• Communications of the Korean Mathematical Society
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• v.31 no.4
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• pp.791-809
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• 2016

By introducing two fractional convolutions, we obtain the convolution theorems for fractional Fourier cosine and sine transforms. Applying these convolutions, we construct two Boehmian spaces and then we extend the fractional Fourier cosine and sine transforms from these Boehmian spaces into another Boehmian space with desired properties.

• Kyungpook Mathematical Journal
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• v.54 no.1
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• pp.43-63
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• 2014

The range spaces of Fourier cosine and sine transforms on $L^1$([0, ${\infty}$)) are characterized. Using Fourier cosine and sine type convolutions, Fourier cosine and sine transformable Boehmian spaces have been constructed, which properly contain $L^1$([0, ${\infty}$)). The Fourier cosine and sine transforms are extended to these Boehmian spaces consistently and their properties are established.

• Bulletin of the Korean Mathematical Society
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• v.43 no.4
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• pp.831-839
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• 2006

By relaxing the requirements for a sequence of functions to be a delta sequence, a space of Boehmians on the torus ${\beta}(T^d)$ is constructed and studied. The space ${\beta}(T^d)$ contains the space of distributions as well as the space of hyperfunctions on the torus. The Fourier transform is a continuous mapping from ${\beta}(T^d)$ onto a subspace of Schwartz distributions. The range of the Fourier transform is characterized. A necessary and sufficient condition for a sequence of Boehmians to converge is that the corresponding sequence of Fourier transforms converges in $D'({\mathbb{R}}^d)$.

• Bulletin of the Korean Mathematical Society
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• v.44 no.3
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• pp.419-428
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• 2007

The space of periodic Boehmians with $\Delta$-convergence is a complete topological algebra which is not locally convex. A family of Boehmians $\{T_\lambda\}$ such that $T_0$ is the identity and $T_{\lambda_1+\lambda_2}=T_\lambda_1*T_\lambda_2$ for all real numbers $\lambda_1$ and $\lambda_2$ is called a one-parameter group of Boehmians. We show that if $\{T_\lambda\}$ is strongly continuous at zero, then $\{T_\lambda\}$ has an exponential representation. We also obtain some results concerning the infinitesimal generator for $\{T_\lambda\}$.

• Communications of the Korean Mathematical Society
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• v.33 no.2
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• pp.515-525
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• 2018

In this paper, we investigate a modified $G^2$ transform on a class of Boehmians. We prove the axioms which are necessary for establishing the $G^2$ class of Boehmians. Addition, scalar multiplication, convolution, differentiation and convergence in the derived spaces have been defined. The extended $G^2$ transform of a Boehmian is given as a one-to-one onto mapping that is continuous with respect to certain convergence in the defined spaces. The inverse problem is also discussed.