• Honam Mathematical Journal
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• v.42 no.2
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• pp.251-268
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• 2020

In this paper, we estimate the degree of approximation by means of the complete modulus of continuity, the partial modulus of continuity, the Lipschitz-type class and Petree's K-functional for the bivariate Szász-Kantorovich operators based on Brenke-type polynomials. Later, we construct Generalized Boolean Sum operators associated with combinations of the Szász-Kantorovich operators based on Brenke-type polynomials. In addition, we obtain the rate of convergence for the GBS operators with the help of the mixed modulus of continuity and the Lipschitz class of the Bögel continuous functions.

• Journal of Korea Multimedia Society
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• v.24 no.1
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• pp.13-20
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• 2021

A Wasserstein GAN(WGAN), optimum in terms of minimizing Wasserstein distance, still suffers from inconsistent convergence or unexpected output due to inherent learning instability. It is widely known some kinds of restriction on the discriminative function should be considered to solve such problems, which implies the importance of Lipschitz continuity. Unfortunately, there are few known methods to satisfactorily maintain the Lipschitz continuity of the discriminative function. In this paper we propose techniques to stably maintain the Lipschitz continuity of the discriminative function by adding effective regularization terms to the objective function, which limit the magnitude of the gradient vectors of the discriminator to one or less. Extensive experiments are conducted to evaluate the performance of the proposed techniques, which shows the single-sided penalty improves convergence compared with the gradient penalty at the early learning process, while the proposed additional penalty increases inception scores by 0.18 after 100,000 number of learning.

• Korean Journal of Mathematics
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• v.24 no.3
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• pp.335-344
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• 2016

In this paper, we obtain quantitative estimates for generalized double Baskakov operators. We calculate global results for these operators using Lipschitz-type spaces and estimate the error using modulus of continuity.

• Journal of applied mathematics & informatics
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• v.31 no.3_4
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• pp.491-498
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• 2013

This note is concerned with the uniform $L^p$-continuity of solution for the stochastic differential equations under Lipschitz condition and linear growth condition. Furthermore, uniform $L^p$-continuity of the solution for the stochastic functional differential equation is given.

• Bulletin of the Korean Mathematical Society
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• v.51 no.2
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• pp.303-315
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• 2014

In this work, we construct Kantorovich type generalization of a class of linear positive operators via Riemann type q-integral. We obtain estimations for the rate of convergence by means of modulus of continuity and the elements of Lipschitz class and also investigate weighted approximation properties.

• East Asian mathematical journal
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• v.19 no.2
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• pp.291-303
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• 2003

A result of Hardy and Littlewood relates Holder continuity of analytic functions in the unit disk with a bound on the derivative. Gehring and Martio extended this result to the class of uniform domains. In this paper we obtain a similar result to the class of uniformly John domains in terms of the inner diameter metric. We give several properties of a domain with the property. Also we show some results on the Holder continuity of conjugate harmonic functions in the above domains.

• 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 $\mathbb{C}^n$.

• Communications of the Korean Mathematical Society
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• v.21 no.3
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• pp.419-428
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• 2006

The Berezin transform is the analogue of the Poisson transform in the Bergman spaces. Dyakonov characterize the holomorphic weighted Lipschitz function in the unit disk in terms of the Possion integral. In this paper, we characterize the harmonic weighted Lispchitz function in terms of the Berezin transform instead of the Poisson integral.

• Kyungpook Mathematical Journal
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• v.62 no.3
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• pp.467-484
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• 2022

In this work, we define bivariate Bernstein-Kantorovich type operators on a triangular domain and obtain some approximation results for these operators. We start off by computing some moment estimates and prove a Korovkin type convergence theorem. Then, we estimate the rate of convergence using the partial and complete modulus of continuity, and derive a Voronovskaya-type asymptotic theorem. Further, we calculate the order of approximation with regard to the Peetre's K-functional and a Lipschitz type class. In addition, we construct the associated GBS type operators and compute the rate of approximation using the mixed modulus of continuity and class of the Lipschitz of Bögel continuous functions for these operators. Finally, we use the two operators to approximate example functions in order to compare their convergence.

• Journal of applied mathematics & informatics
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• v.16 no.1_2
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• pp.1-17
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• 2004

The famous Newton-Kantorovich hypothesis has been used for a long time as a sufficient condition for the convergence of Newton method to a solution of an equation in connection with the Lipschitz continuity of the Frechet-derivative of the operator involved. Using Lipschitz and center-Lipschitz conditions we show that the Newton-Kantorovich hypothesis is weakened. The error bounds obtained under our semilocal convergence result are finer and the information on the location of the solution more precise than the corresponding ones given by the dominating Newton-Kantorovich theorem, and under the same hypotheses/computational cost, since the evaluation of the Lipschitz also requires the evaluation of the center-Lipschitz constant. In the case of local convergence we obtain a larger convergence radius than before. This observation is important in computational mathematics and can be used in connection to projection methods and in the construction of optimum mesh independence refinement strategies.