Let　SF(d)　and　Pi(phi,n,d)　=　{Sigma(n)(j=1)b(j)phi(omega　j.x+theta　j)　:b(j),theta(j)is　an　element　of　R,omega(j)is　an　element　of　R(d)}　be　the　set　of　periodic　and　Lebesgue＇s　square-integrable　functions　and　the　set　of　feedforward　neural　network　(FNN)　functions,　respectively.　Denote　by　dist　(SF(d),　Pi(phi,n,d))　the　deviation　of　the　set　SF(d)　from　the　set　pi(phi,n,d).　A　main　purpose　of　this　paper　is　to　estimate　the　deviation.　In　particular,　based　on　the　Fourier　transforms　and　the　theory　of　approximation,　a　lower　estimation　for　dist　(SF(d),　Pi(phi,n,d))　is　proved.　That　is,　dist(SF(d),Pi(phi,n,d))　＞=　C/(nlog(2)(n))(1/2).　The　obtained　estimation　depends　only　on　the　number　of　neuron　in　the　hidden　layer,　and　is　independent　of　the　approximated　target　functions　and　dimensional　number　of　input.　This　estimation　also　reveals　the　relationship　between　the　approximation　rate　of　FNNs　and　the　topology　structure　of　hidden　layer.