functions and presented through graphical illustrations by implementing the MATHEMATICA program. The outcomes are exhibited with three types of nanomaterials such as copper Cu , copper oxide CuO and copper alumina Cu–Al 2 O 3 with base fluid as water. It is noted that the momentum and heat transport is significant in the case of Casson fluid with porous zone sandwiched between Cu water NFs as compared to Cu–CuO–H 2 O (phase I) and Cu–Al 2 O 3 –H 2 O (phase III) HNFs. Moreover, in the case of injection, it is observed that the effects of Cu nanomaterials are dominant on temperature. On the other side, in suction, the impacts of Cu–Al 2 O 3 nanomaterials are significant on temperature. The results are also compared with a well-known numerical method named as finite element method.

Power of Dirichlet kernels and approximation by discrete linear operators $\textit{I}$: direct results

The second and third powers of the Dirichlet kernel are used to construct discrete linear operators for the approximation of continuous periodic functions. An estimate of the rate of convergence is given. Approximation of non-periodic functions are also considered.

FC-based shock-dynamics solver with neural-network localized artificial-viscosity assignment

This paper presents a spectral scheme for the numerical solution of nonlinear conservation laws in non-periodic domains under arbitrary boundary conditions. The approach relies on the use of the Fourier Continuation (FC) method for spectral representation of non-periodic functions in conjunction with smooth localized artificial viscosity assignments produced by means of a Shock-Detecting Neural Network (SDNN). Like previous shock capturing schemes and artificial viscosity techniques, the combined FC-SDNN strategy effectively controls spurious oscillations in the proximity of discontinuities. Thanks to its use of a localized but smooth artificial viscosity term, whose support is restricted to a vicinity of flow-discontinuity points, the algorithm enjoys spectral accuracy and low dissipation away from flow discontinuities, and, in such regions, it produces smooth numerical solutions—as evidenced by an essential absence of spurious oscillations in level set lines. The FC-SDNN viscosity assignment, which does not require use of problem-dependent algorithmic parameters, induces a significantly lower overall dissipation than other methods, including the Fourier-spectral versions of the previous entropy viscosity method. The character of the proposed algorithm is illustrated with a variety of numerical results for the linear advection, Burgers and Euler equations in one and two-dimensional non-periodic spatial domains.

Grouped Transformations and Regularization in High-Dimensional Explainable ANOVA Approximation

In this paper we propose a tool for high-dimensional approximation based on trigonometric polynomials where we allow only low-dimensional interactions of variables. In a general high-dimensional setting, it is already possible to deal with special sampling sets such as sparse grids or rank-1 lattices. This requires black-box access to the function, i.e., the ability to evaluate it at any point. Here, we focus on scattered data points and grouped frequency index sets along the dimensions. From there we propose a fast matrix-vector multiplication, the grouped Fourier transform, for high-dimensional grouped index sets. Those transformations can be used in the application of the previously introduced method of approximating functions with low superposition dimension based on the analysis of variance (ANOVA) decomposition where there is a one-to-one correspondence from the ANOVA terms to our proposed groups. The method is able to dynamically detected important sets of ANOVA terms in the approximation. In this paper, we consider the involved least-squares problem and add different forms of regularization: Classical Tikhonov-regularization, namely, regularized least squares and the technique of group lasso, which promotes sparsity in the groups. As for the latter, there are no explicit solution formulas which is why we applied the fast iterative shrinking-thresholding algorithm to obtain the minimizer. Moreover, we discuss the possibility of incorporating smoothness information into the least-squares problem. Numerical experiments in under-, overdetermined, and noisy settings indicate the applicability of our algorithms. While we consider periodic functions, the idea can be directly generalized to non-periodic functions as well.

Two-Dimensional Fourier Continuation and Applications

This paper presents a method (2D-FC) for construction of bi-periodic extensions of smooth non-periodic functions defined over general two-dimensional smooth domains. The approach can be directly generalized to domains of any given dimensionality, and even to non-smooth domains, but such generalizations are not considered here. The 2D-FC extensions are produced in a two-step procedure. In the first step the one-dimensional Continuation method is applied along a discrete set of outward boundary-normal directions to produce, along such directions, continuations that vanish outside a narrow interval beyond the boundary. Thus, the first step of the algorithm produces blending-to-zero along normals for the given function values. In the second step, the extended function values are evaluated on an underlying Cartesian grid by means of an efficient, high-order boundary-normal interpolation scheme. A Continuation expansion of the given function can then be obtained by a direct application of the two-dimensional FFT algorithm. Algorithms of arbitrarily high order of accuracy can be obtained by this method. The usefulness and performance of the proposed two-dimensional Continuation method are illustrated with applications to the Poisson equation and the time-domain wave equation within a bounded domain. As part of these examples the novel Fourier Forwarding solver is introduced which, propagating plane waves as they would in free space and relying on certain boundary corrections, can solve the time-domain wave equation and other hyperbolic partial differential equations within general domains at computing costs that grow sublinearly with the size of the spatial discretization.

ОПТИМАЛЬНI МЕТОДИ ВIДНОВЛЕННЯ МIШАНИХ ПОХIДНИХ НЕПЕРIОДИЧНИХ ФУНКЦIЙ

The problem of numerical differentiation for non-periodic bivariate functions is investigated. For the recovering mixed derivatives of such functions an approach on the base of truncation method is proposed. The constructed algorithms deal with Legendere polynomials, the degree of which is chosen so as to minimize the approximation error. It is established that these algorithms are order-optimal both in terms of accuracy and in the sense of the amount of Galerkin information involved.