• Bulletin of the Korean Mathematical Society
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• v.46 no.4
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• pp.701-712
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• 2009

In this paper, we study the simultaneous approximation to functions in $C^m$[0, 1] by neural networks with a squashing function and the complexity related to the simultaneous approximation using a Bernstein polynomial and the modulus of continuity. Our proofs are constructive.

• Honam Mathematical Journal
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• v.31 no.2
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• pp.147-156
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• 2009

In this paper, we actually construct the simultaneous approximation by neural networks to a differentiable function. To do this, we first construct a polynomial approximation using the Fejer sum and then a simultaneous neural network approximation with the squashing activation function. We also give numerical results to support our theory.

• Journal of applied mathematics & informatics
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• v.26 no.1_2
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• pp.317-324
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• 2008

This paper shows the degree of simultaneous neural network approximation for a target function in $C^r$[-1, 1] and its first derivative. We use the Jackson's theorem for differentiable functions to get a degree of approximation to a target function by algebraic polynomials and trigonometric polynomials. We also make use of the de La Vall$\grave{e}$e Poussin sum to get an approximation order by algebraic polynomials to the derivative of a target function. By showing that the divided difference with a generalized translation network can be arbitrarily closed to algebraic polynomials on [-1, 1], we obtain the degree of simultaneous approximation.

• Journal of the Chungcheong Mathematical Society
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• v.23 no.4
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• pp.705-710
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• 2010

Let $C_1(X)$ be a normed linear space over ${\mathbb{R}}^m$, and S be an n-dimensional subspace of $C_1(X)$ with spaned by {$s_1,{\cdots},s_n$}. For each ${\ell}$- tuple vectors F in $C_1(X)$, the two-sided best simultaneous approximation problem is $$\min_{s{\in}S}\;\max\limits_{i=1}^\ell\{{\parallel}f_i-s{\parallel}_1\}$$. A $s{\in}S$ attaining the above minimum is called a two-sided best simultaneous approximation or a Chebyshev center for $F=\{f_1,{\cdots},f_{\ell}\}$ from S. This paper is concerned with algorithm for calculating two-sided best simultaneous approximation, in the case of continuous functions.

• Journal of the Chungcheong Mathematical Society
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• v.22 no.2
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• pp.141-148
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• 2009

We consider various algorithms calculating best onesided simultaneous approximations. We assume that X is a compact subset of $\mathbb{R}^{m}$ satisfying $X=\overline{intX}$, S is an n-dimensional subspace of C(X), and $\mu$ is any 'admissible' measure on X. For any l-tuple $f_1,\;{\cdots},\;f_{\ell}$ in C(X), we present various ideas for best approximation to F from S(F). The problem of best (both one and two-sided) approximation is a linear programming problem.

• Bulletin of the Korean Mathematical Society
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• v.35 no.1
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• pp.127-140
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• 1998

In this paper, we discuss the characterizations and uniqueness of a one-sided best simultaneous approximation for a compact subset from a convex subset of a finite-dimensional subspace of a normed linear space $C_1(X)$. The motivation is furnished by the characterizations of the one-sided best simultaneous approximations for a finite subset ${f_1, \ldots, f_\ell}$ for any $\ell \in N$.

• Bulletin of the Korean Mathematical Society
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• v.48 no.5
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• pp.1105-1109
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• 2011

We establish a result concerning simultaneous approximation and interpolation from certain uniformly dense subsets of the space of vector-valued continuous functions vanishing at infinity on locally compact Hausdorff spaces.

• Journal of the Chungcheong Mathematical Society
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• v.23 no.3
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• pp.435-442
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• 2010

This paper is concerned with algorithm for calculating one-sided best simultaneous approximation, in the case of continuous functions. we will apply any subgradient algorithm. In other words, we consider the algotithms from a mathematical, rather then computational, point of view.

• Journal of applied mathematics & informatics
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• v.39 no.3_4
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• pp.437-442
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• 2021

In this paper, we study the characterizations of two-sided best simultaneous approximations for ℓ-tuple subset from a closed convex subset of ℝ^m with ℓ^m₁(w)-norm. Main fact is, k^* is a two-sided best simultaneous approximation to F from K if and only if there exist f₁, …, f_p in F, for any k ∈ K $${\mid}{\sum\limits_{i=1}^{m}}sgn(f_{ji}-k^*_i)k_iw_i{\mid}{\leq}\;{\sum\limits_{i{\in}Z(f_j-k^*)}}\;{\mid}k_i{\mid}w_i$$ for each j = 1, …, p and 𝐰 ∈ W.

• Honam Mathematical Journal
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• v.27 no.1
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• pp.69-76
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• 2005

The aim of this work is to generalize $L_1$-approximation in order to apply them to a discrete approximation. In $L_1$-approximation, we use the norm given by $${\parallel}f{\parallel}_1={\int}{\mid}f{\mid}d{\mu}$$ where ${\mu}$ a non-atomic positive measure. In this paper, we go to the other extreme and consider measure ${\mu}$ which is purely atomic. In fact we shall assume that ${\mu}$ has exactly m atoms. For any ${\ell}$-tuple $b^1,\;{\cdots},\;b^{\ell}{\in}{\mathbb{R}}^m$, we defined the ${\ell}^m_1{w}$-norn, and consider $s^*{\in}S$ such that, for any $b^1,\;{\cdots},\;b^{\ell}{\in}{\mathbb{R}}^m$, $$\array{min&max\\{s{\in}S}&{1{\leq}i{\leq}{\ell}}}\;{\parallel}b^i-s{\parallel}_w$$, where S is a n-dimensional subspace of ${\mathbb{R}}^m$. The $s^*$ is called the Chebyshev center or a discrete simultaneous ${\ell}^m_1$-approximation from the finite dimensional subspace.