• The Journal of Korean Institute of Communications and Information Sciences
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• v.32 no.11C
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• pp.1073-1078
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• 2007

In this paper, we propose an iterative mixed norm image restoration algorithm using multi regularization parameters. A functional which combines the regularized $l_2$ norm functional and the regularized $l_4$ norm functional is proposed to efficiently remove arbitrary noise. The smoothness of each functional is determined by the regularization parameters. Also, a regularization parameter is used to determine the relative importance between the regularized $l_2$ norm functional and the regularized $l_4$ norm functional using kurtosis. An iterative algorithm is utilized for obtaining a solution and its convergence is analyzed. Experimental results demonstrate the capability of the proposed algorithm.

• Journal of IKEEE
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• v.20 no.2
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• pp.136-142
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• 2016

This paper presents a non-local means (NLM) denoising algorithm based on a new weighting function using a mixed norm. The fidelity of the difference between an anchor patch and the reference patch in the NLM denoising depends on noise level and local activity. This paper introduces a new weighting function based on a mixed norm type of which the order is determined by noise level and local activity of an anchor patch, so that the performance of the NLM denoising can be enhanced. Experimental results demonstrate the objective and subjective capability of the proposed algorithm. In addition, it was verified that the proposed algorithm can be used to improve the performance of the other $l_2$ norm based non-local means denoising algorithms

• The Journal of the Acoustical Society of Korea
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• v.16 no.3E
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• pp.62-68
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• 1997

In this paper, a modified adaptive sign (MAS) algorithm based on a mixed norm error criterion is proposed. The mixed norm error criterion of be minimized is constructed as a combined convex function of the mean-absolute error and the mean-absolute error to the third power. A convergence analysis of the MAS algorithm is also presented. Under a set of mild assumptions, a set of nonlinear evolution equations that characterizes the statistical mean and mean-squared behavior of the algorithm is derived. Computed simulations are carried out to verify the validity of our derivations.

• Journal of applied mathematics & informatics
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• v.7 no.1
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• pp.61-82
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• 2000

In this paper two so-called regularized Green's functions are introduced to derive the optimal maximum norm error estimates for the unknown function and the adjoint vector-valued function for mixed finite element methods of Laplacian operator. One contribution of the paper is a demonstration of how the boundedness of $L^1$-norm estimate for the second Green's function ${\lambda}_2$ and the optimal maximum norm error estimate for the adjoint vector-valued function are proved. These results are seemed to be to be new in the literature of the mixed finite element methods.

• Journal of the Korean Mathematical Society
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• v.45 no.1
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• pp.63-78
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• 2008

We generalize several integral inequalities for analytic functions on the open unit polydisc $U^n={\{}z{\in}C^n||zj|<1,\;j=1,...,n{\}}$. It is shown that if a holomorphic function on $U^n$ belongs to the mixed norm space $A_{\vec{\omega}}^{p,q}(U^n)$, where ${\omega}_j(\cdot)$,j=1,...,n, are admissible weights, then all weighted derivations of order $|k|$ (with positive orders of derivations) belong to a related mixed norm space. The converse of the result is proved when, p, q ${\in}\;[1,\;{\infty})$ and when the order is equal to one. The equivalence of these conditions is given for all p, q ${\in}\;(0,\;{\infty})$ if ${\omega}_j(z_j)=(1-|z_j|^2)^{{\alpha}j},\;{\alpha}_j>-1$, j=1,...,n (the classical weights.) The main results here improve our results in Z. Anal. Anwendungen 23 (3) (2004), no. 3, 577-587 and Z. Anal. Anwendungen 23 (2004), no. 4, 775-782.

• Proceedings of the IEEK Conference
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• 2003.11a
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• pp.489-492
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• 2003

In this paper, we propose an iterative mixed norm image restoration algorithm using multi regularization parameters. A functional which combines the regularized l$_2$ norm functional and the regularized l$_4$ functional is proposed. The smoothness of each functional is determined by the regularization parameters. Also, a regularization parameter is used to determine the relative importance between the regularized l$_2$ functional and the regularized l$_4$ functional. An iterative algorithm is utilized for obtaining a solution and its convergence is analyzed.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.29 no.2C
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• pp.272-282
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• 2004

This paper introduces a regularized mixed norm multi-channel image restoration algorithm using both within-and between- channel deterministic information. For each channel a functional which combines the least mean squares (LMS), the least mean fourth (LMF), and a smoothing functional is proposed. We introduce a mixed norm parameter that controls the relative contribution between the LMS and the LMF, and a regularization parameter defining the degree of smoothness of the solution, where both parameters are updated at each iteration according to the noise characteristics of each channel. The novelty of the proposed algorithm is that no knowledge of the noise distribution for each channel is required and that the parameters mentioned above are adjusted based on the partially restored image.

• Bulletin of the Korean Mathematical Society
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• v.59 no.6
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• pp.1371-1385
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• 2022

For setting a general weight function on n dimensional complex space ℂⁿ, we expand the classical Fock space. We define Fock type space $F^{p,q}_{{\phi},t}({\mathbb{C}}^n)$ of entire functions with a mixed norm, where 0 < p, q < ∞ and t ∈ ℝ and prove that the mixed norm of an entire function is equivalent to the mixed norm of its radial derivative on $F^{p,q}_{{\phi},t}({\mathbb{C}}^n)$. As a result of this application, the space $F^{p,q}_{{\phi},t}({\mathbb{C}}^n)$ is especially characterized by a Lipschitz type condition.

• Journal of the Korean Mathematical Society
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• v.59 no.5
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• pp.927-944
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• 2022

Let $\vec{p}{\in}(0,\;1]^n$ be an n-dimensional vector and A a dilation. Let $H^{\vec{p}}_A(\mathbb{R}^n)$ denote the anisotropic mixed-norm Hardy space defined via the radial maximal function. Using the known atomic characterization of $H^{\vec{p}}_A(\mathbb{R}^n)$ and establishing a uniform estimate for corresponding atoms, the authors prove that the Fourier transform of $f{\in}H^{\vec{p}}_A(\mathbb{R}^n)$ coincides with a continuous function F on ℝⁿ in the sense of tempered distributions. Moreover, the function F can be controlled pointwisely by the product of the Hardy space norm of f and a step function with respect to the transpose matrix of A. As applications, the authors obtain a higher order of convergence for the function F at the origin, and an analogue of Hardy-Littlewood inequalities in the present setting of $H^{\vec{p}}_A(\mathbb{R}^n)$.

• Bulletin of the Korean Mathematical Society
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• v.50 no.5
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• pp.1451-1469
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• 2013

We present various new sharp assertions on multipliers in mixed norm, weighted Hardy and new Lizorkin-Triebel spaces of harmonic functions in higher dimension. Some results are new even in onedimensional case.