• Communications of the Korean Mathematical Society
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• v.25 no.1
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• pp.27-36
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• 2010

We give a simple proof of certain mapping properties of the p-adic Riesz potential and Bessel potential, and the p-adic version of the Sobolev embedding theorem obtained in [6].

• Honam Mathematical Journal
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• v.35 no.4
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• pp.667-677
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• 2013

A remarkably large number of integrals involving a product of certain combinations of Bessel functions of several kinds as well as Bessel functions, themselves, have been investigated by many authors. Motivated the works of both Garg and Mittal and Ali, very recently, Choi and Agarwal gave two interesting unified integrals involving the Bessel function of the first kind $J_{\nu}(z)$. In the present sequel to the aforementioned investigations and some of the earlier works listed in the reference, we present two generalized integral formulas involving a product of Bessel functions of the first kind, which are expressed in terms of the generalized Lauricella series due to Srivastava and Daoust. Some interesting special cases and (potential) usefulness of our main results are also considered and remarked, respectively.

• Communications of the Korean Mathematical Society
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• v.38 no.3
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• pp.787-805
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• 2023

Let (Ω, µ) be a measure space and {τ_α}_α∈Ω be a normalized continuous Bessel family for a finite dimensional Hilbert space 𝓗 of dimension d. If the diagonal ∆ := {(α, α) : α ∈ Ω} is measurable in the measure space Ω × Ω, then we show that $$\sup\limits_{{\alpha},{\beta}{\in}{\Omega},{\alpha}{\neq}{\beta}}\,{\mid}{\langle}{\tau}_{\alpha},\,{\tau}_{\beta}{\rangle}{\mid}^{2m}\,{\geq}\,{\frac{1}{({\mu}{\times}{\mu})(({\Omega}{\times}{\Omega}{\backslash}{\Delta})}\;\[\frac{{\mu}({\Omega})^2}{\({d+m-1 \atop m}\)}-({\mu}{\times}{\mu})({\Delta})\],\;{\forall}m{\in}{\mathbb{N}}.$$ This improves 48 years old celebrated result of Welch [41]. We introduce the notions of continuous cross correlation and frame potential of Bessel family and give applications of continuous Welch bounds to these concepts. We also introduce the notion of continuous Grassmannian frames.

• Korean Journal of Mathematics
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• v.8 no.1
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• pp.73-86
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• 2000

We study the Sobolev spaces to the generalized Sobolev spaces $H^s_{\mathcal{G}}$ of exponential type based on the Silva space $\mathcal{G}$ and investigate its properties such as imbedding theorem and structure theorem. In fact, the imbedding theorem says that for $s$ > 0 $u{\in}H^s_{\mathcal{G}}$ can be analytically continued to the set {$z{\in}\mathbb{C}^n{\mid}{\mid}Im\;z{\mid}$ < $s$}. Also, the structure theorem means that for $s$ > 0 $u{\in}H^{-s}_{\mathcal{G}}$ is of the form $$u={\sum_{\alpha}\frac{s^{{|\alpha|}}}{{\alpha}!}D^{\alpha}g{\alpha}$$ where $g{\alpha}$'s are square integrable functions for ${\alpha}{\in}\mathbb{N}^n_0$. Moreover, we introduce a classes of symbols of exponential type and its associated pseudo-differential operators of exponential type, which naturally act on the generalized Sobolev spaces of exponential type. Finally, a generalized Bessel potential is defined and its properties are investigated.

• Communications of the Korean Mathematical Society
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• v.10 no.1
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• pp.49-55
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• 1995

This paper is the third of a series in which smoothness properties of function in several variables are discussed. The germ of the whole theory was laid in the works by Folland and Stein [4]. On nilpotent Lie groups, they difined analogues of the classical $L^p$ Sobolev or potential spaces in terms of fractional powers of sub-Laplacian, L and extended several basic theorems from the Euclidean theory of differentaiability to these spaces: interpolation properties, boundedness of singular integrals,..., and imbeding theorems. In this paper we study the analogue to the extension theorem for the Folland-Stein spaces. The analogue to Stein's restriction theorem were studied by M. Mekias [5] and Y.M. Kim [6]. First, we have the space of Bessel potentials on the Heisenberg group introduced by Folland [4].

• Proceedings of the KSME Conference
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• 2001.06a
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• pp.628-635
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• 2001

Hydrodynamic behavior and response of vertical-cylindrical liquid-storage tank is considered. The equation of the liquid motion is shown by Laplace's differential equation with the fluid velocity potential. The solution of the Laplace's differential equation of the liquid motion is expressed with the modified Bessel functions. Only rigid tank is studied. The effective masses and heights for the tank contents are presented for engineering design model.

• Advances in materials Research
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• v.9 no.1
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• pp.15-32
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• 2020

In this article, an analytical solution is presented for the steady-state axisymmetric thermal stress distributions in a composite hollow cylinder. The cylinder is composed of two isotropic and anisotropic materials which is subjected to the thermal boundary conditions of convective as well as radiative heating and cooling on the inner and outer surfaces, respectively. The solution of the temperature is obtained by means of Bessel functions and the thermal stresses are developed using Potential functions of displacement. Numerical results are derived for a cylinder which is similar to a gas turbine combustor and showed that the maximum temperature and thermal stresses (radial, hoop, axial) occurred in the middle point of cylinder and the values of thermal stresses in anisotropic cylinder are more than the isotropic cylinder. It is worthy to note that the values of the thermal conditions which estimated in this research, not to be presented in any other papers but these values are very accurate in calculation.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 2005.05a
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• pp.295-300
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• 2005

An analytical method for the hydroelastic vibration of a vessel composed of an upper annular plate and a lower circular plate is developed by the Rayleigh-Ritz method. The two plates are clamped along a rigid cylindrical vessel wall. It is assumed that the fluid bounded by a rigid cylindrical vessel is incompressible and non-viscous. The wet mode shape of the plates is assumed as a combination of the dry mode shapes of the plates. The fluid motion is described by using the fluid displacement potential and determined by using the compatibility conditions along the fluid interface with the plate. Minimizing the Rayleigh quotient based on the energy conservation gives an eigenvalue problem. It is found that the theoretical results can predict well the fluid-coupled natural frequencies comparing with the finite element analysis result.

• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.15 no.8 s.101
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• pp.968-974
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• 2005

An analytical method for the hydroelastic vibration of a vessel composed of an upper annular plate and a lower circular plate is developed by the Rayleigh-Ritz method. The two plates are clamped along a rigid cylindrical vessel wall. It is assumed that the fluid bounded by a rigid cylindrical vessel is incompressible and non-viscous. The wet mode shape of the plates is assumed as a combination of the dry mode shapes of the plates. The fluid motion is described by using the fluid displacement potential and determined by using the compatibility conditions along the fluid interface with the plate. Minimizing the Rayleigh quotient based on the energy conservation gives an eigenvalue problem. It is found that the theoretical results can predict well the fluid-coupled natural frequencies comparing with the finite element analysis result.

• Kyungpook Mathematical Journal
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• v.58 no.1
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• pp.19-35
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• 2018

Motivated mainly by certain interesting recent extensions of the generalized hypergeometric function [30] and the second Appell function [6], we introduce here the incomplete Lauricella functions ${\gamma}^{(n)}_A$ and ${\Gamma}^{(n)}_A$ of n variables. We then systematically investigate several properties of each of these incomplete Lauricella functions including, for example, their various integral representations, finite summation formulas, transformation and derivative formulas, and so on. We provide relevant connections of some of the special cases of the main results presented here with known identities. Several potential areas of application of the incomplete hypergeometric functions in one and more variables are also pointed out.