Π-REGULAR VARIATION

Π-REGULAR VARIATION

August 1981 | J. L. GELUK
The paper by J. L. Geluk focuses on the concept of II-regularly varying functions, which are a specific type of function that exhibits a particular form of asymptotic behavior. A function \( U: R^+ \to R^+ \) is said to be II-regularly varying with exponent \( \alpha \) if \( U(x)x^{-\alpha} \) is nondecreasing and there exists a positive function \( L \) such that a specific limit condition holds as \( x \to \infty \). The author introduces the Laplace-Stieltjes transform \( \hat{U}(t) \) of \( U \) and proves that \( U \) is II-regularly varying if and only if \( \hat{U} \) is II-regularly varying. The paper builds on the theory of regularly varying functions, which were developed by Karamata, and extends it to the class of II-regularly varying functions. It includes definitions, theorems, and proofs that establish the equivalence of various conditions for functions to belong to the class II. Key results include the characterization of II-regularly varying functions using fractional integrals and the relationship between the Laplace-Stieltjes transform and the auxiliary function of the function in question. The author also provides a detailed proof of a theorem that generalizes Karamata's theorems for nondecreasing functions, showing that the conditions for II-regularly varying functions are equivalent to specific limit conditions involving the auxiliary function and the Laplace-Stieltjes transform. The paper concludes with a discussion of the implications of these results and references to relevant literature.The paper by J. L. Geluk focuses on the concept of II-regularly varying functions, which are a specific type of function that exhibits a particular form of asymptotic behavior. A function \( U: R^+ \to R^+ \) is said to be II-regularly varying with exponent \( \alpha \) if \( U(x)x^{-\alpha} \) is nondecreasing and there exists a positive function \( L \) such that a specific limit condition holds as \( x \to \infty \). The author introduces the Laplace-Stieltjes transform \( \hat{U}(t) \) of \( U \) and proves that \( U \) is II-regularly varying if and only if \( \hat{U} \) is II-regularly varying. The paper builds on the theory of regularly varying functions, which were developed by Karamata, and extends it to the class of II-regularly varying functions. It includes definitions, theorems, and proofs that establish the equivalence of various conditions for functions to belong to the class II. Key results include the characterization of II-regularly varying functions using fractional integrals and the relationship between the Laplace-Stieltjes transform and the auxiliary function of the function in question. The author also provides a detailed proof of a theorem that generalizes Karamata's theorems for nondecreasing functions, showing that the conditions for II-regularly varying functions are equivalent to specific limit conditions involving the auxiliary function and the Laplace-Stieltjes transform. The paper concludes with a discussion of the implications of these results and references to relevant literature.
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