On absolute convergence of multiple fourier series

Absolute convergence of multiple Fourier series of functions of $$\phi \left( \varLambda ^1,\ldots ,\varLambda ^N\right) $$ ϕ Λ 1 , … , Λ N -bounded variation

“…Remark 3 We note that, in Theorem 18, the authors of [16] assumed that f ∈ (Λ 1 , Λ 2 ) * BV ( p) (T 2), but looking at the proof we can see that it is true under the weaker hypothesis as in the following theorem.…”

“…241 ', p. 160] and given a multidimensional analogue of the results of Gogoladze and Meskhia [7]. Further, in [16,Theorem 3.5,p. 74], Vyas and Darji have proved a result concerning absolute convergence of multiple Fourier series which includes a multidimensional analogue of one dimensional result of Schramm and Waterman [13,Theorem 1,p.…”

“…Remark 4 We note here that Izumi and Izumi [9], Shiba [14], and Scharamm and Waterman [13] have proved their results by directly estimating the partial sums of the series of absolute values of the Fourier coefficients of the function by using the technique of summation by parts, whereas Vyas [15] and Vyas and Darji [16] have proved their results either by estimating the tail of the series or by estimating sums of dyadic blocks of the series. Here we shall prove our main theorem (Theorem 23 above) by directly estimating the partial sums of the double series of absolute values of the Fourier coefficients of the function by using the technique of summation by parts.…”

“…Here we note the following special cases. Vyas and Darji [16,Theorem 5.5,p. 80] stated the following theorem.…”

“…274]. We note that Schramm and Waterman [13] have proved their results by estimating the partial sums using the technique of summation by parts, whereas Móricz and Veres (see [10,11]) and Vyas and Darji [16] proved their results by estimating the partial sums by partitioning them into dyadic blocks and then using a proper generalization of the Cauchy's condensation test. Interestingly, here, we prove or main result by using the classical technique of summation by parts.…”