  Given a smooth bump function, we consider the multiplier formed by taking the
linear combination of the translations of the bump function and the
corresponding bilinear Fourier multiplier operator. Under certain condition on
the bump function, we give a complete characterization of the coefficients of
the linear combination for which the corresponding bilinear operator defines a
bounded operator from $L^2\times L^2$ to $L^2$-based amalgam spaces.
