Bilinear integral form

The double integral

$$ J ( \phi , \psi ) = \int\limits _ { a } ^ { b } \int\limits _ { a } ^ { b } K(x, s) \phi (x) \overline{ {\psi (s) }}\; dx ds , $$

where $ K(x, s) $ is a given (usually complex-valued) square-integrable function of real variables, and $ \phi (x) $, $ \psi (x) $ are arbitrary (also complex-valued) square-integrable functions, while $ \overline{ {\psi (s) }}\; $ is the complex conjugate function of $ \psi (s) $. If $ \psi (s) = \phi (s) $, $ J( \phi , \phi ) $ is said to be a quadratic integral form.