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Weighted integrability of double cosine series with nonnegative coefficients

Chang-Pao Chen; Ming-Chuan Chen — 2003

Studia Mathematica

Let $f_{c} (x, y) \equiv \sum_{j = 1}^{\infty} \sum_{k = 1}^{\infty} a_{j k} (1 - c o s j x) (1 - c o s k y)$ with $a_{j k} \geq 0$ for all j,k ≥ 1. We estimate the integral $\int_{0}^{π} \int_{0}^{π} x^{α - 1} y^{β - 1} ϕ (f_{c} (x, y)) d x d y$ in terms of the coefficients $a_{j k}$ , 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].

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