Pointwise convergence of cone-like restricted two-dimensional (C,1) means of trigonometric Fourier series

• Autores: György Gát
• Localización: Journal of approximation theory, ISSN 0021-9045, Vol. 149, Nº 1, 2007, págs. 74-102
• Idioma: inglés
• DOI: 10.1016/j.jat.2006.08.006
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• Resumen

  • The aim of this work is to generalize the more than 60 year old celebrated result of Marcinkiewicz and Zygmund on the convergence of the two-dimensional restricted (C,1) means of trigonometric Fourier series. They proved for any integrable function fL1(T2) the a.e. convergences(n1,n2)f?fprovided n1/ß=n2=ßn1, where ß>1 is fixed constant. That is, the set of indices (n1,n2) remains in some positive cone around the identical function. We not only generalize this theorem, but give a necessary and sufficient condition for cone-like sets (of the set of indices) in order to preserve this convergence property.