Uniform convergence of double trigonometric integrals. (English) Zbl 1461.42005
Summary: We study the uniform convergence of double cosine integrals, sine-cosine integrals and double sine integrals of double general monotone functions. We extend the results of F. Móricz [J. Math. Anal. Appl. 424, No. 2, 1530–1543 (2015; Zbl 1321.42020); ibid. 354, No. 1, 213–219 (2009; Zbl 1163.42002)] and of A. Debernardi [Anal. Math. 43, No. 2, 193–217 (2017; Zbl 1389.40010)] concerning the uniform convergence of double sine integrals and the results of M. Dyachenko et al. [J. Math. Anal. Appl. 372, No. 1, 328–338 (2010; Zbl 1201.42003)] regarding the uniform convergence of single trigonometric integrals.
MSC:
| 42B10 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
| 42A20 | Convergence and absolute convergence of Fourier and trigonometric series |
| 40A10 | Convergence and divergence of integrals |
| 26B30 | Absolutely continuous real functions of several variables, functions of bounded variation |
Keywords:
uniform convergence; regular convergence; double cosine integral; sine-cosine integral; double sine integral; general monotonicityReferences:
| [1] | E. Liflyand and S. Tikhonov, The Fourier transforms of general monotone functions, in: Analysis and Mathematical Physics, Trends in Math., Birkh¨auser, 2011, 373-391. [LT3] E. Liflyand and S. Tikhonov, A concept of general monotonicity and applications, Math. Nachr. 284 (2011), 1083-1098. · Zbl 1223.26017 |
| [2] | F. M´oricz, Pointwise convergence of double Fourier integrals of functions of bounded variation over R2, J. Math. Anal. Appl. 424 (2005), 1530-1543. [M2]F. M´oricz, On the uniform convergence of sine integrals, J. Math. Anal. Appl. 354 (2009), 213-219. |
| [3] | F. M´oricz, On the convergence of double integrals and a generalized version of Fubini’s theorem on successive integration, Acta Sci. Math. (Szeged) 78 (2012), 469-487. [T1]S. Tikhonov, Trigonometric series with general monotone coefficients, J. Math. Anal. Appl. 326 (2007), 721-735. |
| [4] | S. Yu. Tikhonov, On the uniform convergence of trigonometric series, Mat. Zametki 81 (2007), 304-310 (in Russian); English transl.: Math. Notes 81 (2007), 268-274. · Zbl 1183.42005 |
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