• Journal of the Chungcheong Mathematical Society
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• v.33 no.4
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• pp.497-504
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• 2020

A weight ω on the positive half real line [0, ∞) is a positive continuous function such that ω(s + t) ≤ ω(s)ω(t), for all s, t ∈ [0, ∞), and ω(0) = 1. The weighted convolution Banach algebra L1(ω) is the algebra of all equivalence classes of Lebesgue measurable functions f such that ‖f‖ = ∫0∞|f(t)|ω(t)dt < ∞, under pointwise addition, scalar multiplication of functions, and the convolution product (f ⁎ g)(t) = ∫0t f(t - s)g(s)ds. We give a sufficient condition on a weight function ω(t) in order that L1(ω) has a bounded approximate identity.

• Korean Journal of Mathematics
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• v.29 no.3
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• pp.581-591
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• 2021

In this paper, we discuss and investigate the existence of the inclusion L^p(.),𝜃 (𝜇) ⊆ L^q(.),𝜃 (𝜈), where 𝜇 and 𝜈 are two finite measures on (X, Σ). Moreover, we show that the generalized variable exponent grand Lebesgue space L^p(.),𝜃 (Ω) has a potential-type approximate identity, where Ω is a bounded open subset of ℝ^d.

• Journal of the Korean Mathematical Society
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• v.56 no.5
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• pp.1333-1354
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• 2019

In authors' paper in 2007, it was shown that the BSE-extension of $C^1_0(R)$, the algebra of continuously differentiable functions f on the real number space R such that f and df /dx vanish at infinity, is the Lipschitz algebra $Lip_1(R)$. This paper extends this result to the case of $C^n_0(R^d)$ and $C^{n-1,1}_b(R^d)$, where n and d represent arbitrary natural numbers. Here $C^n_0(R^d)$ is the space of all n-times continuously differentiable functions f on $R^d$ whose k-times derivatives are vanishing at infinity for k = 0, ${\cdots}$, n, and $C^{n-1,1}_b(R^d)$ is the space of all (n - 1)-times continuously differentiable functions on $R^d$ whose k-times derivatives are bounded for k = 0, ${\cdots}$, n - 1, and (n - 1)-times derivatives are Lipschitz. As a byproduct of our investigation we obtain an important result that $C^{n-1,1}_b(R^d)$ has a predual.

• Journal of Popular Narrative
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• v.25 no.4
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• pp.469-506
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• 2019

More than a few existing studies on popular narratives that pay attention to 'place' tend to adopt as their theoretical framework the celebrated distinction between space and place. According to this distinction, to put it simply, space is allegedly mobile, whereas place is static. Given this distinction, and in this age of high-mobility, where the spaces of mobilities seem to rapidly and extensively undermine the places of immobilities, would studies on popular narratives focusing on 'place' still remain convincing? Referring to David Seamon's recent book Life Takes Place: Phenomenology, Lifeworlds, and Place Making, this article aims to consider the possibility of studies on popular narratives in the era of high-mobility. To explore the concept of 'place' through phenomenological methodology, Seamon's book uses a theoretical framework called the 'progressive approximation,' which is attentive to synergistic relationality. According to this approach, the place should first be put under scrutiny as a whole, i.e. as the monad of place. Phenomenological studies on the monad of place as a whole identify places as the fundamental condition for human beings. Then, in accordance with the 'progressive' order of research, places are studied as dyads, i.e. as binary oppositions. Through these analyses, movement/rest, insideness/outsideness, the ordinary/the extra-ordinary, the within/the without, homeworld/alienworld are identified as the five dyads of place. To make a detour around these binary oppositions and confrontations, however, phenomenological studies on place now advance to the higher order of six place triads including place interaction, place identity, place release, place realization, place intensification, and place creation, whereby the study of place progressively approaches the 'approximate' essence of place. Reflectively asking himself about the idea of 'place' in the high-mobility era, the author of this informative and insightful book submits an answer that place is still the fundamental sine qua non of human beings. However, this answer is more likely to be bounded by the binary opposition of space/place, and movement/rest accordingly. In this article, I suggest as an alternative and hopefully more promising answer a perspective of transcending this kind of a dead-end dichotomy and of performing 'place-making' through the mobilities themselves, while presenting a noticeable example of the manner in which research on popular narratives could begin from this perspective.