Lebesgue points of double Fourier series and strong summability

A general summability method of double Fourier series is given with the help of a double sequence θ(k,n)θ(k,n). Under some conditions on θ we show that the Marcinkiewicz-θ  -means of a function f∈L1(T2)f∈L1(T2) converge to f   at each modified strong Lebesgue point. The same holds for a weaker version of Lebesgue points, for the so-called modified Lebesgue points of f∈Lp(T2)f∈Lp(T2), whenever 1