There is a development of the classical Tauberian theorem for Riesz summabil i ty [5], quoted in this note as Lemma 1, which has some interesting applications. ANANDA RAU has made use of this development in one case ([1], Theorem 4 or Theorem 6) to obtain converse theorems on the abscissae of summabil i ty of general Dirichlet's series ([1], Theorems 7, 8). The present note employs the same development in another case (particularized under Lemma 1) to obtain a precise analogue for multiple Fourier series of a "bestpossible" Tauberian theorem of F. T. WA~o's ([6], Theorem 8) for single Fourier series. This analogue for k-tuple Fourier series, k => 2, s tated below as Theorem I I , improves a result proved by CHANDRASEKHARAN and MI~AXSmSUI~DARA~I {[4], p. 119, Theorem 4.53 ; [4a], p. 110) and it may be of interest in tha t its proof differs in one respect from tha t of its original case k = 1. The proof of the case k = 1 turns on the Riesz summabil i ty (R, exp n c, 5), 0 0, of a class of single Fourier series, while the proof of the case k => 2 rests on the spherical Riesz summabil i ty (R, n 2, 5), with a certain rapidity as in Theorem I, for suitable 5 > 0, of an analogous class of k-tuple Fourier series. Notation. In what follows, we denote a real series, single or k-tuple, by