Difference between revisions of "Bilinear integral form"
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The double integral | 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 | + | 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. | ||
Latest revision as of 10:59, 29 May 2020
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.
How to Cite This Entry:
Bilinear integral form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bilinear_integral_form&oldid=11956
Bilinear integral form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bilinear_integral_form&oldid=11956
This article was adapted from an original article by B.V. Khvedelidze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article