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This paper describes some new methods of synthesizing linear antenna arrays. The methods are developed from a re-examination of the well-known Bernstein polynomials and of the classical theories on approximation and interpolation. Both the ordinary and the trigonometric interpolations are considered. With these methods, one is able to synthesize an array such that 1) an upper bound of the errors between the specified and synthesized patterns can be estimated, 2) either the maximum deviation or the mean-square error can be made to be minimum if the total number of elements in the array is prechosen, or 3) a least required number of elements can be determined if the error specifications are given.