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In this paper, we study the almost everywhere convergence of spherical partial sums of multiple Fourier series of functions from Sobolev classes. It is proved that almost everywhere convergence will take place under the same conditions on the order of smoothness of the expanded function as for multiple Fourier integrals; these conditions were found in a well-known paper of Carbery and Soria (1988). Our reasoning is largely based on the methodology developed in the work of Kenig and Thomas (1980). [ABSTRACT FROM AUTHOR]

We investigate the equiconvergence on T = [−π, π) of expansions in multiple trigonometric Fourier series and in the Fourier integrals of functions f ∈ L(T) and g ∈ L(R), p > 1, N ≥ 3, g( x) = f( x) on T, in the case where the 'partial sums' of these expansions, i.e., S( x; f) and J( x; g), respectively, have 'numbers' n ∈ Z and α ∈ R ( n = [α], j = 1,..., N, [ t] is the integral part of t ∈ R) containing N − 1 components which are elements of 'lacunary sequences.' [ABSTRACT FROM AUTHOR]