| The Journal of the Society for the Foundations of Computational Mathematics Implicit Gamma Theorems (I): Pseudoroots and Pseudospectra (2002) | |||||||||||||||
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| Abstract. Let g: E → F be an analytic function between two Hilbert spaces E and F. We study the set g(B(x,ε)) ⊂ F, the image under g of the closed ball about x ∈ E with radius ε. When g(x) expresses the solution of an equation depending on x, then the elements of g(B(x,ε))are ε-pseudosolutions. Our aim is to investigate the size of the set g(B(x,ε)). We derive upper and lower bounds of the following form: g(x) + Dg(x)(B(0, c1ε)) ⊆ g(B(x,ε)) ⊆ g(x) + Dg(x)(B(0, c2ε)), where Dg(x) denotes the derivative of g at x. We consider both the case where g is given explicitly and the case where g is given implicitly. We apply our results to the | |||||||||||||||
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