FUNCTIONS ON THE REAL LINE WITH NONNEGATIVE FOURIER-TRANSFORMS

Unlike an integrable function on the unit circle which has the nonnega tive Fourier coefficients and is square-integrable near the origin, an integrable function on the real line which has the nonnegative Fourie r transform and is square-integrable near the origin is not always squ are-integrable on the real line. We give some examples, and consider a n additional condition which guarantees the global square-integrabilit y. Moreover, we treat an analogous problem for an integrable function on the real line which has non-negative wavelet coefficients of the Fo urier transform and is square-integrable near the origin.