A nonzero polynomial is said to be positive if its coefficients are nonnegative. Motzkin and Straus [5] observe that if a real polynomial P(Z) has no positive zeros, then there is a positive polynomial Q(Z) such that P(Z).Q(Z) is positive. We interpret this analytically and prove analogous results for some real distributions with compact supports. These support the conjecture that for a real distribution with compact support, its Fourier-Laplace transform being positive on the imaginary axis implies that there is a positive distribution with compact support such that their convolution is positive. This is not proven here, but the converse to this and the next best statement after this are established. Also, some of the propositions enable us to return to prove some theorems on polynomials....[ Read more ]

A nonzero polynomial is said to be positive if its coefficients are nonnegative. Motzkin and Straus [5] observe that if a real polynomial P(Z) has no positive zeros, then there is a positive polynomial Q(Z) such that P(Z).Q(Z) is positive. We interpret this analytically and prove analogous results for some real distributions with compact supports. These support the conjecture that for a real distribution with compact support, its Fourier-Laplace transform being positive on the imaginary axis implies that there is a positive distribution with compact support such that their convolution is positive. This is not proven here, but the converse to this and the next best statement after this are established. Also, some of the propositions enable us to return to prove some theorems on polynomials.