Distribution of zeros of finite Fourier (Laplace) transforms

T h eo rem 1. Suppose that the function f ( t ) is positive and does not decrease in the interval (0,1). Then all zeros of function (1) lie in the half-plane Sftz < 0. Moreover,

1) i f the function f ( t ) is not exceptional, then there are no zeros of function (1) on the imaginary axis;

2) i f the function f ( t ) is exceptional, then the imaginary axis contains an infi­ nite number of zeros of function (1).