Modeling a Periodic Signal Using Fourier Series

This paper covers the concept of Fourier series and its application for a periodic signal. A periodic signal is a signal that repeats its pattern over time at regular intervals. The idea inspiring is to approximate a regular periodic signal, under Dirichlet conditions, via a linear superposition of trigonometric functions, thus Fourier polynomials are constructed. The Dirichlet conditions, are a set of mathematical conditions, providing a foundational framework for the validity of the Fourier series representation. By understanding and applying these conditions, we can accurately represent and process periodic signals, leading to advancements in various areas of signal processing. The resulting Fourier approximation allows complex periodic signals to be expressed as a sum of simpler sinusoidal functions, making it easier to analyze and manipulate such signals.

ASYMPTOTIC BEHAVIOR OF ECKHOFF'S METHOD FOR FOURIER SERIES CONVERGENCE ACCELERATION

The current paper considers the problem of recovering a function using a limited number of its Fourier coefficients. Specifically, a method based on Bernoulli-like polynomials suggested and developed by Krylov, Lanczos, Gottlieb and Eckhoff is examined. Asymptotic behavior of approximate calculation of the so-called ＂jumps＂ is studied and asymptotic L2 constants of the rate of convergence of the method are computed.

Convergence Properties of Generalized Fourier Series on a Parallel Hexagon Domain

A new Rogosinski-type kernel function is constructed using kernel function of partial sums Sn（f; t） of generalized Fourier series on a parallel hexagon domain Ω associating with threedirection partition. We prove that an operator Wn（f; t） with the new kernel function converges uniformly to any continuous function f（t） ∈ Cn（Ω） （the space of all continuous functions with period Ω） on Ω. Moreover, the convergence order of the operator is presented for the smooth approached function.

Alternative Fourier Series Expansions with Accelerated Convergence

The key objective of this paper is to improve the approximation of a sufficiently smooth nonperiodic function defined on a compact interval by proposing alternative forms of Fourier series expansions. Unlike in classical Fourier series, the expansion coefficients herein are explicitly dependent not only on the function itself, but also on its derivatives at the ends of the interval. Each of these series expansions can be made to converge faster at a desired polynomial rate. These results have useful implications to Fourier or harmonic analysis, solutions to differential equations and boundary value problems, data compression, and so on.

APPROXIMATION OF CONVEX TYPE FUNCTION BY PARTIAL SUMS OF FOURIER SERIES

The concept of convex type function is introduced in this paper,from which a kin d of convex decomposition approach is proposed.As one of applications of this a pproach,the approximation of the convex type function by the partial sum of its Fourier series is inves tigated.Moreover,the order of approximation is describe d with the 2th continuous modulus.

Estimation of permafrost thermal behavior using Fourier series model

Permafrost,being an important component of the cryosphere,is sensitive to climate change.Therefore,it is necessary to investigate the change of temperature within permafrost.In this study,we proposed a Fourier series model derived from the conduction equation to simulate permafrost thermal behavior over a year.The boundary condition was represented by the Fourier series and the geothermal gradient.The initial condition was represented as a linear function relative to the geothermal gradient.A comparative study of the different models(sinusoidal model,Fourier series model,and the proposed model)was conducted.Data collected from the northern Da Xing’anling Mountains,Northeast China,were applied for parameterization and validation for these models.These models were compared with daily mean ground temperature from the shallow permafrost layer and annual mean ground temperature from the bottom permafrost layer,respectively.Model performance was assessed using three coefficients of accuracy,i.e.,the mean bias error,the root mean square error,and the coefficient of determination.The comparison results showed that the proposed model was accurate enough to simulate temperature variation in both the shallow and bottom permafrost layer as compared with the other two Fourier series models(sinusoidal model and Fourier model).The proposed model expanded on a previous Fourier series model for which the initial and bottom boundary conditions were restricted to being constant.

Approximation by Nrlund Means of Hexagonal Fourier Series

Let f be an H-periodic HOlder continuous function of two real variables.The error ｜｜f-Nn （p;f）｜｜ is estimated in the uniform norm and in the Holder norm,where p=（pk）k=0∞is a nonincreasing sequence of positive numbers and Nn （p;f） is thenth Norlund mean of hexagonal Fourier series of f with respect to p = （pk）k∞=0.

APPROXIMATION OF SPHERICAL MEANS OF MULTIPLE FOURIER INTEGRALS AND FOURIER SERIES ON SET OF FULL MEASURE

In this paper we consider the approximation for functions in some subspaces of L^2 by spherical means of their Fourier integrals and Fourier series on set of full measure. Two main theorems are obtained.

UNIFORM CONVERGENCE OF CES■RO MEANS OF NEGATIVE ORDER OF DOUBLE TRIGONOMETRIC FOURIER SERIES

In this paper we prove that if f ∈ C （[-π, π]^2） and the function f is bounded partial p-variation for some p ∈[1, ＋∞）, then the double trigonometric Fourier series of a function f is uniformly （C;-α,-β） summable （α＋β 〈 1/p,α,β 〉 0） in the sense of Pringsheim. If α ＋ β ≥ 1/p, then there exists a continuous function f0 of bounded partial p-variation on [-π,π]^2 such that the Cesàro （C;-α,-β） means σn,m^-α,-β（f0;0,0） of the double trigonometric Fourier series of f0 diverge over cubes.

Rainfall Forecasting Using Fourier Series

The need for accurate rainfall prediction is readily apparent when considering many benefits in which such information would provide for river control, reservoir operation, forestry interests, flood mitigation, etc.. Due to importance of rainfall in many aspects, studies on rainfall forecast have been conducted since a few decades ago. Although many methods have been introduced, all the researches describe the study as complex because it involves numerous variables and still need to be improved. Nowadays, there are various traditional techniques and mathematical models available, yet, there are no result on which method provide the most reliable estimation. AR （auto-regressive）, ARMA （auto-regressive moving average）, ARIMA （auto-regressive integrated moving average） and ANNs （artificial neural networks） were introduced as a useful and efficient tool for modeling and forecasting. The conventional time series provide reasonable accuracy but suffer from the assumptions of stationary and linearity. The concept of neurons was introduced first which then developed to ANNs with back propagation training algorithm. Although certain ANNs） models are equivalent to time series model, but it is limited to short term forecasting. This Paper presents a mathematical approach for rainfall forecasting for Iran on monthly basic. The model is trained for monthly rainfall forecasting and tested to evaluate the performance of the model. The result Shows reasonably good accuracy for monthly rainfall forecasting.

A Biproportional Construction Algorithm for Correctly Calculating Fourier Series of Aperiodic Non-Sinusoidal Signal

The Fourier series (FS) applies to a periodic non-sinusoidal function satisfying the Dirichlet conditions, whereas the being-processed function in practical applications is usually an aperiodic non-sinusoidal signal. When is aperiodic, its calculated FS is not correct, which is still a challenging problem. To overcome the problem, we derive a direct calculation algorithm, a constant iteration algorithm, and an optimal iteration algorithm. The direct calculation algorithm correctly calculates its Fourier coefficients (FCs) when is periodic and satisfies the Dirichlet conditions. Both the constant iteration algorithm and the optimal iteration algorithm provide an idea of determining the states of . From the idea, we obtain an algorithm for determining the states of based on the optimal iteration algorithm. In the algorithm, the variable iteration step is introduced; thus, we present an algorithm for determining the states of based on the variable iteration step. The presented algorithm accurately determines the states of . On the basis of these algorithms, we build a biproportional construction theory. The theory consists of a first and a second proportional construction theory. The former correctly calculates the FCs of at the present sampling time

Dribbling Basketball Using Fourier Series

Dribbling a basketball is a fundamental skill in the sport, defined by the rhythmic bouncing of the ball with one hand, regardless of whether the player is stationary or in motion. Mastery of dribbling allows an athlete to maintain control of the ball, maneuver around opponents, and create opportunities for passing, shooting, or driving toward the basket. Additionally, dribbling involves various mathematical principles, such as the physics of motion and the statistical analysis of performance data. One significant mathematical tool in this context is Fourier analysis, which effectively decomposes complex signals, such as the dribbling motion of a basketball, into simpler sinusoidal components. This analysis provides insights into the frequency characteristics of the dribble, enhancing the understanding of a player’s skill and consistency.

MULTIVARIATE FOURIER SERIES OVER A CLASS OF NON TENSOR-PRODUCT PARTITION DOMAINS

This paper finds a way to extend the well-known Fourier methods, to so-called n+1 directions partition domains in n-dimension. In particular, in 2-D and 3-D cases, we study Fourier methods over 3-direction parallel hexagon partitions and 4-direction parallel parallelogram dodecahedron partitions, respectively. It has pointed that, the most concepts and results of Fourier methods on tensor-product case, such as periodicity,orthogonality of Fourier basis system, partial sum of Fourier series and its approximation behavior, can be moved on the new non tensor-product partition case.

A PRELIMINARY STUDY OF FOURIER SERIES ANALYSIS ON CLOUD TRACKING WITH GOES HIGH TEMPORAL RESOLUTION IMAGES

Fourier series analysis is proposed as a new technique to address the problem of“sub-pixel motion”in deriving cloud motion winds(CMW)from high temporal resolution images.Based on a concept different from that of maximum correlation matching technique,the Fourier technique computes phase speed as an estimate of cloud motion.It is very effective for tracking small cellular clouds in 1-min interval images and more efficient for computation than the maximum correlation technique because only two templates in same size are involved in primary tracking procedure. Moreover it obtains not only CMW vectors but potentially also velocity spectrum and variance.A practical example is given to show the cloud motion winds from 1-min interval images with the Fourier method versus those from traditional 30-min interval images with maximum correlation technique.Problems that require further investigation before the Fourier technique can be regarded as a viable technique,especially for cloud tracking with high temporal resolution images,are also revealed.

Fast cooperative trajectory optimization and test verification for close-range satellite formation using Finite Fourier Series method

The process of formation reconfiguration for close-range satellite formation should take into account the risk of collisions between satellites.To this end,this paper presents a method to rapidly generate low-thrust collision-avoidance trajectories in the formation reconfiguration using Finite Fourier Series(FFS).The FFS method can rapidly generate the collision-avoidance threedimensional trajectory.The results obtained by the FFS method are used as an initial guess in the Gauss Pseudospectral Method(GPM)solver to verify the applicability of the results.Compared with the GPM method,the FFS method needs very little computing time to obtain the results with very little difference in performance index.To verify the effectiveness,the proposed method is tested and validated by a formation control testbed.Three satellite simulators in the testbed are used to simulate two-dimensional satellite formation reconfiguration.The simulation and experimental results show that the FFS method can rapidly generate trajectories and effectively reduce the risk of collision between satellites.This fast trajectory generation method has great significance for on-line,constantly satellite formation reconfiguration.

REAL-VALUED PERIODIC WAVELETS: CONSTRUCTION ANDRELATION WITH FOURIER SERIES

In this paper, we construct the real-valued periodic orthogonal wavelets. The method presented here is new. The decomposition and reconstruction formulas involve only 4 terms respectively. It demonstrates that the formulas are simpler than that in other kinds of periodic wavelets. Our wavelets are useful in applications since it is real valued. The relation between the periodic wavelets and the Fourier series is also discussed.

Analyzing China’s OFDI using a novel multivariate grey prediction model with Fourier series

Purpose–With the development of economy,China’s OFDI constantly increase in recent year.Meanwhile,OFDI hasspillovereffectoneconomicdevelopmentandtechnologicaldevelopmentofhomecountry.Thus,accurateOFDI prediction is a prerequisite for the effective development of international investment strategies.The purpose of this paper is to predict China’s OFDI accurately using a novel multivariable grey prediction model with Fourier series.Design/methodology/approach–This paper applied a multivariable grey prediction model,GM(1,N),to forecast China’s OFDI.In order to improve the prediction accuracy and without changing local characteristics of grey model prediction,this paper proposed a novel grey prediction model to improve the performance of the traditionalGM(1,N)modelbycombiningwithresidualmodificationmodelusingGM(1,1)modelandFourierseries.Findings–The coefficients indicate that the export and GDP have positive influence on China’s OFDI,and,according to the prediction result,China’s OFDI shows a growing trend in next five years.Originality/value–This paper proposed an effective multivariable grey prediction model that combined the traditionalGM(1,N)modelwitharesidualmodificationmodelinordertopredictChina’sOFDI.Accurateforecasting of OFDI provides reference for the Chinese Government to implement international investment strategies.

Hybrid Function Projective Synchronization of Chaotic Systems with Uncertain Time-varying Parameters via Fourier Series Expansion

In this paper, the hybrid function projective synchronization (HFPS) of different chaotic systems with uncertain periodically time-varying parameters is carried out by Fourier series expansion and adaptive bounding technique. Fourier series expansion is used to deal with uncertain periodically time-varying parameters. Adaptive bounding technique is used to compensate the bound of truncation errors. Using the Lyapunov stability theory, an adaptive control law and six parameter updating laws are constructed to make the states of two different chaotic systems asymptotically synchronized. The control strategy does not need to know the parameters thoroughly if the time-varying parameters are periodical functions. Finally, in order to verify the effectiveness of the proposed scheme, the HFPS between Lorenz system and Chen system is completed successfully by using this scheme.

On the Gibbs Phenomenon of Fourier Series of a Classical Function

In this paper,we point out that the Fourier series of a classical function∑^∞k=1 sin kx/k has the Gibbs phenomenon in the neighborhood of zero.Furthermore,we estimate the upper bound of its partial sum and get：sup n≥1｜｜∑^n k=1sin kx/k｜｜=∫^x 0sin x/x dx=1.85194, which is better than that in[1].

On Some Diophantine Fourier Series

We continue our study on arithmetical Fourier series by considering two Fourier series which are related to Diophantine analysis. The first one was studied by Hardy and Littlewood in connection with the classification of numbers and the second one was studied by Hartman and Wintner by Lebesgue integration theory.