CONVERGENCE OF LOGARITHMIC MEANS OF MULTIPLE WALSH-FOURIER SERIES

Norlund logarithmic means of multiple Walsh-Fourier series acting from space L In^d-1 L （[0, 1）d）, d≥1 into space weak - LI（[0,1）^d） are studied. The maximal Orlicz space such that the Norlund logarithmic means of multiple Walsh-Fourier series for the functions from this space converge in d-dimensional measure is found.

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.

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.

APPROXIMATION PROPERTIES OF (C, α) MEANS OF DOUBLE WALSH-FOURIER SERIES

We study the rate of Lp approximation by Cesaro means of the quadratic partial sums of double Walsh-Fourier series of functions from Lp.

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.

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.

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.

Homogenized Daily Mean/Maximum/Minimum Temperature Series for China from 1960-2008

Inhomogeneities in the daily mean/maximum/ minimum temperature （Tm/Tmax/Tmin） series from 1960- 2008 at 549 National Standard Stations （NSSs） in China were analyzed by using the Multiple Analysis of Series for Homogenization （MASH） software package. Typical biases in the dataset were illustrated via the cases of Beijing （B J）, Wutaishan （WT）, Urumqi （UR） and Henan （HN） stations. The homogenized dataset shows a mean warming trend of 0.261/0.193/0.344℃/decade for the annual series of Tm/Tmax/Tmin, slightly smaller than that of the original dataset by 0.006/0.009/0.007℃/decade. However, considerable differences between the adjusted and original datasets were found at the local scale. The adjusted Tmin series shows a significant warming trend almost everywhere for all seasons, while there are a number of stations with an insignificant trend in the original dataset. The adjusted Tm data exhibit significant warming trends annually as well as for the autumn and winter seasons in northern China, and cooling trends only for the summer in the middle reaches of the Yangtze River and parts of central China and for the spring in southwestern China, while the original data show cooling trends at several stations for the annual and seasonal scales in the Qinghai, Shanxi, Hebei, and Xinjiang provinces. The adjusted Tmax data exhibit cooling trends for summers at a number of stations in the mid-lower reaches of the Yangtze and Yellow Rivers and for springs and winters at a few stations in southwestern China, while the original data show cooling trends at three/four stations for the annual/autumn periods in the Qinghai and Yunnan provinces. In general, the number of stations with a cooling trend was much smaller in the adjusted Tm and Tmax dataset than in the original dataset. The cooling trend for summers is mainly due to cooling in August. The results of homogenization using MASH appear to be robust; in particular, different groups of stations with consideration of elevation led to minor effects in the results.

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.

The Martingale Hardy Type Inequality for Marcinkiewicz-Fejer Means of Two-dimensional Conjugate Walsh-Fourier Series

The main aim of this paper is to prove that for any 0 〈 p≤ 2/3 there exists a martingale f E Hp such that Marcinkiewicz Fejer means of the two-dimensional conjugate Walsh Fourier series of the martingale f is not uniformly bounded in the space Lp.

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.

ON THE LOCALIZATION AND CONVERGENCE OF MULTIPLE FOURIER INTEGRAL BY BOCHNER-RIESZ MEANS

In this paper we consider lim _(R-) B_R^(f,x_0), in one case that f_x_0 (t) is a ABMV function on [0, ∞], and in another case that f∈L_(m-1)~1(R~) and x^k/~kf∈BV(R) when |k| = m-1 and f(x) = 0 when |x -x_0|<δ for some δ>0. Our theormes improve the results of Pan Wenjie ([1]).

Application of Fourier Series Expansion Method with PMLs to the Microcavities on Two-Dimensional Photonic Crystals

By using a Fourier series expansion method combined with Chew＇s perfectly matched layers （PMLs）, we analyze the frequency and quality factor of a micro-cavity on a two-dimensional photonic crystal is analyzed. Compared with the results by the method without PML and finite-difference time-domain （FDTD） based on supercell approximation, it can be shown that by the present method with PMLs, the resonant frequency and the quality factor values can be calculated satisfyingly and the characteristics of the micro-cavity can be obtained by changing the size and permittivity of the point defect in the micro-cavity.

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

On the Efficacy of Fourier Series Approximations for Pricing European Options

This paper investigates several competing procedures for computing the prices of vanilla European options, such as puts, calls and binaries, in which the underlying model has a characteristic function that is known in semi-closed form. The algorithms investigated here are the half-range Fourier cosine series, the half-range Fourier sine series and the full-range Fourier series. Their performance is assessed in simulation experiments in which an analytical solution is available and also for a simple affine model of stochastic volatility in which there is no closed-form solution. The results suggest that the half-range sine series approximation is the least effective of the three proposed algorithms. It is rather more difficult to distinguish between the performance of the half-range cosine series and the full-range Fourier series. However there are two clear differences. First, when the interval over which the density is approximated is relatively large, the full-range Fourier series is at least as good as the half-range Fourier cosine series, and outperforms the latter in pricing out-of-the-money call options, in particular with maturities of three months or less. Second, the computational time required by the half-range Fourier cosine series is uniformly longer than that required by the full-range Fourier series for an interval of fixed length. Taken together, these two conclusions make a case for pricing options using a full-range range Fourier series as opposed to a half-range Fourier cosine series if a large number of options are to be priced in as short a time as possible.

On the Equiconvergence of the Fourier Series and Integral of Distributions

We prove equiconvergence of the Bochner-Riesz means of the Fourier series and integral of distributions with compact support from the Liouville spaces.

Further Discussion on the Calculation of Fourier Series

Fourier series is an important mathematical concept. It is well known that we need too much computation to expand the function into Fourier series. The existing literature only pointed that its Fourier series is sine series when the function is an odd function and its Fourier series is cosine series when the function is an even function. And on this basis, in this paper, according to the function which satisfies different conditions, we give the different forms of Fourier series and the specific calculation formula of Fourier coefficients, so as to avoid unnecessary calculation. In addition, if a function is defined on [0,a], we can make it have some kind of nature by using the extension method as needed. So we can get the corresponding form of Fourier series.