Find the radial polynomial p(x, y)=a+b r+c r^2, where...

Find the radial polynomial $p(x, y)=a+b r+c r^2$, where $r^2=x^2+y^2$, that best approximates the function $f(x, y)=x$ using the $\mathrm{L}^2$ norm on the unit disk $D=\{r \leq 1\}$ to measure the least squares error.

Key Concepts

Coordinate Transformation to Polar Coordinates

Transforming to polar coordinates can simplify the computation of integrals over symmetric domains such as the unit disk. In polar coordinates, the integration over a circular domain naturally separates the radial and angular components, making it easier to evaluate the norms and inner products required for the least squares approximation.

Projection onto a Subspace

Projection onto a subspace involves finding the component of a function that lies in a given subspace of the function space, often by setting up and solving orthogonality conditions. In many cases, this is done by ensuring the residual (the difference between the function and its approximation) is orthogonal to each function in the chosen basis of the subspace.

Radial Functions

Radial functions depend solely on the distance from the origin, not on the direction. This property is significant because it simplifies integration and analysis in circular or spherical domains. When approximating functions by radial polynomials, the dependence on the angular variable is removed, which poses unique challenges when the target function is not radial.

L2 Norm

The L2 norm measures the size of a function by integrating the square of its absolute value over a domain. In approximation problems, it defines the error metric by computing the square?integrable difference between the target and approximant functions, and the goal is to minimize this squared error over the domain.

Least Squares Approximation

Least squares approximation seeks to find the best approximant within a specified function space by minimizing the sum of the squares of the differences (errors) between the target function and the approximant. This process leads to an optimal solution in the L2 sense, where the error has been minimized with respect to the chosen norm.

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