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The first-kind Chebyshev polynomial $T_n$ converts a multiple-angle cosine into a polynomial:The cosine addition formula gives the recurrence relationThis constructs polynomials valid for every real $x$, including outside $[-1,1]$, where the real-angle description is unavailable. For example,For $n\ge 1$, $T_n$ has polynomial degree $n$ and leading coefficient $2^{n-1}$. On $[-1,1]$ its uniform norm is one, even as its oscillations become more numerous.^[ref][ref]

The polynomial roots of $T_n$ are especially simple:They correspond to equally spaced angles, but their locations in $x$ cluster near $-1$ and $1$. For $T_3$, the polynomial roots are $\sqrt{3}/2,0,-\sqrt{3}/2$. For $T_8$, the positive polynomial roots are approximately $0.980785,0.831470,0.555570,0.195090$, with their negatives completing the set. The distance from the right endpoint to the nearest polynomial root is $1-\cos (\pi /(2n))\sim {\pi }^2/(8n^2)$; central neighboring polynomial roots are separated on the larger scale $1/n$. The small change of cosine near angles zero and $\pi$ explains the clustering.

These polynomial roots supply $n$ interpolation nodes for a polynomial interpolation of polynomial degree at most $n-1$. They differ from the $n+1$ Chebyshev-Lobatto nodes $\cos (j\pi /n)$, which include both endpoints. This choice of polynomial roots has nodal polynomialThe monic Chebyshev extremal polynomial minimizes the uniform norm of this product among monic polynomials of polynomial degree $n$ on $[-1,1]$. Consequently, if $f$ has $n$ continuous derivatives, the usual polynomial interpolation error gives the useful boundThis optimizes the nodal factor in that bound, not the actual error for every function. Endpoint clustering helps control the large endpoint oscillations of equally spaced interpolation associated with Runge's phenomenon. The two node conventions and their geometric construction are discussed in Trefethen's authored text.^[ref]

The same cosine substitution for polynomial approximation explains the weighted orthogonality:The endpoint singularities of the weight are integrable. Thus the Chebyshev polynomials are orthogonal polynomials for this weight, making coefficients in a least-squares polynomial in an orthogonal-polynomial basis separate cleanly. Their polynomial roots also give Chebyshev–Gauss quadrature, connecting polynomial interpolation to Gaussian quadrature.^[ref]

Accuracy still depends on the target function. The absolute value function $|x|$ has a corner at zero, so the hypothesis requiring continuous derivatives in the displayed polynomial interpolation error bound fails. Endpoint clustering cannot remove that corner. A jump poses a stronger obstruction: polynomials are continuous functions, so they cannot converge in the uniform norm to a discontinuous function on the whole interval. Chebfun's documented examples show persistent overshoot for such jumps; these are a limitation of global polynomial interpolation, not a universal promise of rapid accuracy from Chebyshev polynomials.^[ref]

The Gegenbauer differential equation isFor $\alpha >0$, put $p=(1-x^2{)}^{\alpha +1/2}$ and $w=(1-x^2{)}^{\alpha -1/2}$. Multiplying by $w$ gives the Sturm-Liouville equationDifferentiating $p$ verifies the coefficient of $y^{\prime }$. The Gegenbauer polynomial gives a polynomial solution, and the eigenvalue is $n(n+2\alpha )$. Since $p$ vanishes at both endpoints and polynomial values/derivatives remain bounded, the boundary form from integration by parts vanishes for these polynomial solutions. This yields their orthogonality of Sturm-Liouville eigenfunctions with weight $w$.^[ref]

For $\alpha >0$, the Gegenbauer polynomials are normalized by the generating functionnear $t=0$. This fixes the scale left undetermined by the Gegenbauer differential equation. The first examples are $C_0^{\alpha }=1$, $C_1^{\alpha }=2\alpha x$ and $C_2^{\alpha }=2\alpha (\alpha +1)x^2-\alpha$. They are orthogonal polynomials on $(-1,1)$ for the weight $(1-x^2{)}^{\alpha -1/2}$. At $\alpha =1/2$ they are the Legendre polynomials; at $\alpha =1$ they give the Chebyshev polynomial of the second kind family.^[ref]

A Jacobi matrix is a real symmetric tridiagonal matrix with positive off-diagonal entries. Its diagonal entries $b_j$ and off-diagonal entries $a_j>0$ encode the three-term recurrence of orthonormal polynomials:where $p_{-1}=0$ and the $a_0{p}_{-1}$ term is omitted. In the orthonormal polynomial basis this is the matrix of multiplication by $x$, truncated to finitely many basis vectors. Positivity of $a_j$ excludes a reducible split into independent blocks.^[ref]

The determinant $D_n(x)=\det (xI-J_n)$ of the leading $n$-by-$n$ block satisfies $D_{n+1}=(x-b_n)D_n-a_n^2{D}_{n-1}$, with $D_0=1$. Thus its eigenvalues are the zeros of orthogonal polynomials of degree $n$. For unit weight on $[-1,1]$, the first two normalized polynomials give $J_2=\left(\begin{array}{cc}0& 1/\sqrt{3}\\ 1/\sqrt{3}& 0\end{array}\right)$, whose eigenvalues $\pm 1/\sqrt{3}$ are the two-node Gaussian quadrature nodes.

A monic orthogonal polynomial has leading coefficient one and is orthogonal to every lower-degree polynomial in the chosen positive weighted inner product. Suppose the measure has finite moments of the required orders and its inner product is positive definite on polynomials. Subtracting the orthogonal projection onto a finite-dimensional subspace of $x^n$ onto the polynomials of degree below $n$ constructs the degree-$n$ monic orthogonal polynomial.

It is unique: two candidates differ by a lower-degree polynomial $r$ orthogonal to itself, forcing $r=0$ by positive definiteness. With unit weight on $[-1,1]$, the first three are $1,x,x^2-1/3$, since ${\int }_{-1}^1{x}^{2}dx/{\int }_{-1}^{1}1dx=1/3$. A measure on too few support points can make the inner product degenerate and prevent this construction at higher degrees.^[ref]

An orthogonal polynomial family chooses polynomials that do not overlap under a specified inner product. For real polynomials on an interval $(a,b)$, use the weighted inner productAssume $a<b$, the weight $w$ is measurable and positive almost everywhere, and its absolute moments ${\int }_a^b|x{|}^{k}w(x)dx$ are finite for every integer $k\ge 0$. These conditions make the integrals finite and give every nonzero polynomial positive squared norm. A sequence $p_0,p_1,\dots$ is an orthogonal polynomial family when $p_n$ has real coefficients and polynomial degree exactly $n$, and $⟨p_n,p_k{⟩}_w=0$ for $n\ne k$.^[ref]

Here $h_n=\Vert p_n{\Vert }_w^2>0$ and ${\delta }_{nk}$ is the Kronecker delta. Orthogonality does not require $h_n=1$: an orthonormal set is obtained only after replacing $p_n$ by $p_n/\sqrt{h_n}$. Multiplying each member by any nonzero real constant preserves orthogonality, but changes its norm and the corresponding expansion coefficient. The monic orthogonal polynomial convention instead fixes each leading coefficient to one.

The first four Legendre polynomials, named for Adrien-Marie Legendre, use $w=1$ on $[-1,1]$ and the normalization $P_n(1)=1$:Their orthogonality and squared norms areThus their Gram matrix has a positive diagonal and zero entries elsewhere, although their function graphs cross and overlap on the page. Orthogonality measures an integral of products, rather than separation of function graphs.^[ref]

For these four polynomials, a short verification expands each product and uses ${\int }_{-1}^1{x}^{2j+1}dx=0$ and ${\int }_{-1}^1{x}^{2j}dx=2/(2j+1)$. For example, $⟨P_0,P_2⟩=(3(2/3)-2)/2=0$ and $⟨P_1,P_3⟩=(5(2/5)-3(2/3))/2=0$. The same calculation gives the diagonal entries $2,2/3,2/5,2/7$. More generally, apply the Gram-Schmidt process to the monomial basis $1,x,x^2,\dots$: at step $n$, subtract projections onto the earlier polynomials. The result retains its leading term, has degree $n$, and has positive norm, proving existence without a differential equation. Different degrees also give linear independence, so $p_0,\dots ,p_m$ form a basis of the polynomials of degree at most $m$.

This basis turns least-squares solutions for functions into independent scalar calculations, as the decoupled coefficients of orthogonal polynomial projections show. In a spectral method, increasing the degree enlarges the approximation space while keeping earlier projection coefficients fixed. Weighted integrals also connect the same orthogonal polynomials to Gaussian quadrature: its nodes are polynomial roots of a member of the family. The projection problem minimizes an integrated squared error; Gaussian quadrature uses the family to choose evaluation points for an integral.

A real degree-$n$ orthogonal polynomial for a positive weight on an interval, with the required finite moments, has exactly $n$ distinct simple zeros in the interval's interior. This places the quadrature nodes of Gaussian quadrature inside its integration domain.^[ref]

To prove it, list the distinct interior roots where the polynomial $p_n$ changes sign, and let $q$ be the product of their linear factors. Then $p_{n}q$ has one constant sign on the interval and is not identically zero. If there were fewer than $n$ sign-changing roots, $q$ would have lower polynomial degree, so orthogonality would give $\int p_{n}qwdx=0$, contradicting positivity of the integral after a choice of sign. There must therefore be $n$ such roots. They exhaust the polynomial degree, proving both simplicity and the absence of additional roots.