Three classes of Fourier transforms are presented: Fourier (Laplace) transforms on the halfline, Fourier transforms of measures with compact support and Fourier transforms of rapidly decreasing functions (on whole line). The focus is on the behaviour of Fourier transforms in the region of analyticity and the distribution of their zeros. Applications of results are presented: approximation by exponentials on the finite interval; behavior of the nonharmonic Fourier series; Müntz-Szasz's problem of approximation by powers on unit interval; approximation by weighted exponentials on whole line.

Neil Collings is a Principle Research Engineer at Envisics Ltd, developing original ideas for Holographic Projection Systems. He received his BA degree in Natural Sciences from Cambridge University, UK, in 1971, and PhD degree from Salford University, UK, in 1977. He has worked in the field of optics since 1977 and in liquid crystal devices since 1985, and is known internationally for his work on spatial light modulators, programmable optical interconnects, and holographic optical systems. Until 2014, he was Reader in Liquid Crystal Photonics at the Department of Engineering in Cambridge University, UK. Dr. Collings is a Fellow of the Institute of Physics and a Senior Member of the SPIE.