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In this paper, applying Parseval’s formula, we prove a Gθ – summability analogue of Avadhani’s theorem for the Riesz–summability of the eigenfunction expansion. A crucial step in our proof of this theorem was to find a function g(x) that would lead us to the kernel of the Gθ –summability, which is more complex than the kernel of the Riesz – summability.

ON AN APPLICATION OF PARSEVAL’S FORMULA TO PROBLEMS OF Gκ θ -SUMMABILITY OF EIGENFUNCTION EXPANSION OF THE LAPLACIAN OPERATOR