The martingale Hardy space $H_p\left({[0,1)}^2\right)$ and the classical Hardy space $H_p{(}^2)$ are introduced. We prove that certain means of the partial sums of the two-parameter Walsh-Fourier and trigonometric-Fourier series are uniformly bounded operators from $H_p$ to $L_p$ (0 < p ≤ 1). As a consequence we obtain strong convergence theorems for the partial sums. The classical Hardy-Littlewood inequality is extended to two-parameter Walsh-Fourier and trigonometric-Fourier coefficients. The dual inequalities are also verified and a...

We establish two new norm convergence theorems for Henstock-Kurzweil integrals. In particular, we provide a unified approach for extending several results of R. P. Boas and P. Heywood from one-dimensional to multidimensional trigonometric series.

The inequality (*) $({\sum }_{\left|n\right|=1}^{\infty }{\sum }_{\left|m\right|=1}^{\infty }{\left|nm\right|}^{p-2}{\left|f̂(n,m)\right|}^p{)}^{1/p}\le C_p\parallel ƒ{\parallel }_{H_p}$ (0 < p ≤ 2) is proved for two-parameter trigonometric-Fourier coefficients and for the two-dimensional classical Hardy space $H_p$ on the bidisc. The inequality (*) is extended to each p if the Fourier coefficients are monotone. For monotone coefficients and for every p, the supremum of the partial sums of the Fourier series is in $L_p$ whenever the left hand side of (*) is finite. From this it follows that under the same condition the two-dimensional trigonometric-Fourier series...

It is shown that under certain conditions on $c_{jk}$, the rectangular partial sums $s_{mn}(x,y)$ converge uniformly on $T^2$. These conditions include conditions of bounded variation of order (1,0), (0,1), and (1,1) with the weights |j|, |k|, |jk|, respectively. The convergence rate is also established. Corresponding to the mentioned conditions, an analogous condition for single trigonometric series is ${\sum }_{\left|k\right|=n}^{\infty }\left|\Delta c_k\right|=o(1/n)$ (as n → ∞). For O-regularly varying quasimonotone sequences, we prove that it is equivalent to the condition: $n{c}_n=o\left(1\right)$ as...

We prove that if $c_{jk}\to 0$ as max(|j|,|k|) → ∞, and ${\sum }_{\left|j\right|=0\pm }^{\infty }{\sum }_{\left|k\right|=0\pm }^{\infty }{\theta \left(\right|j|}^{\top }{\left)\vartheta \right(\left|k\right|}^{\top }\left)\right|{\Delta }_{12}{c}_{jk}|<\infty$, then f(x,y)ϕ(x)ψ(y) ∈ L¹(T²) and ${\iint }_{T²}|s_{mn}(x,y)-f(x,y)\left|·\right|\varphi \left(x\right)\psi \left(y\right)|dxdy\to 0$ as min(m,n) → ∞, where f(x,y) is the limiting function of the rectangular partial sums $s_{mn}(x,y)$, (ϕ,θ) and (ψ,ϑ) are pairs of type I. A generalization of this result concerning L¹-convergence is also established. Extensions of these results to double series of orthogonal functions are also considered. These results can be extended to n-dimensional case. The aforementioned results generalize work of Balashov [1], Boas [2],...

Let $f_c(x,y)\equiv {\sum }_{j=1}^{\infty }{\sum }_{k=1}^{\infty }{a}_{jk}(1-cosjx)(1-cosky)$ with $a_{jk}\ge 0$ for all j,k ≥ 1. We estimate the integral ${\int }_0^{\pi }{\int }_0^{\pi }{x}^{\alpha -1}{y}^{\beta -1}\varphi \left(f_c(x,y)\right)dxdy$ in terms of the coefficients $a_{jk}$, where α, β ∈ ℝ and ϕ: [0,∞] → [0,∞]. Our results can be regarded as the trigonometric analogues of those of Mazhar and Móricz [MM]. They generalize and extend Boas [B, Theorem 6.7].