Approximation properties of Fourier sums for 2pi-periodic piecewise linear continuous functions

The paper studies the function space of continuous piecewise linear functions in the space of continuous functions on them-dimensional Euclidean space. It also studies the special case of one dimensional continuous piecewise linear functions. The study is based on the theory of Riesz spaces that has many applications in economics. The work also provides the mathematical background to its sister paper Aliprantis, Harris, and Tourky (2006), in which we estimate multivariate continuous piecewise linear regressions by means of Riesz estimators, that is, by estimators of the the Boolean formwhereX=(X1,X2, …,Xm) is some random vector, {Ej}j∈Jis a finite family of finite sets.

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Automatic techniques for economic computer computation of continuous functions

SIMULATION ◽

10.1177/003754976801100116 ◽

1968 ◽

Vol 11 (1) ◽

pp. 37-48 ◽

Cited By ~ 2

Keyword(s):

High Speed ◽

Piecewise Linear ◽

Continuous Functions ◽

Computational Error ◽

Mathematical Function ◽

Chemical Plants ◽

Approximation Techniques ◽

Allowable Error ◽

The Cost ◽

Error Envelope

Methods are presented for the automatic preparation of functions of one or more variables for economical calculation by high-speed digital computers. The cost of calculation is considered according to the factors of number of functions, complexity, requirements for precision, and the frequency with which functions are to be calculated. Contrary to classic approaches, con sideration is not given to minimizing computational error for its own sake. On the contrary, the maximum allowable error may be sought in order to minimize computational costs. In this respect, each function is represented by an error envelope that specifies the required limits of computational precision. It is the error envelope rather than the function itself which is dealt with. The approximation techniques dealt with in this paper are limited to piecewise linear ap proximation of functions of one or two independent variables. Projects requiring the maintaining and computation of large quantities of continuous functions are fre quently to be found in industry and research; for example, in the simulation of real-time processes— aircraft flight and flight trainer simulations, simula tion for control and regulation of continuous pro cesses as in chemical plants, weather calculations, radiation studies, etc. In addition, computer service centers, providing computational services to many users, may extend the range and effectiveness of their mathematical function program library by the use of the economical com putational methods of this paper.

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Approximation by Certain Linear Positive Operators of Two Variables

Abstract and Applied Analysis ◽

10.1155/2014/782080 ◽

2014 ◽

Vol 2014 ◽

pp. 1-6 ◽

Cited By ~ 6

Author(s):

Afşin Kürşat Gazanfer ◽

İbrahim Büyükyazıcı

Keyword(s):

Convergence Rate ◽

Modulus Of Continuity ◽

Weighted Approximation ◽

Continuous Functions ◽

Linear Operators ◽

Positive Linear Operators ◽

Compact Set ◽

Approximation Properties ◽

Linear Positive Operators ◽

Functions Of Two Variables

We introduce positive linear operators which are combined with the Chlodowsky and Szász type operators and study some approximation properties of these operators in the space of continuous functions of two variables on a compact set. The convergence rate of these operators are obtained by means of the modulus of continuity. And we also obtain weighted approximation properties for these positive linear operators in a weighted space of functions of two variables and find the convergence rate for these operators by using the weighted modulus of continuity.

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Approximation of Bounded Continuous Functions by Linear Combinations of Phillips Operators

Demonstratio Mathematica ◽

10.2478/dema-2014-0053 ◽

2014 ◽

Vol 47 (3) ◽

Cited By ~ 2

Author(s):

Gancho Tachev

Keyword(s):

Continuous Functions ◽

Approximation Properties ◽

Linear Combinations ◽

Direct Estimate ◽

Durrmeyer Operators ◽

Rate Of Approximation ◽

Phillips Operators ◽

Bounded Continuous Functions

AbstractWe study the approximation properties of linear combinations of the so-called Phillips operators, which can be considered as genuine Szász-Mirakjan-Durrmeyer operators. As main result, we prove a direct estimate for the rate of approximation of bounded continuous functions f E C[0,x), measured in C|\[0,x)-norm and thus generalizing the results, proved earlier by Gupta, Agrawal, and Gairola in [3]. Our estimates rely on the recent results, obtained in the joint works of M. Heilmann and the author-[10, 11]

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Simple anti-swing feedback control for a gantry crane

Robotica ◽

10.1017/s0263574703005150 ◽

2003 ◽

Vol 21 (6) ◽

pp. 655-666 ◽

Cited By ~ 4

Author(s):

Yannick Aoustin ◽

Alexander Formal'sky

Keyword(s):

Optimal Control ◽

Piecewise Linear ◽

Continuous Functions ◽

Time Optimal Control ◽

Gantry Crane ◽

Control Law ◽

Reference Trajectory ◽

Controlling Parameter ◽

Input Signals ◽

Time Optimal

We propose a simple quasi time optimal control law for a gantry crane with a payload. The force applied to the trolley is a controlling parameter. The control law consists of two parts: a feedforward term and a trolley position and velocity feedback term.Initially, we synthesize the feedforward term and the corresponding reference trajectory by computing the time optimal control for the system mass center. The computed optimal control is a discontinuous function of time with several switching time instants. Undesirable large vibrations due to the payload sway appear under this control. Therefore, we transform this control, replacing its jumps by the piecewise linear continuous functions. The computed feedforward term and the reference trajectory are used as input signals of the PD-controller.

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Approximation Properties of Discrete Fourier Sums in Polynomials Orthogonal on Non-Uniform Grids

Владикавказский математический журнал ◽

10.46698/k4355-6603-4655-y ◽

2020 ◽

Author(s):

А.А. Нурмагомедов

Keyword(s):

Approximation Properties ◽

Uniform Grids ◽

Fourier Sums ◽

Alpha Beta

В данной работе для произвольной непрерывной на отрезке $[-1, 1]$ функции $f(x)$ в~случае целых положительных $\alpha$ и $\beta$ построены дискретные суммы Фурье $S_{n,N}^{\alpha,\beta}(f,x)$ по системе многочленов $\{\hat{p}_{k,N}^{\alpha,\beta}(x)\}_{k=0}^{N-1},$ образующих ортонормированную систему на неравномерных сетках $\Omega_N=\{x_j\}_{j=0}^{N-1},$ состоящих из конечного числа $N$ точек отрезка $[-1, 1]$ с весом типа Якоби. Исследуются аппроксимативные свойства построенных частных сумм $S_{n,N}^{\alpha,\beta}(f,x)$ порядка $n\leq{N-1}$ в~пространстве непрерывных функциий $C[-1, 1].$ А именно, получена двусторонняя поточечная оценка для функции Лебега $L_{n,N}^{\alpha,\beta}(x)$ рассматриваемых дискретных сумм Фурье при $n=O\big(\delta_N^{-1/(\lambda+3)}\big)$, $\lambda=\max\{\alpha, \beta\}$, $\delta_N=\max_{0\leq{j}\leq{N-1}}\Delta{t_j}$. Соответственно, исследован также вопрос сходимости $S_{n,N}^{\alpha,\beta}(f,x)$ к $f(x)$. В частности, получена оценка отклонения частичной суммы $S_{n,N}^{\alpha,\beta}(f,x)$ от $f(x)$ при $n=O\big(\delta_N^{-1/(\lambda+3)}\big),$ которая также зависит от~$n$ и положения точки $x\in[-1, 1].$

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Polynomial approximation on spheres - generalizing de la Vallée-Poussin

Computational Methods in Applied Mathematics ◽

10.2478/cmam-2011-0029 ◽

2011 ◽

Vol 11 (4) ◽

pp. 540-552 ◽

Cited By ~ 12

Author(s):

Ian H. Sloan

Keyword(s):

Trigonometric Polynomial ◽

Polynomial Approximation ◽

Piecewise Linear ◽

Continuous Functions ◽

Fast Convergence ◽

Piecewise Polynomial ◽

Smooth Functions ◽

Higher Dimensional ◽

Trigonometric Polynomial Approximation ◽

De La Vallée Poussin

AbstractFor trigonometric polynomial approximation on a circle, the century-old de la Vallée-Poussin construction has attractive features: it exhibits uniform convergence for all continuous functions as the degree of the trigonometric polynomial goes to infinity, yet it also has arbitrarily fast convergence for sufficiently smooth functions. This paper presents an explicit generalization of the de la Vallée-Poussin construction to higher dimensional spheres S^d ≤ R^{d+1}. The generalization replaces the C^∞ filter introduced by Rustamov by a piecewise polynomial of minimal degree. For the case of the circle the filter is piecewise linear, and recovers the de la Vallée-Poussin construction, while for the general sphere S^d the filter is a piecewise polynomial of degree d and smoothness C^{d−1}. In all cases the approximation converges uniformly for all continuous functions, and has arbitrarily fast convergence for smooth functions.

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