The Least Squares Method and Interpolation of Functions

Consider some vector $x$ and a subspace $L$ in the Euclidean space $\mathbb {R}^{n}$. Prove that among the vectors of the subspace $L$, the orthogonal projection $x^{\| }$ of the vector $x$ onto $L$ ``deviates least'' from the given vector $x$, i.e.,

Prove that a pseudosolution can be found as the solution of a new inhomogeneous system of equations formed using the Gram matrix $G\left(a_{1}, a_{2}, \ldots , a_{n}\right)$:

What is the geometric meaning of the constants $x_{1}, x_{2}, \ldots , x_{n}$ found in this way?

Using the method of least squares, find the pseudosolution of the following inconsistent systems of linear equations:

Prove that the system of polynomials $\varphi_{k}(x), k = 0, \ldots , n$, from (13.1) forms an orthonormal basis of the space of polynomials of degree at most $n$ with respect to the inner product

Given $2 n+1$ pairwise distinct points $x_{i}, i = 0, \ldots , 2 n$, of the interval $[0,2 \pi ]$. In the space $V_{2 n+1}$ of trigonometric polynomials

find trigonometric polynomials $t_{k}$ such that

Show that $t_{k}$ form an orthonormal basis of the space $V_{2 n+1}$ with respect to the inner product

Prove that in the space $V_{2 n+1}$ of trigonometric polynomials there exists a unique solution $\mathscr {T}_{2 n+1}$ of the system of equations

where $x_{0}, \ldots , x_{2 n}-2 n+1$ pairwise distinct points of the interval $[0,2 \pi ]$, and $f:[a, b] \rightarrow \mathbb {R}$ is some function.

Find a polynomial $h$ of minimal degree such that, at the given pairwise distinct points $x_{0}, x_{1}, \ldots , x_{n}$, its values and derivatives coincide with the values and derivatives of a given function:

Using the Lagrange interpolation polynomial $L_{n}^{f}$, one can interpolate an arbitrary function $f$ on the real line from the table of its values $f\left(x_{0}\right), f\left(x_{1}\right), \ldots , f\left(x_{n}\right)$, where $-\infty < x_{0} < \ldots \ldots < x_{n} < +\infty$.

Prove that if the set $\left\{ x_{i}, i = 0, \ldots , n\right\}$ of interpolation nodes is symmetric with respect to zero, then from the evenness (oddness) of the function $f(x)$ it follows that the function $L_{n}^{f}(x)$ is even (odd).

Introduce the space $V_{n+1}$ of discrete functions $f: M_{n+1} \rightarrow \mathbb {R}$ on the finite set $M_{n+1} = \left\{ x_{0}, \ldots , x_{n}\right\}$. Show that $V_{n+1}$ is a Euclidean space with respect to the inner product $(f, g) = f\left(x_{0}\right) g\left(x_{0}\right)+f\left(x_{1}\right) g\left(x_{1}\right)+\ldots +f\left(x_{n}\right) g\left(x_{n}\right)$. Exhibit some orthonormal basis of the space $V_{n+1}$.

Prove that the solution $L_{m}^{f}(x)$ of the least-squares interpolation problem in the space $V_{n+1}$ exists and is unique, and that the coefficients $a_{i}, i = 0, \ldots , m$, are found from the system of inhomogeneous linear equations

where $G\left(\varphi_{0}, \varphi_{1}, \ldots , \varphi_{m}\right)$ is the Gram matrix of the system of functions $\varphi_{0}(x), \ldots \ldots , \varphi_{m}(x)$ in the Euclidean space $V_{n+1}$.

Using the method of least squares, find the interpolation (best mean-square approximation) of the function $f$ given by the values $f(0) = 1, f(1) = 2, f(2) = 3, f(3) = 5$:

Prove that in the case of an orthonormal system of functions $\varphi_{0}(x), \ldots , \varphi_{n}(x)$, the coefficients $a_{i}, i = 0, \ldots , n$, equal the Fourier coefficients of the function $f$, i.e., the linear combination with the Fourier coefficients of the function $f$ is the best interpolation of the function $f$.

Using the method of least squares, find, in the Euclidean space of continuously differentiable functions with the inner product

the interpolation (best mean-square approximation) of a continuously differentiable function $f$ using the system of functions $\varphi_{0}(x) = 1, \varphi_{1}(x) = \cos x, \varphi_{2}(x) = \sin x$.