Best approximations of integrals by integrals of finite rank

We investigate the quantities eσ(f) of the best approximation for integrals of functions from the spaces Lp(A,dμ) by integrals of finite rank σ. We find exact values of these approximations in the case where the function f is a product of two nonnegative functions one of which is fixed and the other varies on the unit ball Up(A) in the space Lp(A,dμ). We also consider applications of the obtained results to the approximation of periodic functions defined by convolutions with summable kernels by functions of exponential type.