Mercer's condition
Encyclopedia
In mathematics
, a real
-valued function
K(x,y) is said to fulfill Mercer's condition if for all square integrable functions g(x) one has
satisfies Mercer's condition, as then the integral becomes by Fubini's theorem
which is indeed non-negative.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a real
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
-valued function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
K(x,y) is said to fulfill Mercer's condition if for all square integrable functions g(x) one has
Examples
The constant functionsatisfies Mercer's condition, as then the integral becomes by Fubini's theorem
Fubini's theorem
In mathematical analysis Fubini's theorem, named after Guido Fubini, is a result which gives conditions under which it is possible to compute a double integral using iterated integrals. As a consequence it allows the order of integration to be changed in iterated integrals.-Theorem...
which is indeed non-negative.