Necessary and Sufficient Condition for Quadratic Functional to be Positive Definite

Theorem
Let


 * $ P \left ( { x } \right ) > 0 \quad \forall x \in \left [ { a \,. \,. \,b } \right ] $

Suppose, the interval $ \left [ { a \,. \,. \,b } \right ] $ contains no points conjugate to $ a $.

Let


 * $ h \left ( { a } \right ) =0, \quad h \left ( { b } \right ) = 0 $

Then $ \displaystyle \int_a^b \left ( { P h'^2 + Q h^2 } \right ) \mathrm d x$ is positive definite.