  The phenomenon in the essence of classical uncertainty principles is well
known since the thirties of the last century. We introduce a new phenomenon
which is in the essence of a new notion that we introduce: "Generalized
Uncertainty Principles". We show the relation between classical uncertainty
principles and generalized uncertainty principles. We generalized
"Landau-Pollak-Slepian" uncertainty principle. Our generalization relates the
following two quantities and two scaling parameters: 1) The weighted time
spreading $\int_{-\infty}^\infty |f(x)|^2w_1(x)dx$, ($w_1(x)$ is a non-negative
function). 2) The weighted frequency spreading $\int_{-\infty}^\infty
|\hat{f}(\omega)|^2w_2(\omega)d\omega$. 3) The time weight scale $a$,
${w_1}_a(x)=w_1(xa^{-1})$ and 4) The frequency weight scale $b$,
${w_2}_b(\omega)=w_2(\omega b^{-1})$. "Generalized Uncertainty Principle" is an
inequality that summarizes the constraints on the relations between the two
spreading quantities and two scaling parameters. For any two reasonable weights
$w_1(x)$ and $w_2(\omega)$, we introduced a three dimensional set in $R^3$ that
is in the essence of many uncertainty principles. The set is called
"possibility body". We showed that classical uncertainty principles (such as
the Heiseneberg-Pauli-Weyl uncertainty principle) stem from lower bounds for
different functions defined on the possibility body. We investigated
qualitative properties of general uncertainty principles and possibility
bodies. Using this approach we derived new (quantitative) uncertainty
principles for Landau-Pollak-Slepian weights. We found the general uncertainty
principles related to homogeneous weights, $w_1(x)=w_2(x)=x^k$, $k\in N$, up to
a constant.
