Upper bounds for the concentration of bandlimited functions in L2(-T,T).

If f is a bandlimited function of the form i) f(t) = 1 over 2pi integral from -1 to 1 f(w)e(iwt)dw, f in L(2)(-1,1), then ii) integral from -T to T|F(t)|(2)dt= lamda(0)(T) integral - infinity to infinity |f(t)|(2)dt, where lamda(0)(T) is the largest eignevalue of a well studied integral equation. It is shown in this paper that iii) lamba(0) (T) tanhT, iv) 1-lamda(0)(T) > 1+2 over pi integral 0 to 1 e (-2xT) over square route 1-x(2) dx over 1 + I(0)(2T), where I(0)(.) is the modified Bessel function. The last inequality compares favorable with the known behavior of lamda(0)(T) for large and small T.