• Bulletin of the Korean Mathematical Society
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• v.59 no.5
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• pp.1215-1235
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• 2022

In this paper we study Szegö kernel projections for Hardy spaces in quaternionic Clifford analysis. At first we introduce the matrix Szegö projection operator for the Hardy space of quaternionic Hermitean monogenic functions by the characterization of the matrix Hilbert transform in the quaternionic Clifford analysis. Then we establish the Kerzman-Stein formula which closely connects the matrix Szegö projection operator with the Hardy projection operator onto the Hardy space, and we get the matrix Szegö projection operator in terms of the Hardy projection operator and its adjoint. At last, we construct the explicit matrix Szegö kernel function for the Hardy space on the sphere as an example, and get the solution to a Diriclet boundary value problem for matrix functions.

• The Pure and Applied Mathematics
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• v.21 no.1
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• pp.51-60
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• 2014

In this paper, we construct a complex reproducing kernel space for singular multi-point BVPs, and skillfully obtain reproducing kernel expressions. Then, we transform the problem into an equivalent operator equation, and give a numerical algorithm to provide the approximate solution. The uniform convergence of this algorithm is proved, and complexity analysis is done. Lastly, we show the validity and feasibility of the numerical algorithm by two numerical examples.

• Journal of the Korean Mathematical Society
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• v.38 no.2
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• pp.469-483
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• 2001

The configuration space and phase space oscillatory path integrals are computed in the case of the stochastic Schrodinger equation for the harmonic oscillator with a stochastic term of the form (K$\psi$(sub)t)(x) o dW(sub)t, where K is either the position operator or the momentum operator, and W(sub)t is the Wiener process. In this way formulae are derived for the stochastic analogues of the Mehler kernel.

• Honam Mathematical Journal
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• v.36 no.4
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• pp.777-786
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• 2014

We construct an orthonormal basis for the Bergman space associated to a simply connected domain. We use the or-thonormal basis for the Hardy space consisting of the Szegő kernel and the Riemann mapping function and rewrite their area integrals in terms of arc length integrals using the complex Green's identity. And we make a note about the matrix of a Toeplitz operator with respect to the orthonormal basis constructed in the paper.

• Kyungpook Mathematical Journal
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• v.50 no.2
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• pp.297-306
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• 2010

In this paper, the integral operator is used. We give a new Hilbert-type integral inequality, whose kernel is the homogeneous form with degree - $\lambda$ and with three pairs of conjugate exponents and the best constant factor and its reverse form are also derived. It is shown that the results of this paper represent an extension as well as some improvements of the earlier results.

• Bulletin of the Korean Mathematical Society
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• v.59 no.5
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• pp.1119-1129
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• 2022

We formulate the matrix representation of a composition operator on the Hardy space of the unit disc with the symbol which is a Riemann map of the unit disc, with respect to a special orthonormal basis.

• Journal of the Korean Mathematical Society
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• v.35 no.3
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• pp.527-538
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• 1998

We study on kernel operators (Wick monomials) on symmetric Fock space. We give optimal conditions on kernels so that the corresponding kernel operators are densely defined linear operators on the Fock space. We try to formulate our results in the framework of white noise analysis as much as possible. The most of the results in this paper can be extended to anti-symmetric Fock space.

• Journal of applied mathematics & informatics
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• v.8 no.2
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• pp.659-668
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• 2001

Let S(z) be a power series with operator coefficients such that multiplication by S(z) is an everywhere defined transformation in the square summable power series C(z). In this paper we show that there exists a reproducing kernel Krein space which is state space of extended canonical linear system with transfer function S(z). Also we characterize the reproducing kernel function of the state space of a linear system.

• Journal of the Korean Institute of Intelligent Systems
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• v.11 no.6
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• pp.454-458
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• 2001

Based on a fuzzy system representation of gray scale images, we derive and edge detection algorithm whose convolution kernel is different from the known kernels such as those of Robert's Prewitt's or Sobel's gradient. Our fuzzy system representation is an exact representation of the bicubic spline function which represents the gray scale image approximately. Hence the fuzzy system is a continuous function and it provides a natural way to define the gradient and the Laplacian operator. We show that the gradient at grid points can be evaluated by taking the convolution of the image with a 3$\times$3 kernel. We also that our gradient coupled with the approximate value of the continuous function generates an edge detection method which creates edge images clearer than those by other methods. A few examples of applying our methods are included.

• The Korean Journal of Applied Statistics
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• v.26 no.1
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• pp.131-139
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• 2013

Support vector machines(SVM) may perform poorly in the presence of noise variables; in addition, it is difficult to identify the importance of each variable in the resulting classifier. A feature selection can improve the interpretability and the accuracy of SVM. Most existing studies concern feature selection in the linear SVM through penalty functions yielding sparse solutions. Note that one usually adopts nonlinear kernels for the accuracy of classification in practice. Hence feature selection is still desirable for nonlinear SVMs. In this paper, we compare the performances of nonlinear feature selection methods such as component selection and smoothing operator(COSSO) and kernel iterative feature extraction(KNIFE) on simulated and real data sets.