• Journal of the Korean Mathematical Society
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• v.42 no.3
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• pp.599-619
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• 2005

We define the convolutions of Fourier hyperfunctions and show that every strongly decreasing Fourier hyperfunction is a convolutor for the space of Fourier hyperfunctions and the converse is true. Also we show that there are no differential operator with constant coefficients which have a fundamental solution in the space of strongly decreasing Fourier hyperfunctions. Lastly we show that the space of multipliers for the space of Fourier hyperfunctions consists of analytic functions extended to any strip in $\mathbb{C}^n$ which are estimated with a special exponential function exp$(\mu|\chi|)$.

• Bulletin of the Korean Mathematical Society
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• v.41 no.4
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• pp.785-803
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• 2004

We research properties of analytic functions which are exponentially decreasing or increasing. Also we show that the space of test functions is dense in the space of extended Fourier hyper-functions, and that the Fourier transform of the space of extended Fourier hyperfunctions into itself is an isomorphism and Parseval's inequality holds.

• Bulletin of the Korean Mathematical Society
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• v.46 no.2
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• pp.295-301
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• 2009

We prove the Hyers-Ulam stability of a Pexiderized exponential equation of mappings f, g, h : $G{\times}S{\rightarrow}{\mathbb{C}}$, where G is an abelian group and S is a commutative semigroup which is divisible by 2. As an application we obtain a stability theorem for Pexiderized exponential equation in Schwartz distributions.

• Communications of the Korean Mathematical Society
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• v.19 no.4
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• pp.661-681
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• 2004

We research properties of the space of measurable functions square integrable with weight exp$2\nu $\mid$x$\mid$$, and those of the space of Fourier hyperfunctions. Also we show that the several embedding theorems hold true, and that the Fourier-Lapace operator is an isomorphism of the space of strongly decreasing Fourier hyperfunctions onto the space of analytic functions extended to any strip in $C^n$ which are estimated with the aid of a special exponential function exp($\mu$｜x｜).

• Bulletin of the Korean Mathematical Society
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• v.33 no.4
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• pp.545-551
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• 1996

It is well known in the theory of distributions and proved in [GS, S] that $$ (i) (Bochner-Schwartz) Every positive definite (tempered) distribution is the Fourier transform of a positive tempered measure \mu. $$ $$ (ii) (Schwartz kernel theorem) Let B(\varphi, \psi) be a bilinear distribution. Then for some u \in D'(R^n \times R^n) B(\varphi, \psi) = u(\varphi(x)\bar{\psi}(y)) for every \varphi, \psi \in C_c^\infty. $$ $$ (iii) A translation invariant positive definite bilinear distribution B(\varphi, \psi) is of the form B(\varphi, \psi) = \smallint \varphi(x)\psi(x) d\mu(x) for every \varphi, \psi \in C_c^\infty (R^n), where \mu is a positive tempered measure.

• Journal of the Chungcheong Mathematical Society
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• v.21 no.3
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• pp.361-370
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• 2008

We consider the Hyers-Ulam type stability of the Cauchy, Jensen, Pexider, Pexider-Jensen differences: $$(0.1){\hspace{55}}C(u):=u{\circ}A-u{\circ}P_1-u{\circ}P_2,\\(0.2){\hspace{55}}J(u):=2u{\circ}\frac{A}{2}-u{\circ}P_1-u{\circ}P_2,\\(0.3){\hspace{18}}P(u,v,w):=u{\circ}A-v{\circ}P_1-w{\circ}P_2,\\(0.4)\;JP(u,v,w):=2u{\circ}\frac{A}{2}-v{\circ}P_1-w{\circ}P_2$$, with respect to bounded distributions.