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.

Estimation of Winds at Different Isobaric Levels Based on the Observed Winds at 850 hPa Level Using Double Fourier Series

A technique based on the double Fourier series is developed to estimate the winds at different isobaric levels forthe limited area domain, 35°E to 140°E and 30°S to 40°N, using the observed winds at 850 hPa lcvcl for the month ofJune. For this purpose the wind field at a level under consideration is taken in the ratio form with that of 850 hPa level and the coefficients of the double Fouricr series are computed. These coefficients are subsequently used to computethe winds which are compared with the actual winds. The results of the double Fourier series technique are comparedwith those of the polynomial surface fitting method developed by Bavadekar and Khaladkar (1 992). The technique isalso applied for the daily wind data of 11. June, 1979 and the validation of the technique is tested for a few radiosondestations of india. The computed winds for these radiosonde stations arc quite close to observed winds.

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.

SUMMATION OF TRIGONOMETRIC SERIES BY FOURIER TRANSFORMS

This paper gives the theorems concerning the summation of trigonometric series with the help of Fourier transforms. By means of the known results of Fourier transforms, many difficult and complex problems of summation of trigonometric series can be solved. This method is a comparatively unusual way to find the summation of trigonometric series, and has been used to establish the comprehensive table of summation of trigonometric series. In this table 10 thousand scries arc given, and most of them are new.

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

Super-Fast Approximation Algorithms Using Classical Fourier Tools

In the author’s recent publications, a parametric system biorthogonal to the corresponding segment of the exponential Fourier system was unusually effective. On its basis, it was discovered that knowledge of a finite number of Fourier coefficients of function f from an infinite-dimensional set of elementary functions allows f to be accurately restored (the phenomenon of over-convergence). Below, parametric biorthogonal systems are constructed for classical trigonometric Fourier series, and the corresponding phenomena of over-convergence are discovered. The decisive role here was played by representing the space L2 as an orthogonal sum of two corresponding subspaces. As a result, fast parallel algorithms for reconstructing a function from its truncated trigonometric Fourier series are proposed. The presented numerical experiments confirm the high efficiency of these convergence accelerations for smooth functions. In conclusion, the main results of the work are summarized, and some prospects for the development and generalization of the proposed approaches are discussed.

On the Fourier approximation method for steady water waves

A computational method for steady water waves is presented on the basis of potential theory in the physical plane with spatial variables as independent quantities. The finite Fourier series are applied to approximating the free surface and potential function. A set of nonlinear algebraic equations for the Fourier coefficients are derived from the free surface kinetic and dynamic boundary conditions. These algebraic equations are numerically solved through Newton＇s iterative method, and the iterative stability is further improved by a relaxation technology. The integral properties of steady water waves are numerically analyzed, showing that （1） the set-up and the set-down are both non-monotonic quantities with the wave steepness, and （2） the Fourier spectrum of the free surface is broader than that of the potential function. The latter further leads us to explore a modification for the present method by approximating the free surface and potential function through different Fourier series, with the truncation of the former higher than that of the latter. Numerical tests show that this modification is effective, and can notably reduce the errors of the free surface boundary conditions.

Mode analysis of structures using the Fourier p-element method

The Fourier p-element method is an improvement to the finite element method,and is particularly suitable for vibration analysis due to the well-behaved Fourier series.In this paper,an iteration procedure is presented for solving the resulting nonlinear eigenvalue problem.Three types of Fourier version shape functions are constructed for analyzing the circular shaft torsional vibration,the plate in-plane vibration and annular plate flexural vibration modes,respectively. The numerical results show that this method can achieve higher accuracy and converge much faster than the FEM based on polynomial interpolation,especially for higher mode analysis.

ON THE ORDER OF SUMMABILITY OF THE FOURIER INVERSION FORMULA

In this article we show that the order of the point value, in the sense of Lojasiewicz, of a tempered distribution and the order of summability of the pointwise Fourier inversion formula are closely related. Assuming that the order of the point values and certain order of growth at infinity are given for a tempered distribution, we estimate the order of summability of the Fourier inversion formula. For Fourier series, and in other cases, it is shown that if the distribution has a distributional point value of order k, then its Fourier series is e.v. Cesaro summable to the distributional point value of order k＋1. Conversely, we also show that if the pointwise Fourier inversion formula is e.v. Cesaro summable of order k, then the distribution is the （k ＋ 1）-th derivative of a locally integrable function, and the distribution has a distributional point value of order k ＋ 2. We also establish connections between orders of summability and local behavior for other Fourier inversion problems.

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.

Fourier Coefficients of a Class of Eta Quotients of Weight 16 with Level 12

Recently, Williams [1] and then Yao, Xia and Jin [2] discovered explicit formulas for the coefficients of the Fourier series expansions of a class of eta quotients. Williams expressed all coefficients of 126 eta quotients in terms of and and Yao, Xia and Jin, following the method of proof of Williams, expressed only even coefficients of 104 eta quotients in terms of and . Here, by using the method of proof of Williams, we will express the even Fourier coefficients of 360 eta quotients i.e., the Fourier coefficients of the sum, f(q) + f(?q), of 360 eta quotients in terms of and .

Dynamic characteristics of multi-span spinning beams with elastic constraints under an axial compressive force

A theoretical model for the multi-span spinning beams with elastic constraints under an axial compressive force is proposed.The displacement and bending angle functions are represented through an improved Fourier series,which ensures the continuity of the derivative at the boundary and enhances the convergence.The exact characteristic equations of the multi-span spinning beams with elastic constraints under an axial compressive force are derived by the Lagrange equation.The efficiency and accuracy of the present method are validated in comparison with the finite element method(FEM)and other methods.The effects of the boundary spring stiffness,the number of spans,the spinning velocity,and the axial compressive force on the dynamic characteristics of the multi-span spinning beams are studied.The results show that the present method can freely simulate any boundary constraints without modifying the solution process.The elastic range of linear springs is larger than that of torsion springs,and it is not affected by the number of spans.With an increase in the axial compressive force,the attenuation rate of the natural frequency of a spinning beam with a large number of spans becomes larger,while the attenuation rate with an elastic boundary is lower than that under a classic simply supported boundary.