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- Chicago Review Press
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- English (Publication Language)
- 336 Pages - 07/01/1989 (Publication Date) - Lawrence Hill Books (Publisher)
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- Mercer, Peter R. (Author)
- English (Publication Language)
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- Feldman, Burton (Author)
- English (Publication Language)
- 264 Pages - 09/01/2011 (Publication Date) - Arcade (Publisher)
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FAQ:
Q: What is Mercer’s theorem?
A: In mathematics, specifically functional analysis, Mercer’s theorem is a representation of a symmetric positive-definite function on a square as a sum of a convergent sequence of product functions. This theorem, presented in ( Mercer 1909 ), is one of the most notable results of the work of James Mercer (1883–1932).
Q: What is Mercer’s condition for K to be positive definite?
A: If K is a symmetric positive-definite kernel on ( X, M, μ), then where the convergence in the L2 norm. Note that when continuity of the kernel is not assumed, the expansion no longer converges uniformly. 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
Q: What is Mercer’s condition for a function?
A: 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 ∬ g ( x ) K ( x , y ) g ( y ) d x d y ≥ 0. {\\displaystyle \\iint g (x)K (x,y)g (y)\\,dx\\,dy\\geq 0.}
Q: Does truncation of the Mercer sum lead to an underestimate?
A: The truncation of the Mercer sum does lead to an underestimate of the values of the kernel K ( s, t). For our example, we should get K ( s, s) = 1, but we get noticeably less. In fact, the eigenvalues decrease only like O ( n − 2), which makes the pointwise convergence in the number of terms rather slow.