Polynomial Inner Products and Approximation

Lecture 4 · Reading guide

Read the symbols first

Wooseok Ha, MAS110 Lecture 4 (Fall 2026), lec04.pdf: printed slides 7–11, 18, 29, 34–39 (PDF pages 9–13, 21, 33, 39–44).

A polynomial is a vector in a finite-dimensional function space. Its coefficient array is a coordinate representation, not the definition of its length. We use the lecture’s inner product $\langle f,g\rangle = \int_{-1}^{1} f(t)g(t) \,dt$ and start with $\mathcal P_{2}$, the polynomials of degree at most two, including the zero polynomial.

$\langle u,v\rangle;\quad \lVert u\rVert;\quad u\perp v$

An inner product gives one real number, $\Vert u\Vert = \sqrt{\langle u,u\rangle}$ is its induced norm, and $u \perp v$ means $\langle u,v\rangle = 0$. The lecture writes |u| for this norm. In Euclidean space the default is $u^{\mathsf T}v$ and the norm is $\Vert u\Vert _{2}$; for polynomials it is the stated integral norm.

$\varphi;\quad \theta;\quad \hat\theta$

This chapter uses $\varphi$ for an angle and $\theta$ for model coefficients in Lab 4.4. The lecture sometimes uses $\theta$ for the angle too; the distinction here prevents mixing an angle with a coefficient vector. A hat marks a minimizing coefficient, not a unit vector.

$A;\quad G;\quad Q;\quad R;\quad P$

A stores input columns in the Euclidean QR and least-squares labs. G denotes a Gram matrix of inner products, including the polynomial metric (which the lecture locally calls A). Q stores orthonormal columns, R stores their reconstruction coefficients, and P projects onto a subspace. These matrices have different jobs and shapes.

$\mathcal P_n;\quad B=(1,t,\ldots,t^n)$

$\mathcal P_{n}$ has dimension n + 1. “Degree at most n” includes lower degrees and the zero polynomial; polynomials of degree exactly n do not form a vector space. B is an ordered basis, so order determines which coefficient goes with each term.

$f(t)=c_0+c_1t+c_2t^2;\quad [f]^B=c$

The coordinate vector $c = (c_{0},c_{1},c_{2})^{\mathsf T}$ has three entries. The scalar t is the argument; f(t) is the value at that argument. This page uses t instead of the lecture’s occasional x to distinguish an argument from a coordinate vector.

$G_{ij}=\int_{-1}^1t^{i+j}\,dt,\quad i,j=0,\ldots,n$

These indices label monomial degrees, starting at 0. $G_{ij} = 0$ for odd i+j and 2/(i+j+1) for even i+j. The lecture calls this Gram matrix A locally; we call it G to keep A for the design matrix in 4.3–4.4.

$\langle f,g\rangle=c^{\mathsf T}Gd;\quad \lVert f\rVert^2=c^{\mathsf T}Gc$

c and d are the coordinates of f and g in B. The plain coordinate dot product $c^{\mathsf T}d$ is generally different. For example, the coordinate arrays for 1 and $t^{2}$ have dot product 0, but their integral inner product is 2/3.

$q_0,q_1,q_2;\quad C=\bigl[\,[q_0]^B\mid[q_1]^B\mid[q_2]^B\,\bigr]$

We label polynomial basis vectors by degree, beginning at 0; these correspond to the lecture’s $w_{1},w_{2},w_{3}$. C has coefficient columns. Orthonormality for this metric means $C^{\mathsf T}GC = I_{3}$, not $C^{\mathsf T}C = I_{3}$.

$\text{Legendre }P_k(t);\quad \text{normalized }q_k(t)$

Standard Legendre polynomials satisfy $P_{k}(1) = 1$ and $\int_{-1}^{1} P_{k}^{2} = 2/(2k+1)$. Multiplying by $\sqrt{(2k+1)/2}$ gives $q_{k}$ of unit integral norm. Endpoint normalization and unit-norm normalization are different.

NumPy ascending powers: $[c_{0},c_{1},c_{2}]$

numpy.polynomial.polynomial uses increasing powers: [0,1] is t and [0,0,1] is $t^{2}$. The original notebook uses legacy np.polyval/np.polyint, whose arrays run in the opposite order. We use the modern ascending convention consistently and do not mix the two APIs.