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The book is devoted to the synthesis problems that arise in the theory and design of radiating systems (antennas). The characteristics of desired amplitude are data placed into a synthesis problem. A synthesis problem belongs to a class of inverse problems and its aim is to determine a distribution of current or fields in an antenna, which produces the amplitude radiation characteristic as close as possible to the desired one. Freedom of choice of phase distribution of the desired radiation pattern (RP) is used as an additional possibility of better approximation to such RPs. This book studies various different types of antennas and arrays as the radiation systems under consideration. A special class of problems related to acoustic and electromagnetic scattering on a set of bodies (particles) of small size is also discussed, while the constructive procedures of creating inhomogeneous materials with specific properties are proposed