For double Walsh-Fourier series and with f ∈ L2([0, 1) × [0, 1)) we prove two almost orthogonality results relative to the linearized maximal square partial sums operator SN(x,y)f (x, y). Assumptions are N(x, y) non-decreasing as a function of x and of y and, roughly speaking, partial derivatives with approximately constant ratio $${{N_y^\prime \left({x,y} \right)} \over {N_x^\prime \left({x,y} \right)}} \cong {2^{{n_0}}}$$ for all x and y, where n0 is any fixed non-negative integer. Estimates, independent of N(x, y) and n0, are then extended to Lr, 1 10.

中文翻译：

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Walsh-Fourier 级数平方偏和的新异常集和收敛

对于双沃尔什 - 傅立叶级数和 f ∈ L2([0, 1) × [0, 1)) 我们证明了相对于线性化最大平方偏和算子 SN(x,y)f (x, y) 的两个几乎正交的结果）。假设 N(x, y) 作为 x 和 y 的函数不递减，粗略地说，偏导数具有近似恒定的比率 $${{N_y^\prime \left({x,y} \right)} \over {N_x^\prime \left({x,y} \right)}} \cong {2^{{n_0}}}$$ 对于所有 x 和 y，其中 n0 是任何固定的非负整数。与 N(x, y) 和 n0 无关的估计值然后扩展到 Lr, 1 10。