• Bulletin of the Korean Mathematical Society
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• v.26 no.1
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• pp.11-17
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• 1989

In [1], Johnson and Lapidus introduced a family { $A_{t}$ :t>0} of Banach algebras of functionals on Wiener space and showed that for every F in $A_{t}$ , the analytic operator-valued function space integral $K_{\lambda}$$^{t}$ (F) exists for all nonzero complex numbers .lambda. with nonnegative real part. In [2,3] Johnson and Lapidus introduced a noncommtative multiplication having the property that if F.mem. $A_{t}$ $_{1}$ and G.mem. $A_{t}$ $_{2}$ then $F^{*}$G.mem. A$t_{1}$+$_{t}$ $_{2}$ and (Fig.) Note that for F, G in $A_{t}$ , $F^{*}$G is not in $A_{t}$ but rather is in $A_{2t}$ and so the multiplication * is not internal to the Banach algebra $A_{t}$ . In this paper we introduce an internal noncommutative multiplication on $A_{t}$ having the property that for F, G in $A_{t}$ , F G is in $A_{t}$ and (Fig.) for all nonzero .lambda. with nonnegative real part. Thus is an auxiliary binary operator on $A_{t}$ .TEX> .

• 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 the Chungcheong Mathematical Society
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• v.35 no.4
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• pp.315-327
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• 2022

In this paper, we investigate the transferred superstability for the p-radical sine functional equation $$f\(\sqrt[p]{\frac{x^p+y^p}{2}}\)^2-f\(\sqrt[p]{\frac{x^p-y^p}{2}}\)^2=f(x)f(y)$$ from the p-radical functional equations: $$f({\sqrt[p]{x^p+y^p}})+f({\sqrt[p]{x^p-y^p}})={\lambda}g(x)g(y),\;\\f({\sqrt[p]{x^p+y^p}})+f({\sqrt[p]{x^p-y^p}})={\lambda}g(x)h(y),$$ where p is an odd positive integer, λ is a positive real number, and f is a complex valued function. Furthermore, the results are extended to Banach algebras. Therefore, the obtained result will be forced to the pre-results(p=1) for this type's equations, and will serve as a sample to apply it to the extension of the other known equations.