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Classes of -functions with fixed rate of decrease of their Fourier coefficients are considered. For each class , it is shown that there exists a (recovery) set with arbitrarily small measure such that any function in can be recovered from its values on . A formula for evaluation of the Fourier coefficients of such a function from its values on is given. In addition, it is shown that, for any -function, a function-specific recovery set can be found. The problem of recovery of general trigonometric series from the Zygmund classes which converge to summable functions on such sets is also solved. Bibliography: 10 titles.

We find sufficient conditions which are in a sense best possible that must be satisfied by the functions of an orthonormal system in order for the Fourier coefficients of functions of bounded variation to satisfy the hypotheses of the Men’shov–Rademacher theorem. We also prove a theorem saying that every system contains a subsystem with respect to which the Fourier coefficients of functions of bounded variation satisfy those hypotheses. The results obtained complement and generalize the corresponding results in [1].

We construct an integrable function whose Fourier series possesses the following property. After an appropriate choice of signs of the coefficients of this series, the partial sums of the resulting series are dense in , .

We develop a systematic approach for proving the existence of choreographic solutions in the gravitational $n$ body problem. Our main focus is on spatial torus knots: that is, periodic motions where the positions of all $n$ bodies follow a single closed which winds around a $2$-torus in $\mathbb{R}^3$. After changing to rotating coordinates and exploiting symmetries, the equation of a choreographic configuration is reduced to a delay differential equation (DDE) describing the position and velocity of a single body. We study periodic solutions of this DDE in a Banach space of rapidly decaying Fourier coefficients. Imposing appropriate constraint equations lets us isolate choreographies having prescribed symmetries and topological properties. Our argument is constructive and makes extensive use of the digital computer. We provide all the necessary analytic estimates as well as a working implementation for any number of bodies. We illustrate the utility of the approach by proving the existence of some spatial choreographies for $n=4,5,7$, and $9$ bodies.

We study problems of the absolute convergence of Fourier series of differentiable functions with respect to general orthonormal systems (ONS). We obtain results which show that under certain conditions on functions in the ONS, the Fourier series of an arbitrary function in a certain differentiability class is absolutely convergent. We finally note that our results cannot be improved.

We establish an asymptotic formula for the rate of approximation of Fourier series of individual periodic functions by linear averages with an error , . This formula is applicable to the means of Riesz, Gauss–Weierstrass, Picard and others. The result is new even for the arithmetic means of partial Fourier sums. We use the formula to determine the asymptotic behaviour of functions in a certain class. Separately, we consider the case of positive integral convolution operators.

We consider the nonlinear Schroedinger equation on the one dimensional torus, with a defocousing polynomial nonlinearity and study the dynamics corresponding to initial data in a set of large measure with respect to the Gibbs measure. We prove that along the corresponding solutions the modulus of the Fourier coefficients is approximately constant for times of order $\beta^{2+\varsigma}$, $\beta$ being the inverse of the temperature and $\varsigma$ a positive number (we prove $\varsigma= 1/10$). The proof is obtained by adapting to the context of Gibbs measure for PDEs some tools of Hamiltonian perturbation theory.

We consider functions represented as trigonometric series with general monotone Fourier coefficients. The main result of the paper is the equivalence of the modulus of smoothness, , of such functions to certain sums of their Fourier coefficients. As applications, for such functions we give a description of the norm in the Besov space and sharp direct and inverse theorems in approximation theory. Bibliography: 34 titles.

We prove an exponential integral estimate for the quadratic partial sums of multiple Fourier series on large sets that implies some new properties of Fourier series., 16 pages

We consider the semilinear wave equation $V(x) u_{tt} -u_{xx}+q(x)u = \pm f(x,u)$ for three different classes (P1), (P2), (P3) of periodic potentials $V,q$. (P1) consists of periodically extended delta-distributions, (P2) of periodic step potentials and (P3) contains certain periodic potentials $V,q\in H^r_{\per}(\R)$ for $r\in [1,3/2)$. Among other assumptions we suppose that $|f(x,s)|\leq c(1+ |s|^p)$ for some $c>0$ and $p>1$. In each class we can find suitable potentials that give rise to a critical exponent $p^\ast$ such that for $p\in (1,p^\ast)$ both in the "+" and the "-" case we can use variational methods to prove existence of time-periodic real-valued solutions that are localized in the space direction. The potentials are constructed explicitely in class (P1) and (P2) and are found by a recent result from inverse spectral theory in class (P3). The critical exponent $p^\ast$ depends on the regularity of $V, q$. Our result builds upon a Fourier expansion of the solution and a detailed analysis of the spectrum of the wave operator. In fact, it turns out that by a careful choice of the potentials and the spatial and temporal periods, the spectrum of the wave operator $V(x)\partial_t^2-\partial_x^2+q(x)$ (considered on suitable space of time-periodic functions) is bounded away from $0$. This allows to find weak solutions as critical points of a functional on a suitable Hilbert space and to apply tools for strongly indefinite variational problems.

Regression analysis is a method of analysis to determine the relationship between the response and the predictor variables. There are three approaches in regression analysis, namely the parametric, nonparametric, and semiparametric approaches. Biresponse Semiparametric regression model is a regression model that uses a combination approach between parametric and nonparametric components, where two response variables are correlated with each other. For data cases with several predictor variables, different estimation technique approaches can be used for each variable. In this study, the parametric component is assumed to be linear. At the same time, the nonparametric part is approached using a mixture of three estimation techniques, namely, spline truncated, Fourier series, and the kernel. The unknown data pattern is assumed to follow the criteria of each of these estimation techniques. The spline is used when the data pattern tends to change at certain time intervals, the Fourier series is used when the data pattern tends to repeat itself, and the kernel is used when the data does not have a specific way. This study aims to obtain parameter estimates for the mixed semiparametric regression model of spline truncated, Fourier series, and the kernel on the biresponse data using the Weighted Least Square (WLS) method. The formed model depends on the selection of knot points, oscillation parameters, and optimal bandwidth. The best model is based on the smallest Generalized Cross Validation (GCV).

A new method for designing stellarators with variable-thickness perpendicular permanent magnets has been developed, based on a two-step magnet design strategy that we previously proposed for designing stellarators with standardized magnet blocks. Our design strategy uses a ‘local compensation’ method to obtain an initial magnet design and a ‘global fine-tuning’ method to further optimize the initial design. The new method is compared to the previously proposed Fourier decomposition method, and the results indicate that this new method can yield a similar magnet design for an N fp = 2 quasi-axisymmetric stellarator at a lower computational cost. High accuracy is achieved, as demonstrated by a normal magnetic field square on the plasma surface of χ B 2 = 3.43 × 1 0 − 8 T 2 m 2 , a maximum residual of B n = 3.44 G s for a ∼1 T total field, and a flux-surface-averaged residual B n of B n / B = 3.16 × 1 0 − 5 relative to the total field. Furthermore, the new method can automatically yield a magnet design with large areas of continuous vacancies for ports and plasma access, which is more flexible than the Fourier decomposition method. These results indicate that the new method is robust and effective when used to design stellarators with variable-thickness perpendicular permanent magnets, and, most importantly, that the two-step magnet design strategy has great potential to develop permanent magnet design methods suitable for various magnet layouts.

A Fourier series is one of the methods to represent a function as a summation of orthogonal functions. In this paper, we give a detailed proof of a property regarding the behavior of a function at its continuity points. Finally, we show that an integrable function vanishes at its continuity points if all of its Fourier coefficients vanish.

The a im for this paper is to fit Fourier series on the daily rainfall data to describe the rainfall patterns and to identify the time frame of the wettest and the driest period for the studied stations. The estimated parameter of the Fourier series in modelling the mean daily rainfall amount received were obtained using the method of Generalized Linear Model and the performance was evaluated using the analysis of deviance. Following the objective, the daily rainfall data ranges from 41 to 58 years over four stations in Sarawak namely Kuching, Miri, Bintulu a nd Sibu are studied. Results showed three harmonics is appropriate in modelling the mean daily rainfall received for Kuching, Sibu and Miri stations. However, only two harmonics was found to be fitted the mean daily rainfall amount received significantly for Bintulu station. The wettest month of the studied stations wa s estimated to occur in late December to early January, whereas the driest month is in March for Miri station, April for Bintulu, July for Kuching a nd June for Sibu station. The information gained from this study can be used to monitor the occurrence of flood and drought events in the area so that future action can be taken to reduce the impacts.

This paper is devoted to the determination of the trigonometric Fourier series by the method of four-dimensional mathematics. In this paper, a new four-dimensional method for evaluation of the sums containing trigonometric function is proposed. Current work provides a first study and finding towards Fourier’s series in four-dimensional space.

In this article, we use a new multiplication to propose a definition of fractional Fourier series, regarding the Jumarie type of modified Riemann-Liouville fractional derivatives. Furthermore, several examples are provided to illustrate the applications of fractional Fourier series.

Steady state (SS) and transient temperature-rise in tissue from radiofrequency exposure forms the underlying basis for limits in international exposure guidelines. Periodically pulsed or intermittent exposures form a special case of having both peak and average levels, producing temperature-rise oscillations in the SS. Presented here is a method for determining tissue temperature-rise for periodic specific absorption rate (SAR) modulation having arbitrary waveform. It involves the finite difference solution of a form of the Pennes Bioheat Transfer equation (BHTE) and uses the concept of the transfer function and the Fast Fourier Transform (FFT). The time-dependent BHTE is converted to a SS harmonic version by assuming that the time-dependent SAR waveform and tissue temperature can both be represented by Fourier series. The transfer function is obtained from solutions of the harmonic BHTE for an assumed SAR waveform consisting of periodic impulses. The temperature versus time response for an arbitrary periodic SAR waveform is obtained from the inverse FFT of the product of the transfer function and the FFT of the actual SAR waveform. This method takes advantage of existing FFT algorithms on most computational platforms and the ability to store the transfer function for later re-use. The transfer function varies slowly with harmonic number, allowing interpolation and extrapolation to reduce the computational effort. The method is highly efficient for the case where repeated temperature-rise calculations for parameter variations in the SAR waveform are sought. Examples are given for a narrow, circularly symmetric beam incident on a planar skin/fat/muscle model with rectangular, triangular and cosine-pulsed SAR modulation waveforms. Calculations of temperature-rise crest factor as a function of rectangular pulse duty factor and pulse repetition frequency for the same exposure/tissue model are also presented as an example of the versatility of the method.

This report mainly explores three major parts, namely, periodic discrete-time signals, generation of periodic Fourier signals and analysis of quasi-periodic Fourier signals. We use Matlab to analyze the signal and generate Excel spreadsheet. The waveforms of different frequencies are drawn by Matlab, and the waveforms are analyzed. In addition, the formula for synthesizing periodic signals is simplified and demonstrated here.

It is well-known that grey system theory was put forward in the statistical literature to overcome the problem of partially unknown parameters, therefore it is given the attention of the researchers in many scientific areas. Therefore, many methods have been presented in which the grey system is combined with the Fourier series and others for more accurate prediction. This paper presents a methodology to modify the method of least squares based on Fourier series residuals that are computed by using grey system theory. The objective of this methodology is to improve the estimates of a modified exponential growth model. The performance of the proposed method by using simulation is compared with two famous methods and the results show our proposed method outperforms more than others and it is highly efficient and reliable. On the practical, the new method was used to predict the price of a barrel of crude oil for the OPEC basket for the period from Jul 2019 to Dec 2019, and It can be seen that OPEC basket prices will decline in the end of 2019.

We present results describing some properties of the Fourier coefficients of functions with respect to general orthonormal systems (ONS). We note that good differential properties of the functions do not ensure the `good' behaviour of the Fourier coefficients (in the sense of convergence to zero) of these functions with respect to general ONS. We find conditions on the functions forming an ONS (, , for which the series of Fourier coefficients of the functions , where , are absolutely convergent. We consider relationships between ONS, that is, problems of absolute independence for orthonormal systems.

While resistors with memory, sometimes called memristive elements (such as ReRAM cells), are often studied under conditions of periodic driving, little attention has been paid to the Fourier features of their memory response (hysteresis). Here we demonstrate experimentally that the hysteresis of memristive systems can be unambiguously distinguished from the linear or non-linear response of systems without hysteresis by the values of certain Fourier series coefficients. We also show that the Fourier series convergence depends on driving conditions, and introduce a measure of hysteresis. These results may be used to quantify the memory content of resistive memories, and tune their Fourier spectrum according to the excitation signal.

We present a simple approach, different from approaches in the literature, for calculating the magnetic field of a finite solenoid with a circular cross section, carrying an ideal surface current with an ordinary azimuthal component and a small axial component due to nonideal winding. The results are given as an infinite series whose terms are elementary functions or simple polynomials rather than other infinite series. It is easy to deal with the results because the polynomials involved are partial binomial expansions. These series converge rapidly everywhere, except near the edges of the solenoid. From these results, it is very easy to derive some general properties of the magnetic field, to reduce the field to greatly simplified forms in various special regions, and to address the special cases of infinite and semi-infinite solenoids. The azimuthal field due to nonideal winding is shown to be small everywhere. A solenoid with a small but finite thickness in the radial direction is investigated in detail, and we present simple analytic approximate results for its magnetic field. A comparison with an exact numerical evaluation shows that these are very good approximations.

The pattern of the relationship between the response variable and the unknown predictor can be determined using a nonparametric regression approach. Nonparametric regression allows it to be used for response variables following different curves between one variable and another. In this regression, there are several types of approaches including the kernel, spline, and Fourier series. In its use, there is not only one type of approach, but can be in the form of a mixture, such as a mixture of a spline and Fourier series, a kernel and a Fourier series, and so on. In this study, in modeling the HDI cases in East Java Province, nonparametric regression mixed kernels, Spline Truncated, and Fourier series were used. The results of research that have been applied to HDI in East Java with the predictor variables for APM SMA, Morbidity, and GRDP per Capita are using the mixed kernel, Spline Truncated, and Fourier series nonparametric regression approach with three-knot points and three oscillations. Good because the coefficient of determination of the estimator is 75.1041%.

Regression modelling is one Statistical methods that be used to investigate the relationship between predictor variable and response variable. In regression modelling can be estimated with three approaches, such as parametric, nonparametric and semiparametric regression. In this case, we concentrated to elaborate semiparametric regression. Semiparametric regression consists of parametric and nonparametric component. This research examined semiparametric regression model with Fourier series estimator for longitudinal data. By minimizing Weighted Least Square (WLS), the Fourier series estimator depends on the oscillation parameter. The result is the estimator for parameter and curve regression, that be used to model with real data. The optimal model is selected based on minimum Generalized Cross Validation (GCV) which affects the small value of Mean Square Error (MSE) and high determination coefficient so that the model can be used further as estimation and prediction.

Nonparametric regression aims to determine the relationship between response and predictor when the data does not follow a specific pattern. Fourier series is a nonparametric regression approach which has the flexibility to follow the characteristics of data. The purpose of this study is to obtain the estimator of the nonparametric regression using Fourier series and apply the model to the fertility data. The fertility rate represented by children ever born is one of the demographic factors that determine the decline in population growth rate. The data were obtained from the Indonesia Demographic and Health Survey 2017 with children ever born as a response. The predictors are the proportion of women graduating from junior high school, the proportion of women having sex before the age of 18, the proportion of women using a modern contraceptive method, and the infant mortality rate. The relationship between response and predictors tends to have a repetitive pattern with a certain trend. The best nonparametric regression model of children ever born in Indonesia is obtained by using 3 oscillation parameters for each predictor variable with GCV = 0.0534 and R-square = 80.04%.

To obtain engineering-feasible designs of stellarators with permanent magnets and simplified coils, a new algorithm has been developed based on Fourier decomposition and surface magnetic charges method. The strong toroidal fields in a quasi-axisymmetric stellarator are still generated by coils. The permanent magnets are designed to compensate the normal magnetic field B n on the plasma surface ∂P created by the coils and plasma. The normal magnetic fields created by the permanent magnets B pmn are calculated as the difference between the magnetic fields created by the surface magnetic charges on the inner surface ∂D and the outer surface ∂D h of the magnets. The Fourier coefficients of the magnet thickness function h θ , ϕ are computed through matrix division operation based on the least square principle with dominant Fourier components selected through 2D Fourier transformation of B pmn. The residual uncompensated B n is minimized through iteration to progressively optimize the thickness function. This new algorithm has been successfully applied to design the permanent magnets of an l = 2 quasi-axisymmetric stellarator with background magnetic field created by 12 identical circular planar coils for demonstrations. High accuracy has been achieved, allowing for a flux-surface-averaged residual B n relative to the total field B n / B ∼ 1.3 × 1 0 − 5 and a maximum residual B n of less than 2 Gs for ∼1 T total field. This new design has some advantages in engineering implementations: all permanent magnet pieces have the same remanence B r; only one single layer of magnets are mounted perpendicular to the winding surface; the magnets can be easily inserted into the cells of a gridded frame attached to the winding surface and fixed with springs from the back, which greatly simplifies the manufacture, assembly and maintenance of the magnets, and thus facilitates precision control and cost reduction.

A quantitative model of the penetration of a quasistationary electric field from the Earth’s surface into the ionosphere with an inclined magnetic field is developed. A two-dimensional model is constructed and applied when the earthquake preparation zone is extended along a magnetic meridian. In the main part of the atmosphere up to a height of 50 km, we use an empirical conductivity model. Above 90 km we use the model, which is based on the empirical models IRI, MSISE and IGRF. In the rest layer the values of the components of the conductivity tensor are interpolated. The conjugate ionosphere is included to the model since the field-aligned conductivity in the magnetosphere is regarded as an infinite one. We solve the electric conductivity problem using Fourier transform at each height and numerically solve the systems of ordinary differential equations for the Fourier coefficients. The space distributions of the electric field strength and the current density are calculated. The current lines in both hemispheres are plotted. The electric field strength in the ionosphere is in the microvolt per meter range when it is about 100 mV/m in the air near ground. The known approximate estimates of the decrease of the electric field penetrating into the ionosphere with the decrease of the magnetic latitude are confirmed and detailed. The penetration of the electric field and current from the ground to the ionosphere through the atmospheric conductor cannot be a physical process which creates observed ionospheric precursors of earthquakes.

This paper aims to investigate the response mechanism of faulty gears, and further proposes a resonance-based bandwidth Fourier decomposition (RBBFD) method for gearbox fault identification. To reveal the vibration patterns of the gear-pair system, a dynamic model with time-varying stiffness is established to obtain the transient response excited by local damage. The analysis results demonstrate that the impulse response of a faulty gear is essentially carried by the natural frequency instead of the widely recognized meshing frequency and its harmonies. After that, the RBBFD method is developed based on the dynamic characteristics. In the proposed method, the resonance frequencies of the gear system are estimated according to the maximum point distribution of the Fourier spectrum, and then the Fourier spectrum bandwidth optimization algorithm is introduced to decompose the vibration signal into mono-components centered on the estimated natural frequencies. A new criterion termed the characteristic frequency ratio is utilized to measure the fault-related degree of the decomposed modes, and the fault type is identified by Hilbert demodulation. The effectiveness of RBBFD is verified through two diagnostic cases. The comparison with several state-of-art works validates the superiority of the proposed method in gearbox fault diagnosis.

With Aperture synthesis (AS) technique, a number of small antennas can assemble to form a large telescope which spatial resolution is determined by the distance of two farthest antennas instead of the diameter of a single-dish antenna. Different from direct imaging system, an AS telescope captures the Fourier coefficients of a spatial object, and then implement inverse Fourier transform to reconstruct the spatial image. Due to the limited number of antennas, the Fourier coefficients are extremely sparse in practice, resulting in a very blurry image. To remove/reduce blur, "CLEAN" deconvolution was widely used in the literature. However, it was initially designed for point source. For extended source, like the sun, its efficiency is unsatisfied. In this study, a deep neural network, referring to Generative Adversarial Network (GAN), is proposed for solar image deconvolution. The experimental results demonstrate that the proposed model is markedly better than traditional CLEAN on solar images., 14 pages, 6 figures, 2 tables

Dynamic electrical impedance tomography-based image reconstruction using conventional algorithms such as the extended Kalman filter often exhibits inferior performance due to the presence of measurement noise, the inherent ill-posed nature of the problem and its critical dependence on the selection of the initial guess as well as the state evolution model. Moreover, many of these conventional algorithms require the calculation of a Jacobian matrix. This paper proposes a dynamic oppositional biogeography-based optimization (OBBO) technique to estimate the shape, size and location of the non-stationary region boundaries, expressed as coefficients of truncated Fourier series, inside an object domain using electrical impedance tomography. The conductivity of the object domain is assumed to be known a priori. Dynamic OBBO is a novel addition to the family of dynamic evolutionary algorithms. Moreover, it is the first such study on the application of dynamic evolutionary algorithms for dynamic electrical impedance tomography-based image reconstruction. The performance of the algorithm is tested through numerical simulations and experimental study and is compared with state-of-the-art gradient-based extended Kalman filter. The dynamic OBBO is shown to be far superior compared to the extended Kalman filter. It is found to be robust to measurement noise as well as the initial guess, and does not rely on a priori knowledge of the state evolution model.