Find the plane z=α+βx+γy that best approximates the...

Find the plane $z=\alpha+\beta x+\gamma y$ that best approximates the following functions on the square $S=\{0 \leq x \leq 1,0 \leq y \leq 1\}$ using the $\mathrm{L}^2$ norm $\|f\|^2=\iint_S|f(x, y)|^2 d x d y$ to measure the least squares error: (a) $x^2+y^2$, (b) $x^3-y^3$, (c) $\sin \pi x \sin \pi y$.

Find the plane $z=\alpha+\beta x+\gamma y$ that best approximates the following functions on the square $S=\{0 \leq x \leq 1,0 \leq y \leq 1\}$ using the $\mathrm{L}^2$ norm $\|f\|^2=\iint_S|f(x, y)|^2 d x d y$ to measure the least squares error:
(a) $x^2+y^2$,
(b) $x^3-y^3$,
(c) $\sin \pi x \sin \pi y$.

Key Concepts

Basis Functions and Subspace Identification

Choosing an appropriate subspace is key in approximation problems. In this case, the subspace consists of constant and linear functions (1, x, y) which define the family of planes. Properties of these basis functions, including their orthogonality and symmetry over the integration domain, can simplify the calculations and determine the coefficients of the best-fit plane.

L² Norm and Inner Product

The L² norm, defined as the square root of the integral of the square of a function over a domain, is a measure of the size or energy of a function. In this least squares problem, the error between the function and its approximation is measured using the L² norm, and the corresponding inner product (integral of the product of two functions) is used to derive conditions for optimality based on orthogonality.

Orthogonal Projection in Function Spaces

In the context of L² spaces, the best approximation can be viewed as the orthogonal projection of the function onto the subspace of functions spanned by the basis functions (here, 1, x, y). The condition for a best approximation is that the error (difference) is orthogonal to every function in the subspace used for the approximation, leading to a system of equations commonly referred to as the normal equations.

Least Squares Approximation

This concept involves finding the function within a given class (in this case, the set of all planes defined by z = ? + ?x + ?y) that minimizes the L² error between the original function and its approximation. It is a method for reducing overall error by minimizing the sum (or integral) of the squared differences over the domain.

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