• The Pure and Applied Mathematics
• /
• v.11 no.1
• /
• pp.89-96
• /
• 2004

We prove de Leeuw's restriction theorem result Jodeit, Jr. [4] for multipliers on $H^{p}$ spaces, p＜1.

• Journal of the Korean Mathematical Society
• /
• v.30 no.1
• /
• pp.25-39
• /
• 1993

• Bulletin of the Korean Mathematical Society
• /
• v.22 no.2
• /
• pp.83-85
• /
• 1985

Let K be a real function field over a real closed field F. Then there exists an F-place .phi.:K.rarw.F.cup.{.inf.}. This is Lang's Existence of Rational Place Theorem (6). There is an equivalent version of Lang's Theorem in (4). That is, if K is a function field over a field F, then, for any ordering P$_{0}$ on F which extends to K, there exists an F-place .phi.: K.rarw.F'.cup.{.inf.} where F' is a real closure of (F, P$_{0}$). In [2], Knebusch pointed out the converse of the version of Lang's Theorem is also true. By a valuation theoretic approach to Lang's Theorem, we have found out the following generalization of Lang and Knebusch's Theorem. Let K be an arbitrary extension field of a field F. then an ordering P$_{0}$ on F can be extended to an ordering P on K if there exists an F-place of K into some real closed field R containing F. Of course R$^{2}$.cap.F=P$_{0}$. The restriction K being a function field of F is vanished, though the codomain of the F-place is slightly varied. Therefore our theorem is a generalization of Lang and Knebusch's theorem.

• Bulletin of the Korean Mathematical Society
• /
• v.44 no.2
• /
• pp.351-358
• /
• 2007

For an entire function f whose Fourier transform has a compact support confined to $[-{\pi},\;{\pi}]$ and restriction to ${\mathbb{R}}$ belongs to $L^2{\mathbb{R}}$, we derive a nonuniform sampling theorem of Lagrange interpolation type with sampling points ${\lambda}_n{\in}{\mathbb{R}},\;n{\in}{\mathbb{Z}}$, under the condition that $$\frac{lim\;sup}{n{\rightarrow}{\infty}}|{\lambda}_n-n|<\frac {1}{4}$.

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

• Journal of the Korean Statistical Society
• /
• v.4 no.1
• /
• pp.67-78
• /
• 1975

Over twenty-five years ago, Professor Klein and Rubin presented the linear expenditure system. That system was first estimated by Stone. Subsequently many investigators have estimated that system. In this paper, many points of the error structure shown by Pollak and Wales are referred to. Barten presented an estimation theorem on a singular covariance matrix. In order to estimate parameters, we place an emphasis on the maximum likihood method which we believe to be most appropriate. As we have one linear restriction on parameters to be estimated, we maximized the associated likelihood function subject to that linear restriction through the well-known lagrange multiplier method. This paper is organized in the following fashion : (1) a brief description on classical consumer theory, (2) a linear expenditure system and its constraint, (3) dyanmic specification and stochastic specification, (4) estimation method, and (5) conclusion.

• Journal of the Korean Mathematical Society
• /
• v.41 no.4
• /
• pp.667-680
• /
• 2004

We generalize a theorem of W. Benz by proving the following result: Let $H_{\theta}$ be a half space of a real Hilbert space with dimension $\geq$ 3 and let Y be a real normed space which is strictly convex. If a distance $\rho$ > 0 is contractive and another distance N$\rho$ (N $\geq$ 2) is extensive by a mapping f : $H_{\theta}$ \longrightarrow Y, then the restriction f│$_{\theta}$ $H_{+}$$\rho$/2// is an isometry, where $H_{\theta}$＋$\rho$/2/ is also a half space which is a proper subset of $H_{\theta}$. Applying the above result, we also generalize a classical theorem of Beckman and Quarles.

• Journal of the Korean Mathematical Society
• /
• v.58 no.1
• /
• pp.123-132
• /
• 2021

Let K be a number field and L a finite abelian extension of K. Let E be an elliptic curve defined over K. The restriction of scalars ResKLE decomposes (up to isogeny) into abelian varieties over K $$Res^L_KE{\sim}{\bigoplus_{F{\in}S}}A_F,$$ where S is the set of cyclic extensions of K in L. It is known that if L is a quadratic extension, then AL is the quadratic twist of E. In this paper, we consider the case that K is a number field containing a primitive third root of unity, $L=K({\sqrt[3]{D}})$ is the cyclic cubic extension of K for some D ∈ K×/(K×)3, E = Ea : y2 = x3 + a is an elliptic curve with j-invariant 0 defined over K, and EaD : y2 = x3 + aD2 is the cubic twist of Ea. In this case, we prove AL is isogenous over K to $E_a^D{\times}E_a^{D^2}$ and a property of the Selmer rank of AL, which is a cubic analogue of a theorem of Mazur and Rubin on quadratic twists.

• Bulletin of the Korean Mathematical Society
• /
• v.52 no.5
• /
• pp.1401-1422
• /
• 2015

Under the restriction of finite order, we prove two uniqueness theorems of nonconstant meromorphic functions sharing three values with their difference operators, which are counterparts of Theorem 2.1 in [6] for a finite-order meromorphic function and its shift operator.

• The Pure and Applied Mathematics
• /
• v.10 no.3
• /
• pp.193-198
• /
• 2003

Let X and Y be n-dimensional Euclidean spaces with $n\;{\geq}\;3$. In this paper, we generalize a classical theorem of Bookman and Quarles by proving that if a mapping, from a half space of X into Y, preserves a distance $\rho$, then the restriction of f to a subset of the half space is an isometry.