We study the properties of the chebyshev approximation by the Gompertz function. The condition under which this approximation exists (with a certain relative error) and is unique is established. A method for determina...
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In many situations, exact solutions to complex problems may be challenging or impossible to obtain, making approximation techniques necessary for making informed decisions. Since function implementation on FPGAs can b...
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In many situations, exact solutions to complex problems may be challenging or impossible to obtain, making approximation techniques necessary for making informed decisions. Since function implementation on FPGAs can be difficult and resource-consuming, therefore it would be a better idea to approximate them. In this study, the chebyshev approximation technique is thoroughly investigated, and an FPGA implementation is proposed and analyzed. This implementation will be compared to other implementations of approximation techniques with parameters such as accuracy, speed, and design size taken into consideration. Applications of this FPGA design are also discussed and shown such as approximating the sigmoid function for machine learning in an efficient manner. This study proved the adequacy of the chebyshev approximation and its accelerated FPGA implementation for various applications including machine learning, filter design, signal processing, and many other practical applications in engineering and science.
This paper investigates the problem of simultaneous approximation of a prescribed multidimensional frequency response. The frequency responses of multidimensional IIR digital filters are used as nonlinear approximatin...
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This paper investigates the problem of simultaneous approximation of a prescribed multidimensional frequency response. The frequency responses of multidimensional IIR digital filters are used as nonlinear approximating functions. chebyshev approximation theory and the notion of line homotopy are used to reveal the approximation properties of this set of IIR functions. A sign condition is derived to characterize a convex stable domain in this set. This sign condition can be incorporated into the optimization of the chebyshev simultaneous approximation. The generally sufficient global Kolmogorov criterion is shown to be a necessary condition, for the characterization of best approximation, in the considered set of approximating functions. Thus, it can be incorporated, as a stopping constraint, in the design of the optimal filter. Moreover, the local Kolmogorov criterion is shown to be also necessary for the set of approximating functions. Finally, it is proved that the best approximation is a global minimum.
Where previous authors have considered linear approximations with a minimum sum of squared differences, we consider, instead, Chehyshev linear approximations, which minimize the maximum deviation. We obtain thus: 1) A...
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Where previous authors have considered linear approximations with a minimum sum of squared differences, we consider, instead, Chehyshev linear approximations, which minimize the maximum deviation. We obtain thus: 1) A new characterization of threshold functions, 2) A characterization of optimal threshold realizations as being virtually identical to the chebyshev-best linear approximations, and 3) A new insight into the test-synthesis problem with which we opened this note.
The problem of calculating the best approximating straight line-in the sense of chebyshev-to a finite set of points in R(n) is considered. First- and second-order optimality conditions are derived and analysed. Lipsch...
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The problem of calculating the best approximating straight line-in the sense of chebyshev-to a finite set of points in R(n) is considered. First- and second-order optimality conditions are derived and analysed. Lipschitz optimization techniques can be used to find a global minimizer.
This paper is concerned with chebyshev approximation by spline functions with free knots. Necessary and sufficient conditions for the best approximations are derived. It is shown by examples that the gap between these...
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This paper is concerned with chebyshev approximation by spline functions with free knots. Necessary and sufficient conditions for the best approximations are derived. It is shown by examples that the gap between these conditions cannot be bridged. The situation is less complicated, if the given function satisfies a generalized convexity condition.
The method for constructing the chebyshev approximation of a discrete multivariable function with reproducing the values of the function and its partial derivatives at given points is proposed. The method is based on ...
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The method for constructing the chebyshev approximation of a discrete multivariable function with reproducing the values of the function and its partial derivatives at given points is proposed. The method is based on constructing a boundary mean-power approximation with appropriate interpolation conditions. The authors use an iterative scheme based on the least squares method with a variable weight function for constructing the mean-power approximation. The results of approximating the one-variable function confirm the fulfillment of the characteristic feature of the chebyshev approximation with the reproduction of the function and its derivative values at given points. The test examples instantiate the fast convergence of the proposed method.
We describe the theoretical solution of an approximation problem that uses a finite weighted sum of complex exponential functions. The problem arises in an optimization model for the design of a telescope array occurr...
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We describe the theoretical solution of an approximation problem that uses a finite weighted sum of complex exponential functions. The problem arises in an optimization model for the design of a telescope array occurring within optical interferometry for direct imaging in astronomy. The problem is to find the optimal weights and the optimal positions of a regularly spaced array of aligned telescopes, so that the resulting interference function approximates the zero function on a given interval. The solution is given by means of chebyshev polynomials. (C) 2010 Elsevier Inc. All rights reserved.
A sufficient condition is given for best chebyshev approximations of the form a+bϕ(cx) to be characterized by alternation of their error curve. Several examples are given of ϕ for which alternation occurs. The problem...
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A sufficient condition is given for best chebyshev approximations of the form a+bϕ(cx) to be characterized by alternation of their error curve. Several examples are given of ϕ for which alternation occurs. The problem of computing a best approximation is considered. It is shown for some ϕ that adding all first degree polynomials to the family of approximations still gives an alternating theory.
chebyshev approximation by nonlinear families on a general compact space is studied. Attention is restricted to approximants satisfying a local Haar condition. A necessary and sufficient condition for the approximant ...
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chebyshev approximation by nonlinear families on a general compact space is studied. Attention is restricted to approximants satisfying a local Haar condition. A necessary and sufficient condition for the approximant to be locally best is given. A linear approximation problem is given which is equivalent to the nonlinear problem of locally best approximation. The existence of a minimal set on which a locally best approximation is locally best is shown. An alternation result is given for approximation on an interval.
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