Abstract :
Letfbe the function periodic with period 2πinxandywhich extends the indicator function of the parallelogramA={(x, y): 0⩽y⩽π, y/c⩽x⩽y/c+π} (0≠c∈R). The partial sums of the Fourier series offof order 2M+1, say, evaluated at (πx/(2M+1), πy/(2M+1)), converge forM→∞ to a sum of integrals of the functions sin t/t, sin s/s sin t/t, cos s/s cos t/tover domains depending onx y, andc. This limit appears to depend only on the part ofAinside an arbitrarily small circle about 0.
