Polynomial interpolation - Revision history

en>Janus.debondt: /* Lebesgue constants */ "makes" added

2014-12-19T14:49:54Z

Lebesgue constants: "makes" added

← Older revision		Revision as of 16:49, 19 December 2014
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en>Kusma: /* Proof 2 */ splines are typically not polynomials globally, only piecewise, so I don't see how we get the same result for splines

2014-02-28T12:36:19Z

Proof 2: splines are typically not polynomials globally, only piecewise, so I don't see how we get the same result for splines

← Older revision		Revision as of 14:36, 28 February 2014
Line 1:		Line 1:
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	In [~~[mathematics~~]], the ~~'''Liouville–Neumann series'''~~ is ~~an [[infinite series]] that corresponds to~~ the ~~[[resolvent formalism]] technique~~ of ~~solving~~ the ~~[[Fredholm integral equation]]~~s in [~~[Fredholm theory]]~~.

	==~~Definition~~==
	~~The Liouville–Neumann series is defined as~~

	:~~<math>\phi\left(x\right)~~ = ~~\sum^\infty_{n~~=~~0} \lambda^n \phi_n \left(x\right)</math>~~

	~~which~~ is a ~~unique~~, ~~[[continuous function\|continuous]] solution of~~ a ~~[[Fredholm integral equation]]~~ of ~~the second kind~~:

	:~~<math>f(t)= \phi(t) - \lambda \int_a^bK(t~~,~~s)\phi(s)\~~,ds.~~</math>~~

	If ~~the ''n''th iterated [[Kernel (integral operator)\|kernel]] is defined as~~

	~~:<math>K_n\left(x~~,~~z\right) = \int\int\cdots\int K\left(x~~,~~y_1\right)K\left(y_1~~,~~y_2\right) \cdots K\left(y_{n~~-1}, ~~z\right) dy_1 dy_2 \cdots dy_{n-1}~~<~~/math~~>

	~~then~~

	:<~~math~~>~~\phi_n\left(x\~~right~~) = \int K_n\left(x~~,~~z\right)f\left(z\right)dz</math>~~

	~~with~~

	~~:<math>\phi_0\left(x\right) = f\left(x\right)~~.<~~/math~~>

	~~The [[Resolvent formalism\|resolvent]] or solving kernel is given by~~

	:<~~math~~>~~K\left(x~~, ~~z;\lambda\right) = \sum^\infty_{n=0} \lambda^n K_{n+1} \left(x~~, ~~z\right)~~.
	~~</math>~~

	~~The solution~~ of the ~~integral equation becomes~~

	:<~~math~~>~~\phi\left(x\right) = \int K \left( x, z;\lambda\right) f\left(z\right)dz.~~<~~/math~~>

	~~Similar methods may be used~~ to ~~solve the~~ [~~[Volterra equation~~]]s.

	~~{{DEFAULTSORT:Liouville-Neumann Series}}~~
	~~[[Category:Fredholm theory]]~~
	~~[[Category:Mathematical series]]~~

en>Glrx: rvt last 4; divided diffs are not defined as coefficients; method to calc coefficients

2014-02-03T23:18:47Z

rvt last 4; divided diffs are not defined as coefficients; method to calc coefficients

← Older revision		Revision as of 01:18, 4 February 2014
Line 1:		Line 1:
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			In [[mathematics]], the '''Liouville–Neumann series''' is an [[infinite series]] that corresponds to the [[resolvent formalism]] technique of solving the [[Fredholm integral equation]]s in [[Fredholm theory]].

			==Definition==
			The Liouville–Neumann series is defined as

			:<math>\phi\left(x\right) = \sum^\infty_{n=0} \lambda^n \phi_n \left(x\right)</math>

			which is a unique, [[continuous function\|continuous]] solution of a [[Fredholm integral equation]] of the second kind:

			:<math>f(t)= \phi(t) - \lambda \int_a^bK(t,s)\phi(s)\,ds.</math>

			If the ''n''th iterated [[Kernel (integral operator)\|kernel]] is defined as

			:<math>K_n\left(x,z\right) = \int\int\cdots\int K\left(x,y_1\right)K\left(y_1,y_2\right) \cdots K\left(y_{n-1}, z\right) dy_1 dy_2 \cdots dy_{n-1}</math>

			then

			:<math>\phi_n\left(x\right) = \int K_n\left(x,z\right)f\left(z\right)dz</math>

			with

			:<math>\phi_0\left(x\right) = f\left(x\right).</math>

			The [[Resolvent formalism\|resolvent]] or solving kernel is given by

			:<math>K\left(x, z;\lambda\right) = \sum^\infty_{n=0} \lambda^n K_{n+1} \left(x, z\right).
			</math>

			The solution of the integral equation becomes

			:<math>\phi\left(x\right) = \int K \left( x, z;\lambda\right) f\left(z\right)dz.</math>

			Similar methods may be used to solve the [[Volterra equation]]s.

			{{DEFAULTSORT:Liouville-Neumann Series}}
			[[Category:Fredholm theory]]
			[[Category:Mathematical series]]

en>Amit Moscovich: /* Proof 2 */

2012-08-06T19:15:57Z

Proof 2

New page

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