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单词 BestApproximationInInnerProductSpaces
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best approximation in inner product spaces


The study of best approximations in inner product spacesMathworldPlanetmath has a very elegant treatment with profound consequences. Most of the theory of Hilbert spacesMathworldPlanetmath depends on this study and several approximation problems are better understood using this techniques and results.

For example: least square fitting, linear regression, approximation of functions by polynomials, among many other problems, can be seen as particular cases of the general study of best approximation in inner product spaces.

Some of the above problems are going to be discussed later in this entry.

1 Existence and Uniqueness

Our fundamental result on the existence and uniqueness of best approximations is the following (we postpone its proof to this attached entry (http://planetmath.org/ProofOfExistenceAndUniquenessOfBestApproximations)):

Theorem - Let X be an inner product space and AX a completePlanetmathPlanetmathPlanetmathPlanetmath (http://planetmath.org/Complete), convex and non-empty subset. Then for every xX there exists a unique best approximation (http://planetmath.org/BestApproximation) of x in A, i.e. there exists a unique element a0A such that

x-a0=d(x,A)=infaAx-a.

2 Geometric Interpretation

The following result gives a very geometric interpretationMathworldPlanetmathPlanetmath of the best approximation when A is a subspacePlanetmathPlanetmath of X. We also postpone its proof to an attached entry.

Theorem - Let X be an inner product space, AX a subspace and xX. The following statements are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath:

  • a0A is the best approximation of x in A.

  • a0A and x-a0A.

Thus, the best approximation of x in a subspace A is just the orthogonal projection of x in A.

3 Calculation of Best Approximations

When the A is a complete subspace of X, the best approximation can be ”calculated” explicitly. Recall that, in this case, A becomes an Hilbert space (since it is complete) and therefore it has an orthonormal basisMathworldPlanetmath.

Again, we postpone the proof of the next result to an attached entry.

Theorem - Let X be an inner product space and AX a complete subspace. Let (ei)iJ be an orthonormal basis for A. Then for every xX the best approximation a0A of x in A is given by

a0=iJx,eiei.

One can also write the best approximation in of any other basis (not necessarily an orthonormal one). For simplicity we present here how that can be done when A is a finite dimensional subspace of X.

Theorem - Let X be an inner product space and AX a finite dimensional subspace. Let v1,,vn be a basis for A. Then for every xX the best approximation a0A of x in A is given by

a0=i=1na0ivi

where the coefficients a0i are the solutions of the system of equations

(v1,v1v1,vnvn,v1vn,vn)(a01a0n)=(x,v1x,vn).

Remark- The above matrix is a symmetricPlanetmathPlanetmathPlanetmathPlanetmath positive definitePlanetmathPlanetmath (http://planetmath.org/PositiveDefinite) matrix, which implies that the system has a unique solution as expected.

4 Applications

There are several applications of the above results. We explore two of them in the following.

4.0.1 - Approximation of functions by polynomials :

Suppose we want to find a polynomial of degree n that approximates in the best possible way a given function f. We are in fact trying to find a point in the subspace of polynomials of degree n that is closest to f, i.e. we are trying to find the best approximation of f in that subspace.

For example, let fL2([0,1]). Consider the basis vk(t)=tk,0kn, of the subspace of polynomials of degree n.

The best approximation of f by these polynomials is the function a0(t)=a01+a01t++a0ntn, where the coefficients a01,,a0n are the solutions of the system

(11n+11n+112n+1)(a01a0n)=(01f(t)𝑑t01tnf(t)𝑑t).

Remark- Instead of polynomials we could approximate f by any other of functions using the same procedure.

4.0.2 - Best Fitting Lines :

Suppose we want to find the line that best fits some given points (t1,y1),,(tn,yn), i.e. the affine function a0(t)=αt+β that minimizes k=1n|a0(tk)-yk|2.

We are then led to consider the inner productMathworldPlanetmath

f,g=k=1nf(tk)g(tk)

in the space of functions h:{t1,,tk}.

With this setting we are then looking for the best approximation of the function f(tk)=yk on the subspace of affine functions.

A base for the subspace of affine functions is given by the functions v1(t)=1 and v2(t)=t.

The best approximation of f on this space is the function a0(t)=β+αt, where the coefficients β,α are the solutions of the system

(nk=1ntkk=1ntkk=1ntk2)(βα)=(k=1nykk=1nyktk).

Thus, the function a0(t)=β+αt obtained by the above procedure provides the line that best fits the data (t1,y1),,(tn,yn).

Titlebest approximation in inner product spaces
Canonical nameBestApproximationInInnerProductSpaces
Date of creation2013-03-22 17:32:16
Last modified on2013-03-22 17:32:16
Ownerasteroid (17536)
Last modified byasteroid (17536)
Numerical id12
Authorasteroid (17536)
Entry typeFeature
Classificationmsc 41A65
Classificationmsc 46C05
Classificationmsc 46N10
Classificationmsc 49J27
Classificationmsc 41A52
Classificationmsc 41A50
Definesapproximation by polynomials
Definesbest fitting lines
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