• 대한수학회보
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• 제37권3호
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• pp.601-618
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• 2000

Let (and $g^*$) be the space of regular test (and generalized, resp.) white noise functions. The integral kernel operators acting on and transformation groups of operators on are studied, and then every integral kernel operator acting on can be extended to continuous linear operator on $g^*$. The existence and uniqueness of solutions of Cauchy problems associated with certain integral kernel operators with intial data in $g^*$ are investigated.

• 대한수학회보
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• 제55권6호
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• pp.1791-1809
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• 2018

Let T be the singular integral operator with nonsmooth kernel which was introduced by Duong and McIntosh, and $T_q(q{\in}(1,{\infty}))$ be the vector-valued operator defined by $T_qf(x)=({\sum}_{k=1}^{\infty}{\mid}T\;f_k(x){\mid}^q)^{1/q}$. In this paper, by proving certain weak type endpoint estimate of L log L type for the grand maximal operator of T, the author establishes some quantitative weighted bounds for $T_q$ and the corresponding vector-valued maximal singular integral operator.

• Kyungpook Mathematical Journal
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• 제50권2호
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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.

• 대한수학회논문집
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• 제11권2호
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• pp.367-375
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• 1996

Suppose the kernel function $\kappa$ belongs to $S(R^2)$ and is symmetric such that $ < \otimes x, \kappa >\geq 0$ for all $x \in S'(R)$. Let A be the class of functions f such that the function f is measurable on $S'(R)$ with $\int_{S'(R)}$\mid$f((I + tK)^{\frac{1}{2}}x$\mid$^2d\mu(x) < M$ for some $M > 0$ and for all t > 0, where K is the integral operator with kernel function $\kappa$. We show that the \lambda$-potential $G_Kf$ of f is a weak solution of $(\lambda I - \frac{1}{2} \tilde{\Xi}_{0,2}(\kappa))_u = f$.

• Journal of applied mathematics & informatics
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• 제6권1호
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• pp.163-174
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• 1999

The purpose of this paper is to obtain the solution of Fredholm-Volterra integral equation with singular kernel in the space $L_2(-1, 1)\times C(0,T), 0 \leq t \leq T< \infty$, under certain conditions,. The numerical method is used to solve the Fredholm integral equation of the second kind with weak singular kernel using the Toeplitz matrices. Also the error estimate is computed and some numerical examples are computed using the MathCad package.

• 대한수학회논문집
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• 제33권3호
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• pp.787-798
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• 2018

For $1{\leq}p{\leq}{\infty}$, let $H^p$ be the usual Hardy space on the unit circle. When ${\alpha}$ and ${\beta}$ are bounded functions, a singular integral operator $S_{{\alpha},{\beta}}$ is defined as the following: $S_{{\alpha},{\beta}}(f+{\bar{g}})={\alpha}f+{\beta}{\bar{g}}(f{\in}H^p,\;g{\in}zH^p)$. When p = 2, we study the hyponormality of $S_{{\alpha},{\beta}}$ when ${\alpha}$ and ${\beta}$ are some special functions.

• Kyungpook Mathematical Journal
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• 제59권3호
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• pp.433-443
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• 2019

In this work, we evaluate the Mellin transform of the Marichev-Saigo-Maeda fractional integral operator with Appell's function $F_3$ type kernel. We then discuss six special cases of the result involving the Saigo fractional integral operator, the $Erd{\acute{e}}lyi$-Kober fractional integral operator, the Riemann-Liouville fractional integral operator and the Weyl fractional integral operator. We obtain new and known results as special cases of our main results. Finally, we obtain the images of S-generalized Gauss hypergeometric function under the operators of our study.

• Journal of applied mathematics & informatics
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• 제32권5_6호
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• pp.567-584
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• 2014

In this paper, we consider approximation of eigenelements of a two dimensional compact integral operator with a smooth kernel by discrete collocation and iterated discrete collocation methods. By choosing numerical quadrature appropriately, we obtain convergence rates for gap between the spectral subspaces, and also we obtain superconvergence rates for eigenvalues and iterated eigenvectors. We then apply Richardson extrapolation to obtain further improved error bounds for the eigenvalues. Numerical examples are presented to illustrate theoretical estimates.

• 대한수학회지
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• 제33권4호
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• pp.929-954
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• 1996

• 대한수학회논문집
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• 제35권1호
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• pp.125-136
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• 2020

In this article, we discuss classes of generalized functions for certain modified integral operator of Bessel-type involving Fox's H-function kernel. We employ a known differentiation formula of Fox's H-function to obtain the definition and properties of the distributional modified Bessel-type integral. Further, we derive a smoothness theorem for its kernel in a complete countably multi-normed space. On the other hand, using an appropriate class of convolution products, we derive axioms and establish spaces of modified Boehmians which are generalized distributions. On the defined spaces, we introduce addition, convolution, differentiation and scalar multiplication and further properties of the extended integral.