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Kórus, Péter; Szal, Bogdan

On the uniform convergence and integrability of special trigonometric integrals. (English) Zbl 07926893

Math. Inequal. Appl. 27, No. 1, 1-18 (2024).
Summary: Necessary and sufficient conditions for the uniform convergence of trigonometric Fourier integrals are well-established when admissible monotone or general monotone functions are considered. In this paper, we generalize these main results by giving such conditions for the uniform convergence of sine and cosine integrals \(\int^\infty_0 f_1(x) \sin (ux^p) dx\) and \(\int^\infty_0 f_2 (x) \cos (ux^p) dx\) in case of admissible general monotone functions \(f_1\) and \(f_2\). Moreover, we give necessary and sufficient conditions for the \(L^q\)-integrability with the power weights of these integrals when non-negative functions \(f_1\) and \(f_2\) belong to the class \(\overline{GM}_{p \theta}\).

MSC:

42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
26A48 Monotonic functions, generalizations
26A45 Functions of bounded variation, generalizations

Keywords:

sine and cosine integrals; uniform convergence; weighted \(L^q\)-convergence; general monotone functions
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References:

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