MATH 590: Meshfree Methods - Chapter 6 - Applied Mathematics

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ProofInterpolation with Multivariate PolynomialsWe use induction on m.For m = 0 the result is trivial.Take R to be the matrix arising from polynomial interpolation at thepoints u j ∈ U, i.e.,R jk = p k (u j ), j, k = 1, . . . , M,where the p k form a basis of Π 2 m.We want to show that the only possible solution to Rc = 0 is c = 0.fasshauer@iit.edu MATH 590 – Chapter 6 8
ProofInterpolation with Multivariate PolynomialsWe use induction on m.For m = 0 the result is trivial.Take R to be the matrix arising from polynomial interpolation at thepoints u j ∈ U, i.e.,R jk = p k (u j ), j, k = 1, . . . , M,where the p k form a basis of Π 2 m.We want to show that the only possible solution to Rc = 0 is c = 0.This is equivalent to showing that if p ∈ Π 2 m satisfiesthen p is the zero polynomial.p(u i ) = 0, i = 1, . . . , M,fasshauer@iit.edu MATH 590 – Chapter 6 8
Page 3 and 4: OutlineInterpolation with Multivari
Page 6 and 7: Interpolation with Multivariate Pol
Page 16 and 17: ProofInterpolation with Multivariat
Page 38 and 39: Example: Reproduction of Linear Fun
Page 54 and 55: Scattered Data Fitting w/ More Gene
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Almost Negative Definite Matrices a
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References IAppendixReferencesBuhma
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MATH 590: Meshfree Methods - Chapter 6 - Applied Mathematics

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