• 대한수학회보
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• 제59권6호
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• pp.1539-1555
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• 2022

In this paper, we establish the boundedness and continuity for variation operators for θ-type Calderón-Zygmund singular integrals and their commutators on the Triebel-Lizorkin spaces. As applications, we obtain the corresponding results for the Hilbert transform, the Hermit Riesz transform, Riesz transforms and rough singular integrals as well as their commutators.

• KSII Transactions on Internet and Information Systems (TIIS)
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• 제8권11호
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• pp.4118-4136
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• 2014

To preserve the spatial consistency of low-level features, generalized Riesz-wavelet transform (GRWT) is adopted for fusing multi-modality images. The proposed method can capture the directional image structure arbitrarily by exploiting a suitable parameterization fusion model and additional structural information. Its fusion patterns are controlled by a heuristic fusion model based on image phase and coherence features. It can explore and keep the structural information efficiently and consistently. A performance analysis of the proposed method applied to real-world images demonstrates that it is competitive with the state-of-art fusion methods, especially in combining structural information.

• 대한수학회보
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• 제58권1호
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• pp.235-251
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• 2021

We consider the Schrödinger type operator ��k = (-∆)k+Vk on ℝn(n ≥ 2k + 1), where k = 1, 2 and the nonnegative potential V belongs to the reverse Hölder class RHs with n/2 < s < n. In this paper, we establish the (Lp, Lq)-boundedness of the higher order Riesz transform T��,�� = V2��∇2��-��2 (0 ≤ �� ≤ 1/2 < �� ≤ 1, �� - �� ≥ 1/2) and its adjoint operator T∗��,�� respectively. We show that T��,�� is bounded from Hardy type space $H^1_{\mathcal{L}_2}({\mathbb{R}}_n)$ into Lp2 (ℝn) and T∗��,�� is bounded from ��p1 (ℝn) into BMO type space $BMO_{\mathcal{L}_1}$ (ℝn) when �� - �� > 1/2, where $p_1={\frac{n}{4({\beta}-{\alpha})-2}}$, $p_2={\frac{n}{n-4({\beta}-{\alpha})+2}}$. Moreover, we prove that T��,�� is bounded from $BMO_{\mathcal{L}_1}({\mathbb{R}}_n)$ to itself when �� - �� = 1/2.

• Kyungpook Mathematical Journal
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• 제61권2호
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• pp.371-381
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• 2021

In this paper, we attempt to obtain sufficient conditions for the existence of tight wavelet frame sets in locally compact abelian groups. The condition is generated by modulating a collection of characteristic functions that correspond to a generalized shift-invariant system via the Fourier transform. We present two approaches (for stationary and non-stationary wavelets) to construct the scaling function for L²(G) and, using the scaling function, we construct an orthonormal wavelet basis for L²(G). We propose an open problem related to the extension principle for Riesz wavelets in locally compact abelian groups.

• 대한수학회보
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• 제59권5호
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• pp.1255-1268
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• 2022

We consider the Schrödinger type operator 𝓛 = (-𝚫_ℍⁿ)² + V² on the Heisenberg group ℍⁿ, where 𝚫_ℍⁿ is the sub-Laplacian and the non-negative potential V belongs to the reverse Hölder class RH_s for s ≥ Q/2 and Q ≥ 6. We shall establish the (L^p, L^q) estimates for the Riesz transforms T_𝛼,𝛽,j = V^2𝛼𝛁^j_ℍⁿ𝓛^-𝛽, j = 0, 1, 2, 3, where 𝛁_ℍⁿ is the gradient operator on ℍⁿ, 0 < α ≤ 1-j/4, j/4 < 𝛽 ≤ 1, and 𝛽 - 𝛼 ≥ j/4.

• Journal of applied mathematics & informatics
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• 제18권1_2호
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• pp.339-350
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• 2005

A space-time fractional advection-dispersion equation (ADE) is a generalization of the classical ADE in which the first-order time derivative is replaced with Caputo derivative of order $\alpha{\in}(0,1]$, and the second-order space derivative is replaced with a Riesz-Feller derivative of order $\beta{\in}0,2]$. We derive the solution of its Cauchy problem in terms of the Green functions and the representations of the Green function by applying its Fourier-Laplace transforms. The Green function also can be interpreted as a spatial probability density function (pdf) evolving in time. We do the same on another kind of space-time fractional advection-dispersion equation whose space and time derivatives both replacing with Caputo derivatives.

• 대한수학회보
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• 제56권5호
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• pp.1117-1127
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• 2019

Let ${\mathcal{L}}_2=(-{\Delta})^2+V^2$ be the $Schr{\ddot{o}}dinger$ type operator, where nonnegative potential V belongs to the reverse $H{\ddot{o}}lder$ class $RH_s$, s > n/2. In this paper, we consider the operator $T_{{\alpha},{\beta}}=V^{2{\alpha}}{\mathcal{L}}^{-{\beta}}_2$ and its conjugate $T^*_{{\alpha},{\beta}}$, where $0<{\alpha}{\leq}{\beta}{\leq}1$. We establish the $(L^p,\;L^q)$-boundedness of operator $T_{{\alpha},{\beta}}$ and $T^*_{{\alpha},{\beta}}$, respectively, we also show that $T_{{\alpha},{\beta}}$ is bounded from Hardy type space $H^1_{L_2}({\mathbb{R}}^n)$ into $L^{p_2}({\mathbb{R}}^n)$ and $T^*_{{\alpha},{\beta}}$ is bounded from $L^{p_1}({\mathbb{R}}^n)$ into BMO type space $BMO_{{\mathcal{L}}1}({\mathbb{R}}^n)$, where $p_1={\frac{n}{4({\beta}-{\alpha})}}$, $p_2={\frac{n}{n-4({\beta}-{\alpha})}}$.

• 대한수학회지
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• 제51권5호
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• pp.897-918
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• 2014

This paper addresses multi-window Gabor frames with rational time-frequency product. Such issue was considered by Zibulski and Zeevi (Appl. Comput. Harmonic Anal. 4 (1997), 188-221) in terms of Zak transform matrix (so-called Zibuski-Zeevi matrix), and by many others. In this paper, we introduce of a new Zak transform matrix. It is different from Zibulski-Zeevi matrix, but more direct and convenient for our purpose. Using such Zak transform matrix we characterize rational time-frequency multi-window Gabor frames (Riesz bases and orthonormal bases), and Gabor duals for a Gabor frame. Some examples are also provided, which show that our Zak transform matrix method is efficient.

• 대한수학회보
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• 제44권2호
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• pp.351-358
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• 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}$.

• 대한수학회지
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• 제59권2호
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• pp.235-254
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• 2022

Let ℍⁿ be the Heisenberg group and Q = 2n + 2 be its homogeneous dimension. Let 𝓛 = -∆_ℍⁿ + V be the Schrödinger operator on ℍⁿ, where ∆_ℍⁿ is the sub-Laplacian and the nonnegative potential V belongs to the reverse Hölder class $B_{q_1}$ for q₁ ≥ Q/2. Let H^p_𝓛(ℍⁿ) be the Hardy space associated with the Schrödinger operator 𝓛 for Q/(Q+𝛿₀) < p ≤ 1, where 𝛿₀ = min{1, 2 - Q/q₁}. In this paper, we consider the Hardy type estimates for the operator T_𝛼 = V^𝛼(-∆_ℍⁿ + V )^-𝛼, and the commutator [b, T_𝛼], where 0 < 𝛼 < Q/2. We prove that T_𝛼 is bounded from H^p_𝓛(ℍⁿ) into L^p(ℍⁿ). Suppose that b ∈ BMO^𝜃_𝓛(ℍⁿ), which is larger than BMO(ℍⁿ). We show that the commutator [b, T_𝛼] is bounded from H¹_𝓛(ℍⁿ) into weak L¹(ℍⁿ).