Convergence of summability means of higher dimensional Fourier series and Lebesgue points

Abstract We introduce a new concept of Lebesgue points for higher dimensional functions. Every continuity point is a Lebesgue point and almost every point is a Lebesgue point of an integrable function. Given a strictly increasing continuous function $$\delta$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>δ</mml:mi> </mml:math> , we prove that the Fejér or Cesàro means $$\sigma_n^{\alpha}f$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>σ</mml:mi> <mml:mi>n</mml:mi> <mml:mi>α</mml:mi> </mml:msubsup> <mml:mi>f</mml:mi> </mml:mrow> </mml:math> of the Fourier series of a two-dimensional function $$f\in L_1(\mathbb{T}^2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>∈</mml:mo> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mrow> <mml:mi>T</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> converge to $$f$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>f</mml:mi> </mml:math> at each Lebesgue point as $$n\to \infty$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>→</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> and n is in the cone around the graph of $$\delta$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>δ</mml:mi> </mml:math> . We also prove this result for higher dimensional functions and for other summability means. This is a generalization of the classical one-dimensional Lebesgue’s theorem for the Fejér means.