• Journal of the Korean Society for Industrial and Applied Mathematics
• /
• v.9 no.1
• /
• pp.111-123
• /
• 2005

Here, the Fredholm integral equation with Hilbert kernel is solved numerically, using two different methods. Also the error, in each case, is estimated.

• Kyungpook Mathematical Journal
• /
• v.54 no.1
• /
• pp.1-9
• /
• 2014

We build new Hilbert-type integral inequalities in the whole plane with the non-homogeneous kernel involving some parameters and the best constant factors. We also consider their reverse.

• Kyungpook Mathematical Journal
• /
• v.50 no.2
• /
• pp.307-314
• /
• 2010

By introducing a homogeneous kernel of 0-degree with an independent parameter and estimating the weight coefficient, a bilateral form of the Hilbert-type series inequality with a best constant factor is established.

• Kyungpook Mathematical Journal
• /
• v.49 no.3
• /
• pp.521-532
• /
• 2009

In this paper, by introducing a homogenous kernel of -4-degree, we establish a new Hilbert-type integral inequality with multi-parameter and a best constant factor. As applications, the equivalent form, the reverse forms and some particular results are given correspondingly.

• Honam Mathematical Journal
• /
• v.43 no.3
• /
• pp.523-537
• /
• 2021

In this paper, we obtain some new inequalities for the Berezin number of operators on reproducing kernel Hilbert spaces by using the Hölder-McCarthy operator inequality. Also, we give refine generalized inequalities involving powers of the Berezin number for sums and products of operators on the reproducing kernel Hilbert spaces.

• Journal of the Korean Mathematical Society
• /
• v.47 no.3
• /
• pp.537-546
• /
• 2010

Multiple discrete Hilbert type inequalities are established in the case of non-homogeneous kernel function by means of Laplace integral representation of associated Dirichlet series. Using newly derived integral expressions for the Mordell-Tornheim Zeta function a set of subsequent special cases, interesting by themselves, are obtained as corollaries of the main inequality.

• Kyungpook Mathematical Journal
• /
• v.50 no.2
• /
• pp.281-288
• /
• 2010

By introducing some parameters and using the way of weight function and the technic of real analysis and complex analysis, a new Hilbert-type integral inequality with a best constant factor and a combination kernel involving two mean values is given, which is an extension of Hilbert's integral inequality. As applications, the equivalent form and the reverse forms are considered.

• Korean Journal of Mathematics
• /
• v.31 no.2
• /
• pp.189-201
• /
• 2023

In this article, we establish Hilbert-type integral inequalities with the help of a non-homogeneous kernel of hyperbolic function with best constant factor. We also study the obtained inequalities's equivalent form. Additionaly, several specific Hilbert's type inequalities with constant factors in the term of the rational fraction expansion of higher order derivatives of cotangent and cosine functions are presented.

• Communications for Statistical Applications and Methods
• /
• v.21 no.4
• /
• pp.317-325
• /
• 2014

Consider a nonlinear transform ${\Phi}(x)$ of x in $\mathbb{R}^p$ to Hilbert space H and assume that the dot product between ${\Phi}(x)$ and ${\Phi}(x^{\prime})$ in H is given by < ${\Phi}(x)$, ${\Phi}(x^{\prime})$ >= K(x, x'). The aim of this paper is to propose a mathematical technique to take screen shots of the multivariate dataset mapped to Hilbert space H, particularly suited to Gaussian kernel $K({\cdot},{\cdot})$, which is defined by $K(x,x^{\prime})={\exp}(-{\sigma}{\parallel}x-x^{\prime}{\parallel}^2)$, ${\sigma}$ > 0. Several numerical examples are given.

• Journal of the Korean Data and Information Science Society
• /
• v.16 no.2
• /
• pp.411-418
• /
• 2005

We review support vector and wavelet density estimation. The relationship between support vector and wavelet density estimation in reproducing kernel Hilbert space (RKHS) is investigated in order to use wavelets as a variety of support vector kernels in support vector density estimation.