We use the concept of moduli of continuity to give some generalised characterisation theorems in some generalised Hölder type spaces of functions over \({\mathbb {R}}\), characterising certain Lipschitz type conditions on functions in terms of their Fourier transforms. Our main results are inspired by a theorem of Titchmarsh, combined with the notion of moduli of continuity, for characterising Jacobi–Lipschitz conditions in \(L^2({\mathbb {R}}^+, d\mu (t))\) using approximation by a generalised translation operator.

中文翻译：

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使用连续性模量表征扩展 Lipschitz 类型条件

我们使用连续性模的概念在\({\mathbb {R}}\) 上的一些广义 Hölder 类型函数空间中给出一些广义表征定理，根据傅立叶变换表征函数的某些 Lipschitz 类型条件。我们的主要结果受 Titchmarsh 定理的启发，结合连续性模的概念，用于表征\(L^2({\mathbb {R}}^+, d\mu (t)) 中的Jacobi-Lipschitz 条件\)使用广义翻译算子的近似。