• The Pure and Applied Mathematics
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• v.8 no.1
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• pp.71-78
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• 2001

In 1984, Johnson[A bounded convergence theorem for the Feynman in-tegral, J, Math. Phys, 25(1984), 1323-1326] proved a bounded convergence theorem for hte Feynman integral. This is the first stability theorem of the Feynman integral as an $L(L_2 (\mathbb{R}^N), L_2(\mathbb{R}^{N}))$ theory. Johnson and Lapidus [Generalized Dyson series, generalized Feynman digrams, the Feynman integral and Feynmans operational calculus. Mem, Amer, Math, Soc. 62(1986), no 351] studied stability theorems for the Feynman integral as an $L(L_2 (\mathbb{R}^N), L_2(\mathbb{R}^{N}))$ theory for the functional with arbitrary Borel measure. These papers treat functionals which involve only a single integral. In this paper, we obtain the stability theorems for the Feynman integral as an $L(L_1 (\mathbb{R}^N), L_{\infty}(\mathbb{R}^{N}))$theory for the functionals which involve double integral with some Borel measures.

• Journal of the Chungcheong Mathematical Society
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• v.31 no.4
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• pp.487-509
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• 2018

The Truncated Moment Problem (TMP) entails finding a positive Borel measure to represent all moments in a finite sequence as an integral; once the sequence admits one or more such measures, it is known that at least one of the measures must be finitely atomic with positive densities (equivalently, a linear combination of Dirac point masses with positive coefficients). On the contrary, there are more general moment problems for which we aim to find a "signed" measure to represent a sequence; that is, the measure may have some negative densities. This type of problem is referred to as the General Truncated Moment Problem (GTMP). The Jordan Decomposition Theorem states that any (signed) measure can be written as a difference of two positive measures, and hence, in the view of this theorem, we are able to apply results for TMP to study GTMP. In this note we observe differences between TMP and GTMP; for example, we cannot have an analogous to the Flat Extension Theorem for GTMP. We then present concrete solutions to lower-degree problems.

• Journal of the Chungcheong Mathematical Society
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• v.20 no.3
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• pp.333-342
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• 2007

Suppose that ${\mu}$ is a finite positive Borel measure on the unit ball $B{\subset}C^n$. The boundary of B is the unit sphere $S=\{z:{\mid}z{\mid}=1\}$. Let ${\sigma}$ be the rotation-invariant measure on S such that ${\sigma}(S)=1$. In this paper, we will show that if $sup_{{\zeta}{\in}S}\;{\int}_{B}\;P(z,{\zeta})d{\mu}(z)$<${\infty}$ where $P(z,{\zeta})$ is the Poission-Szeg$\ddot{o}$ kernel for B, then ${\mu}$ is a Carleson measure. We will also show that if $sup_{{\zeta}{\in}S}\;{\int}_{B}\;P(z,{\zeta})d{\mu}(z)$<${\infty}$, then the operator T such that T(f) = P[f] is compact as a mapping from $L^p(\sigma)$ into $L^p(B,d{\mu})$.

• Journal of the Chungcheong Mathematical Society
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• v.22 no.1
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• pp.89-99
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• 2009

Let $\mu$ be a finite positive Borel measure on the unit ball $B{\subset}\mathbb{C}^n$ and $\nu$ be the Euclidean volume measure such that ${\nu}(B)=1$. For the unit sphere $S=\{z:{\mid}z{\mid}=1\}$, $\sigma$ is the rotation-invariant measure on S such that ${\sigma}(S)=1$. Let $\mathcal{P}[f]$ be the invariant Poisson integral of f. We will show that there is a constant M > 0 such that $\int_B{\mid}{\mathcal{P}}[f](z){\mid}^{p}d{\mu}(z){\leq}M\;{\int}_B{\mid}{\mathcal{P}}[f](z)^pd{\nu}(z)$ for all $f{\in}L^p({\sigma})$ if and only if ${\parallel}{\mu}{\parallel_r}\;=\;sup_{z{\in}B}\;\frac{\mu(E(z,r))}{\nu(E(z,r))}\;<\;\infty$.

• Journal of the Korean Mathematical Society
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• v.32 no.4
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• pp.697-712
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• 1995

In the present article we consider multiparameter maximal averages and discover the crucial roles played by the number of parameters in their boundedness properties. The problem we shall deal with is initiated by Rubio de Francia [8] and will be in the spirit of an inductive extension to multiparameter cases, in which tools of our study rely on the theory of Harmonic Analysis on product spaces. Suppose that $d_\mu$ is a complex Borel measure supported on a compact subset S of $R^N$ having total mass one, $\smallint_S d_\mu = 1$.

• Honam Mathematical Journal
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• v.34 no.2
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• pp.219-230
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• 2012

In the paper, a symbolic construction is considered to define generalized Cantor-like sets. Lower and upper bounds for the correlation dimension of the sets with a regular condition are obtained with respect to a probability Borel measure. Especially, for some special cases of the sets, the exact formulas of the correlation dimension are established and we show that the correlation dimension and the Hausdorff dimension of some of them are the same. Finally, we find a condition which guarantees the positive correlation dimension of the generalized Cantor-like sets.

• Journal of applied mathematics & informatics
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• v.23 no.1_2
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• pp.489-496
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• 2007

Let g be a continuous function on an interval I which is not constant on any subinterval of I, and let ${\mu}$ be a Borel measure on I. In this paper we give a necessary and sufficient conditions guaranteeing, for the strongly measurable function f on I with values in a Banach space X, the existence of a continuous primitive function F on I with respect to g.

• Journal of the Chungcheong Mathematical Society
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• v.23 no.3
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• pp.471-479
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• 2010

For a complex regular Borel measure ${\mu}$ on ${\Omega}$ which is a subset of ${\mathbb{C}}^k$, where k is a positive integer we define the Toeplitz operator $T_{\mu}$ on a reproducing analytic space which comtains polynomials. Using every symmetric polynomial is a polynomial of elementary polynomials, we show that if $T_{\mu}$ has finite rank then ${\mu}$ is a finite linear combination of point masses.

• Bulletin of the Korean Mathematical Society
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• v.33 no.1
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• pp.135-170
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• 1996

Recently many people have studied the Sobolev orthogonal polynomials, that is, polynomials which are orthogonal relative to a symmetric bilinear form $\phi(\cdot,\cdot)$ defined by $$ (1.1) $\phi(p,q) := (p,q)_N = \sum_{k=0}^{N} \int_{R}p^(k) (x)q^(k) (x) d\mu_k, $$ where each $d\mu_k$ is a signed Borel measure on the real line $R$ with finite moments of all orders. For the brief history on this subject, we refer to the survey article Ronveaux [13] and Marcellan and et al [10].

• Communications of the Korean Mathematical Society
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• v.13 no.4
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• pp.781-789
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• 1998

Let K be a loosely self-similar set. Then a-dimensional packing measure of K is the same as that of a Borel subset K( $r_1^{\alpha}$ㆍㆍㆍ$r_{m}$ $^{\alpha}$/) of K. And packing dimension of K is equal to that of K\K( $r_1^{\alpha}$ㆍㆍㆍ $r_{m}$ $^{\alpha}$/) and K( $r_1^{\alpha}$ㆍㆍㆍ $r_{m}$ $^{\alpha}$/).X> $^{\alpha}$/)).