Book contents
- Frontmatter
- Contents
- Foreword
- Introduction
- 1 Inner product spaces
- 2 Normed spaces
- 3 Hilbert and Banach spaces
- 4 Orthogonal expansions
- 5 Classical Fourier series
- 6 Dual spaces
- 7 Linear operators
- 8 Compact operators
- 9 Sturm–Liouville systems
- 10 Green's functions
- 11 Eigenfunction expansions
- 12 Positive operators and contractions
- 13 Hardy spaces
- 14 Interlude: complex analysis and operators in engineering
- 15 Approximation by analytic functions
- 16 Approximation by meromorphic functions
- Appendix: square roots of positive operators
- References
- Answers to selected problems
- Afterword
- Index of notation
- Subject index
15 - Approximation by analytic functions
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Foreword
- Introduction
- 1 Inner product spaces
- 2 Normed spaces
- 3 Hilbert and Banach spaces
- 4 Orthogonal expansions
- 5 Classical Fourier series
- 6 Dual spaces
- 7 Linear operators
- 8 Compact operators
- 9 Sturm–Liouville systems
- 10 Green's functions
- 11 Eigenfunction expansions
- 12 Positive operators and contractions
- 13 Hardy spaces
- 14 Interlude: complex analysis and operators in engineering
- 15 Approximation by analytic functions
- 16 Approximation by meromorphic functions
- Appendix: square roots of positive operators
- References
- Answers to selected problems
- Afterword
- Index of notation
- Subject index
Summary
A problem which commonly arises in statistics, physics and engineering is to find the function in a given class which approximates some known function as well as possible. To turn this vague notion into a well-posed mathematical question we need a numerical criterion of how good an approximation is. A popular choice is to look for a best ‘least squares’ fit. That is, we ask that the norm of the error function (the difference between the given function and the chosen approximation) be as small as possible, where the norm used is a Hilbert space norm. The advantage of this type of criterion is the obliging nature of Hilbert space geometry: such problems are usually comparatively easy to solve. Let us check this in a particular case.
Problem Given a function ϕ ∈ L2, find a function g∈H2 such that ∥ϕ - g∥L2 is minimized.
If we were to write out the definition of ∥ϕ - g∥ the reason for the expression ‘least squares’ would be apparent. In this instance the class of functions from which the approximation is to be chosen is a closed subspace of L2, and in this case the best approximation problem was essentially solved in Chapter 4. Let us recall the relevant facts and introduce some notation. If M is a closed subspace of a Hilbert space H and x∈H then, by Theorem 4.24, there exist y∈M, z∈M⊥ such that x = y + z.
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- Information
- An Introduction to Hilbert Space , pp. 187 - 202Publisher: Cambridge University PressPrint publication year: 1988