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Обозначим: A1(x, t) = A(x, t), если {0 ≤ t ≤ 1/2 − x, 0 ≤ x ≤ 1/2}, A2(x, t) = A(x, t), если {1/2 + x ≤ t ≤ 1, 0 ≤ x ≤ 1/2}, A3(x, t) = A(x, t), если {0 ≤ t ≤ −1/2 + x, 1/2 ≤ x ≤ 1}, A4(x, t) = A(x, t), если {3/2 − x ≤ t ≤ 1, 1/2 ≤ x ≤ 1}, A5(x, t) = A(x, t), если {1/2 − x ≤ t ≤ 1/2 + x, 0 ≤ x ≤ 1/2} и {−1/2 + x ≤ t ≤ 3/2 − x, 1/2 ≤ x ≤ 1}. Предположим, что Ai(x, t), i = 1, . . . , 5 непрерывно дифференцируемые в своих областях, причем A5(x, 1/2 − x + 0) − A1(x, 1/2 − x − 0) = a, A5(x, 1/2 + x − 0) − A2(x, 1/2 + x + 0) = b, A5(x,−1/2 + x + 0) − A3(x,−1/2 + x − 0) = c, A5(x, 3/2 − x − 0) − A4(x, 3/2 − x + 0) = d, где a, b, c, d — постоянные. Частный случай оператора (1) впервые рассматривался в [1]. Рассмотрим следующий оператор:

On Convergence of Riesz Means of the Expansions in Eigen and Associated Functions of Integral Operator with Kernel Having Jumps on Broken Lines