History of Lebesgue constants of the classical interpolation node families

 Title: Lebesgue constants of the classical interpolation node families-Definition: For a named family $T_n=\{x_0,\ldots,x_n\}$ of interpolation nodes in-  $[-1,1]$, this table gives the Lebesgue constant $\Lambda_n(T_n)=\max_{-1\leq x\leq1}\sum_{j=0}^n-  |\ell_j(x)|$ CITE{WikiLebesgue}, where $\ell_j$ is the Lagrange basis polynomial-  for the nodes and $n$ is the interpolation degree.+Definition: For a named family $X_n=\{x_0,\ldots,x_n\}$ of interpolation nodes in+  $[-1,1]$, this table gives the Lebesgue constant $\Lambda_n(X_n)$ CITE{WikiLebesgue},+  where $n$ is the interpolation degree. Keywords: - Lebesgue constant

     type: Symbolic     title: node family-    display: $T_n$-    constraints: one of the named interpolation node families listed in the comments+    display: $X_n$+    constraints: equally spaced, Chebyshev first kind, Chebyshev–Lobatto, stretched+      Chebyshev, Gauss–Legendre, or Gauss–Lobatto     values:       equally-spaced: equally spaced nodes including the endpoints       chebyshev-first-kind: Chebyshev points of the first kind-      chebyshev-lobatto: Chebyshev-Lobatto points+      chebyshev-lobatto: Chebyshev–Lobatto points       stretched-chebyshev: stretched Chebyshev points-      gauss-legendre: Gauss-Legendre points-      gauss-lobatto: Gauss-Lobatto points+      gauss-legendre: Gauss–Legendre points+      gauss-lobatto: Gauss–Lobatto points   n:     type: Z

   comment-nodes: The nodes are written on $[-1,1]$. The equally spaced family is $x_j=-1+2j/n$.     The Chebyshev points of the first kind CITE{WikiChebyshevNodes} are $x_j=\cos((2j+1)\pi/(2n+2))$,-    the roots of HREF{Chebyshev_polynomials_of_the_first_kind}[$T_{n+1}$]. The Chebyshev-Lobatto-    points are $x_j=\cos(j\pi/n)$, the extrema of $T_n$. The stretched Chebyshev points-    divide the Chebyshev points of the first kind by $\cos(\pi/(2n+2))$. Here $0\leq-    j\leq n$.-  comment-gauss: The Gauss-Legendre family uses the nodes of the $n+1$ point HREF{Nodes_and_weights_of_Gauss_Legendre_quadrature}[Gauss-Legendre-    quadrature] rule. The Gauss-Lobatto family uses the nodes of the $n+1$ point HREF{Nodes_and_weights_of_Gauss_Lobatto_quadrature}[Gauss-Lobatto-    quadrature] rule, namely the endpoints $-1,1$ and the roots of the derivative-    of the HREF{Legendre_polynomials}[Legendre polynomial] $P_n$ when $n>1$.+    the roots of HREF{Chebyshev_polynomials_of_the_first_kind}[$T_{n+1}$]. The Chebyshev–Lobatto+    points are $x_j=\cos(j\pi/n)$, the extrema of the Chebyshev polynomial $T_n$.+    The cosines in these two Chebyshev families are entries of HREF{Cos_pi_times_x_at_rational_numbers}[$\cos(\pi+    x)$ for rational $x$]. The stretched Chebyshev points divide the Chebyshev points+    of the first kind by $\cos(\pi/(2n+2))$. Here $0\leq j\leq n$.+  comment-gauss: The Gauss–Legendre family uses the nodes of the $n+1$ point HREF{Nodes_and_weights_of_Gauss_Legendre_quadrature}[Nodes+    and weights of Gauss–Legendre quadrature] rule. The Gauss–Lobatto family uses+    the nodes of the $n+1$ point HREF{Nodes_and_weights_of_Gauss_Lobatto_quadrature}[Nodes+    and weights of Gauss–Lobatto quadrature] rule, namely the endpoints $-1,1$ and+    the roots of the derivative of the HREF{Legendre_polynomials}[Legendre polynomial]+    $P_n$ when $n>1$.   comment-degree: The parameter $n$ is the interpolation degree, so each row uses     $n+1$ nodes. Some sources index the same constants by the number of nodes.

     not give the minimal Lebesgue constant obtained by optimising over all node sets     of a fixed size.+  comment-small-degrees: At $n=1$, the equally spaced, Chebyshev–Lobatto, stretched+    Chebyshev, and Gauss–Lobatto families all have nodes $\{-1,1\}$. At $n=2$, the+    same four families all have nodes $\{-1,0,1\}$, so their Lebesgue constants agree+    in those rows. Formulas:-  formula-definition: $\Lambda_n(T_n)=\max_{-1\leq x\leq1}\sum_{j=0}^n |\ell_j(x)|$,-    where $\ell_j(x)=\prod_{m\neq j}(x-x_m)/(x_j-x_m)$.+  formula-definition: $\lambda_n(x;X_n)=\sum_{j=0}^n |\ell_j(x)|$, where $\ell_j(x)=\prod_{m\neq+    j}(x-x_m)/(x_j-x_m)$. Then $\Lambda_n(X_n)=\max_{-1\leq x\leq1}\lambda_n(x;X_n)$. Programs:   program-sage:

 - table: HREF{Lagrange_basis_polynomials_for_equally_spaced_nodes}[Lagrange basis     polynomials for equally spaced nodes]-  relation: the equally spaced rows use these nodes, and the Lebesgue constant is-    the uniform-norm condition number of the corresponding Lagrange basis+  relation: the equally spaced rows use nodes affinely equivalent to these, and the+    Lebesgue constant is the uniform norm of the corresponding interpolation operator - table: HREF{Chebyshev_polynomials_of_the_first_kind}[Chebyshev polynomials of the     first kind]

     of these polynomials - table: HREF{Nodes_and_weights_of_Gauss_Legendre_quadrature}[Nodes and weights of-    Gauss-Legendre quadrature]-  relation: the Gauss-Legendre rows use the same nodes and omit the quadrature weights+    Gauss–Legendre quadrature]+  relation: the Gauss–Legendre rows use the same nodes and omit the quadrature weights - table: HREF{Nodes_and_weights_of_Gauss_Lobatto_quadrature}[Nodes and weights of-    Gauss-Lobatto quadrature]-  relation: the Gauss-Lobatto rows use the same nodes and omit the quadrature weights-- table: HREF{Newton_Cotes_weights}[Newton-Cotes weights]-  relation: the closed Newton-Cotes rules use the same equally spaced endpoint nodes+    Gauss–Lobatto quadrature]+  relation: the Gauss–Lobatto rows use the same nodes and omit the quadrature weights+- table: HREF{Newton_Cotes_weights}[Newton–Cotes weights]+  relation: the closed Newton–Cotes rules use equally spaced endpoint nodes affinely+    equivalent to the equally spaced rows here Links:   WikiLebesgue:

   complete: 'no'   complete-note: it holds all six listed node families for interpolation degrees $1\leq-    n\leq4$+    n\leq4$, as a rectangular range for comparing the families; the next Gauss–Legendre+    row is already a multi-minute exact computation with the maximisation method used+    here   rigour details: 'The generator builds the node polynomials over $\mathbb{Q}$ and     isolates their roots in Sage''s algebraic real field. It forms the Lagrange basis

     when it is proved rational and is returned exactly. -    Before the entries were written, `agents/table-build/dry_run.py` computed all-    24 entries. The exact values $\Lambda_2=5/4$ for equally spaced nodes, $\Lambda_1=\sqrt2$-    for Chebyshev points of the first kind, $\Lambda_1=1$ for endpoint node families,-    and $\Lambda_1=\sqrt3$ for Gauss-Legendre nodes were checked independently. The-    values $\Lambda_4=2.2078243973258429982911088644463\ldots$ for equally spaced-    nodes and $\Lambda_4=1.9888543819998317571273389349850\ldots$ for Chebyshev points-    of the first kind were checked against the screening calculation for numberdb-data-    issue 162.'++    As independent checks, the small-degree closed forms $\Lambda_1=1$ for the endpoint+    node families, $\Lambda_2=5/4$ for the four families with nodes $\{-1,0,1\}$,+    $\Lambda_1=\sqrt2$ for Chebyshev points of the first kind, and $\Lambda_1=\sqrt3$+    for Gauss–Legendre nodes were derived directly from the node sets.' Display properties:-  number-header: $\Lambda_n(T_n)$+  number-header: $\Lambda_n(X_n)$ Numbers:   equally-spaced: