It is shown that, theoretically it is possible to approach the mainlobe width of a rectangular window (+/-1/T), while maintaining any desired high asymptotic decay, with the proposed polynomial series window. A synthesis procedure for the polynomial window is described to obtain a practically narrowest possible mainlobe width with a desired high asymptotic decay. Time domain and frequency domain plots are shown for a 10th order polynomial window with 24dB/octave decay of sidelobes and a mainlobe width of +/-1.43/T.