A general summability method of multi-dimensional Fourier transforms, the so called θ\documentclass[12pt]{minimal}
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\begin{document}$${\theta}$$\end{document}-summability is investigated. Under some conditions on θ\documentclass[12pt]{minimal}
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\begin{document}$${\theta}$$\end{document} we show that the Marcinkiewicz-θ\documentclass[12pt]{minimal}
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\begin{document}$${\theta}$$\end{document}-means of a function f∈W(L1,ℓ∞)(Rd)\documentclass[12pt]{minimal}
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\begin{document}$${f\in
W(L_1,\ell_\infty)({\mathbb{R}}^d)}$$\end{document} converge to f\documentclass[12pt]{minimal}
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\begin{document}$${f}$$\end{document} 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∈W(Lp,ℓ∞)(Rd)\documentclass[12pt]{minimal}
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\begin{document}$${f\in
W(L_p,\ell_\infty)({\mathbb{R}}^d)}$$\end{document}, whenever 1<p<∞\documentclass[12pt]{minimal}
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\begin{document}$${1 < p < \infty}$$\end{document}. As an application we generalize the classical one-dimensional strong summability results of Hardy and Littlewood, Marcinkiewicz, Zygmund and Gabisoniya for f∈W(L1,ℓ∞)(R)\documentclass[12pt]{minimal}
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\begin{document}$${f\in W(L_1,\ell_\infty)({\mathbb{R}})}$$\end{document} and for strong θ\documentclass[12pt]{minimal}
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\begin{document}$${\theta}$$\end{document}-summability.
