• Journal of the Korean Mathematical Society
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• v.50 no.1
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• pp.189-202
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• 2013

On the unit ball of $\mathbb{C}^n$, the space of those holomorphic functions satisfying the mean Lipschitz condition $${\int}_0^1\;{\omega}_p(t,f)^q\frac{dt}{t^1+{\alpha}q}\;<\;{\infty}$$ is characterized by integral growth conditions of the tangential derivatives as well as the radial derivatives, where ${\omega}_p(t,f)$ denotes the $L^p$ modulus of continuity defined in terms of the unitary transformations of $\mathbb{C}^n$.

• Journal of Applied and Pure Mathematics
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• v.4 no.3_4
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• pp.185-209
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• 2022

Here we exhibit multivariate quantitative approximations of Banach space valued continuous multivariate functions on a box or ℝ^N, N ∈ ℕ, by the multivariate normalized, quasi-interpolation, Kantorovich type and quadrature type neural network operators. We treat also the case of approximation by iterated operators of the last four types. These approximations are achieved by establishing multidimensional Jackson type inequalities involving the multivariate modulus of continuity of the engaged function or its high order Fréchet derivatives. Our multivariate operators are defined by using a multidimensional density function induced by the generalized symmetrical sigmoid function. The approximations are point-wise and uniform. The related feed-forward neural network is with one hidden layer.

• Journal of Applied and Pure Mathematics
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• v.5 no.1_2
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• pp.69-93
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• 2023

Here we give multivariate quantitative approximations of Banach space valued continuous multivariate functions on a box or ℝ^N, N ∈ ℕ, by the multivariate normalized, quasi-interpolation, Kantorovich type and quadrature type neural network operators. We treat also the case of approximation by iterated operators of the last four types. These approximations are derived by establishing multidimensional Jackson type inequalities involving the multivariate modulus of continuity of the engaged function or its high order Fréchet derivatives. Our multivariate operators are defined by using a multidimensional density function induced by a parametrized Gudermannian sigmoid function. The approximations are pointwise and uniform. The related feed-forward neural network is with one hidden layer.

• Honam Mathematical Journal
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• v.28 no.1
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• pp.125-133
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• 2006

We investigate the approximation order to a function in $L_p$[-1, 1] for $0{\leq}p<{\infty}$ by generalized translation networks. In most papers related to neural network approximation, sigmoidal functions are adapted as an activation function. In our research, we choose an infinitely many times continuously differentiable function as an activation function. Using the integral modulus of continuity and the divided difference formula, we get the approximation order to a function in $L_p$[-1, 1].

• Kyungpook Mathematical Journal
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• v.27 no.2
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• pp.173-179
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• 1987

In this article the absolute convergence of lacunary Fourier Coefficients Series is studied for Hilbert space valued functions. The considered functions arc assumed to be of either the modulus of continuity or the modulus of smoothness of order l which are considered only at a fixed point in [$-{\pi},{\pi}$]. On the other hand for values in weakly sequentially complete Banach space, the lacunary Fourier coefficients series is strongly unconditionally convergent. The results obtained here are a kind of a generalization of the results due to Kandil [4].

• 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.

• Kyungpook Mathematical Journal
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• v.51 no.1
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• pp.87-91
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• 2011

In this paper we obtain a Voronvskaya type theorem for modi ed Post-Widde operators.

• Korean Journal of Mathematics
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• v.25 no.1
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• pp.99-116
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• 2017

The aim of this article is to introduce a new form of Kantorovich $Sz{\acute{a}}sz$-type operators involving Charlier polynomials. In this manuscript, we discuss the rate of convergence, better error estimates. Further, we investigate order of approximation in the sense of local approximation results with the help of Ditzian-Totik modulus of smoothness, second order modulus of continuity, Peetre's K-functional and Lipschitz class.

• Kyungpook Mathematical Journal
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• v.59 no.3
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• pp.465-480
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• 2019

In the present article, we analyse the behaviour of a new family of Kantorovich type sampling operators $(K^{\varphi}_wf)_{w>0}$. First, we give a Voronovskaya type theorem for these Kantorovich generalized sampling series and a corresponding quantitative version in terms of the first order of modulus of continuity. Further, we study the order of approximation in $C({\mathbb{R}})$, the set of all uniformly continuous and bounded functions on ${\mathbb{R}}$ for the family $(K^{\varphi}_wf)_{w>0}$. Finally, we give some examples of kernels such as B-spline kernels and the Blackman-Harris kernel to which the theory can be applied.

• Structural Engineering and Mechanics
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• v.43 no.1
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• pp.15-30
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• 2012

A damped Timoshenko beam element is introduced for the DOF-efficient forced vibration analysis of beam-like structures coated with viscoelastic damping layers. The rotary inertia as well as the shear deformation is considered, and the damping effect of viscoelastic layers is modeled as an imaginary loss factor in the complex shear modulus. A complex composite cross-section of structures is replaced with a homogeneous one by means of the transformed section approach in order to construct an equivalent single-layer finite element model capable of employing the standard $C^{0}$-continuity basis functions. The numerical reliability and the DOF-efficiency are explored through the comparative numerical experiments.