• Korean Journal of Mathematics
• /
• v.27 no.1
• /
• pp.221-267
• /
• 2019

In the five first sections of this paper we define and study the hypergeometric transmutation operators $V^W_k$ and $^tV^W_k$ called also the trigonometric Dunkl intertwining operator and its dual corresponding to the Heckman-Opdam's theory on ${\mathbb{R}}^d$. By using these operators we define the hypergeometric translation operator ${\mathcal{T}}^W_x$, $x{\in}{\mathbb{R}}^d$, and its dual $^t{\mathcal{T}}^W_x$, $x{\in}{\mathbb{R}}^d$, we express them in terms of the hypergeometric Fourier transform ${\mathcal{H}}^W$, we give their properties and we deduce simple proofs of the Plancherel formula and the Plancherel theorem for the transform ${\mathcal{H}}^W$. We study also the hypergeometric convolution product on W-invariant $L^p_{\mathcal{A}k}$-spaces, and we obtain some interesting results. In the sixth section we consider a some root system of type $BC_d$ (see [17]) of whom the corresponding hypergeometric translation operator is a positive integral operator. By using this positivity we improve the results of the previous sections and we prove others more general results.

• Journal for History of Mathematics
• /
• v.33 no.5
• /
• pp.277-287
• /
• 2020

There are many results for Zadeh's max-min composition operators based on Zadeh's extension principle. We calculate Zadeh's max-min composition operators for two triangular fuzzy numbers defined on ℝ.

• Management Science and Financial Engineering
• /
• v.2 no.1
• /
• pp.73-101
• /
• 1996

We show assets can be classified into diversifiable risks and non-diversifiable risks based on aggregate endowment and spanning so that in equilibrium agents eliminate diversifiable risks which must have zero values. Consequently, the benchmark portfolio that represents a pricing operator should have only a non-diversifiable risk, aggregate endowment should earn a positive risk premium over a riskless asset, and, even in incomplete markets, there should be a pricing operator represented by a function of aggregate endowment if any asset mean-independent of aggregate endowment is diversifiable. These results apply to both the CAPM and a representative agent model.

• Bulletin of the Korean Mathematical Society
• /
• v.53 no.3
• /
• pp.711-722
• /
• 2016

We study Toeplitz operators $T_{\nu}$ on generalized Fock spaces $F^2_{\phi}$ with a locally finite positive Borel measures ${\nu}$ as symbols. We characterize operator-theoretic properties (boundedness and compactness) of $T_{\nu}$ in terms of the Fock-Carleson measure and the Berezin transform ${\tilde{\nu}}$.

• Bulletin of the Korean Mathematical Society
• /
• v.52 no.5
• /
• pp.1649-1660
• /
• 2015

In this paper, we completely characterize the nontrivial reducing subspaces of the Toeplitz operator $T{_{z{^N_1{\bar{z}}^M_2}}$ on the Bergman space $A^2(\mathbb{D}^2)$, where N and M are positive integers.

• Kyungpook Mathematical Journal
• /
• v.49 no.1
• /
• pp.57-65
• /
• 2009

In the papers [5]-[7] was examined approximation of functions by the modified Sz$\'{a}$sz-Mrakyan operators and other positive linear operators preserving $e_2(x)=x^2$. In this paper we introduce the Post-Widder and Stancu operators preserving $x^2$ in polynomial weighted spaces. We show that these operators have better approximation properties than classical Post-Widder and Stancu operators.

• Journal of applied mathematics & informatics
• /
• v.20 no.1_2
• /
• pp.557-567
• /
• 2006

This paper proves the invariability of reachable sets for the linear control system with positive isolated spectrum points in case where the principal operator generates $C_0-semigroup$ and derives the approximate controllability for the semilinear control system by using spectral operators with respect to isolated spectrum points.

• Bulletin of the Korean Mathematical Society
• /
• v.34 no.1
• /
• pp.103-114
• /
• 1997

The asymptotic Dirichlet problem for harmonic functions on a noncompact complete Riemannian manifold has a long history. It is to find the harmonic function satisfying the given Dirichlet boundary condition at infinity. By now, it is well understood [A, AS, Ch, S], when M is a Cartan-Hadamard manifold with sectional curvature $-b^2 \leq K_M \leq -a^2 < 0$. (By a Cartan-Hadamard manifold, we mean a complete simply connected manifold of non-positive sectional curvature.)

• Journal of Applied and Pure Mathematics
• /
• v.5 no.5_6
• /
• pp.407-414
• /
• 2023

In this article we continue the study of smooth Picard singular integral operators that started in [2], see there chapters 10-14. This time the foundation of our research is a trigonometric Taylor's formula. We establish the convergence of our operators to the unit operator with rates via Jackson type inequalities engaging the first modulus of continuity. Of interest here is a residual appearing term. Note that our operators are not positive. Our results are pointwise and uniform.

• Communications of the Korean Mathematical Society
• /
• v.13 no.3
• /
• pp.525-538
• /
• 1998

Recently, a stability theorem for the Feynman integral as a bounded linear operator on$ L_2$($R^{d}$ /) with respect to measures whose positive and negative variations are in the generalized Kato class was proved. We study a stability theorem for the Feynman integral with respect to measures whose positive variations are in the class of $\sigma$-finite smooth measures and negative variations are in the generalized Kato class. This extends the recent result in the sense that the class of $\sigma$-finite smooth measures properly contains the generalized Kato class.