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Use the nonnegative Laplacian convention $\tilde{\Delta }=-{\sum }_{j=1}^{d+1}{\partial }_j^2$ and $\Delta =-\mathrm{div}\mathrm{grad}$ on the unit sphere. In polar coordinates the Euclidean Riemannian metric is $d{r}^2+r^2{g}_{S^d}$, its volume form is $r^{d}drd{V}_{S^d}$, and therefore the Laplacian in polar coordinates isConsequently the required restriction identity isFor the opposite sign convention all eigenvalues below change sign, and the restriction identity changes accordingly.

Let ${\mathcal{P}}_n$ be the vector space of degree-$n$ homogeneous polynomials in $d+1$ variables, and let ${\mathcal{H}}_n$ be its subspace of harmonic polynomials. Write $H(r\omega )=r^{n}Y(\omega )$ for $H\in {\mathcal{H}}_n$. Substitution in the Laplacian identity givesThus each restricted harmonic polynomial is a spherical harmonic with eigenvalue ${\lambda }_n=n(n+d-1)$.

To count these spherical harmonics, give polynomials the Fischer inner product $⟨x^{\alpha },x^{\beta }{⟩}_F=\alpha !{\delta }_{\alpha \beta }$. Multiplication by $x_j$ is adjoint to ${\partial }_j$, so multiplication by $r^2$ is adjoint to the ordinary signed Laplacian ${\sum }_j{\partial }_j^2$. In these finite-dimensional inner product spaces, the orthogonal complement of $r^2{\mathcal{P}}_{n-2}$ is exactly ${\mathcal{H}}_n$. Hence the Fischer decomposition isMultiplication by $r^2$ is injective, while restriction of a homogeneous polynomial to the sphere is injective: homogeneity recovers its value at every nonzero point. Counting monomials now gives the eigenvalue multiplicitywhere the second term is zero for $n<2$.

It remains to establish that these are all the eigenvalues and their full multiplicities. Iterating the Fischer decomposition expresses every restricted polynomial as a sum of the restricted harmonic polynomials. Restricted polynomials separate points and contain the constants, so the Stone-Weierstrass theorem makes their span dense in continuous functions on the sphere, hence dense in its L2 space. The Laplace-Beltrami operator is self-adjoint, and the distinct numbers ${\lambda }_n$ strictly increase for $d\ge 1$. Its different eigenspaces are therefore orthogonal. Any additional eigenfunction, or any additional vector in one of these eigenspaces orthogonal to its known spherical harmonics, would be orthogonal to a dense subspace and hence zero. The standard compact elliptic spectral theorem then shows that the spectrum of the round sphere is precisely ${\lambda }_n=n(n+d-1)$ with multiplicity $m_n$. For $S^1$ this gives multiplicity one at zero and two at $n^2$ for $n\ge 1$. If the degenerate case $d=0$ is included, $S^0$ has two points and its Laplacian is zero on a two-dimensional function space.

A Riemann surface is a connected one-complex-dimensional complex manifold: it has local complex coordinates with holomorphic transition maps. These coordinates induce an orientation and a conformal structure. Fix a closed oriented topological surface $S_g$ of genus $g$. A point of Teichmüller space is a marked Riemann surface $(X,f)$, where $f:S_g\to X$ is an orientation-preserving marking. Two pairs are equivalent if there is a biholomorphism $h:X\to X^{\prime }$ with $h\circ f$ isotopic to $f^{\prime }$. Thus the marking remembers how curves on the reference surface sit in $X$; forgetting it gives the moduli space of Riemann surfaces, obtained by quotienting by the mapping class group.

For $g\ge 2$, the uniformization theorem supplies a unique curvature-$-1$ hyperbolic metric in each conformal class. A pants decomposition has $3g-3$ cuffs, and their positive lengths together with their real twists give Fenchel–Nielsen coordinates. HenceFor completeness, ${\mathcal{T}}_0$ is a point, whereas ${\mathcal{T}}_1$ is the upper half-plane, parametrizing marked complex tori $\mathbb{C}/(\mathbb{Z}+\tau \mathbb{Z})$ with $\mathrm{Im}\tau >0$.

The Wolpert generic spectral rigidity theorem says that, for fixed $g\ge 2$, there is a closed proper real-analytic exceptional subset of ${\mathcal{T}}_g$ such that a hyperbolic surface outside that subset is determined up to isometry by its unmarked length spectrum, equivalently its Laplacian spectrum. This is an unoriented isometry statement: reversing orientation cannot be detected by either spectrum.

The Huber spectral equivalence theorem says that two closed curvature-$-1$ hyperbolic surfaces have the same Laplacian spectrum, with multiplicities, if and only if they have the same unmarked length spectrum, with multiplicities. One may consistently count unoriented closed geodesics including their iterates; alternatively, the primitive multiset determines the multiset with iterates and conversely. This equivalence comes from the Selberg trace formula.

The Buser finite length spectrum theorem is the finite-determination ingredient: for each $g\ge 2$ and $\epsilon >0$ there is a finite number $L(g,\epsilon )$ such that, for any two closed hyperbolic surfaces of genus $g$ with hyperbolic systole at least $\epsilon$, equality of their length multisets through $L(g,\epsilon )$ implies equality of their entire length spectra. The finite-length statement is used in Buser's chapter on Wolpert's theorem. It determines the full spectrum, not necessarily the isometry class.

Here is a proof of the Buser finite length spectrum theorem through finite spectral matching by polynomial invariants. We use the following subsidiary results, stating the required content. First, Mumford's compactness theorem makes the part of the genus-$g$ moduli space of Riemann surfaces with hyperbolic systole at least $\epsilon$ compact. Second, marked hyperbolic holonomy representations of a closed oriented surface lift from ${\mathrm{PSL}}_2(\mathbb{R})$ to ${\mathrm{SL}}_2(\mathbb{R})$, and normalized representatives can be chosen continuously on sufficiently small parameter charts. Their hyperbolic translation length satisfiesThird, the length spectrum of a compact hyperbolic surface is locally finite, and metrics on compact marked parameter charts are uniformly bilipschitz equivalent. Finally, the Hilbert basis theorem implies that a descending sequence of algebraic sets in a finite-dimensional affine space stabilizes. These are compactness, holonomy, elementary geometric finiteness and algebraic finiteness results; none is the finite unmarked-spectrum assertion we are proving.

By compactness, finitely many relatively compact marked charts cover the thick moduli space of Riemann surfaces. Choose normalized lifted hyperbolic holonomy representations on these charts and take their compact closures. Their finite union $K$ is a compact set of determinant-one matrix tuples representing every surface in the thick part, possibly more than once. Each tuple records the images of the $2g$ generators of the fundamental group. Thus $K$ sits in a fixed finite-dimensional real matrix space. Every group word is a polynomial in these matrix entries: multiplication is polynomial, and for a determinant-one two-by-two matrix its inverse is its adjugate, also polynomial. Consequently each $q_{\gamma }$ is a polynomial on the ambient space. Squared traces remove the sign ambiguity of the lift, and $4{\cosh }^2(\ell /2)$ is strictly increasing for positive $\ell$.

Enumerate the unoriented nontrivial free homotopy classes as ${\gamma }_1,{\gamma }_2,\dots$, including the iterates, and write ${\ell }_j(x)$ for their geodesic lengths at $x\in K$. The uniform finiteness of short geodesic classes follows here directly: on each of the finitely many compact marked charts, choose a fixed reference metric $h$ and a uniform bilipschitz constant $C$. A class with ${\ell }_j(x)\le R$ somewhere on that chart has reference length at most $CR$. Only finitely many classes have that reference length, and a finite union of these finite collections is finite. In particular, for every $j$ the candidate setis finite. If two parameters in $K$ have equal lengths for classes $j,k$, then $k\in F_j$. These candidate sets depend on the fixed compact set $K$ and the source label $j$, not on the number of labels currently being matched.

For every $N$, define $E_N$ in the full ambient space of pairs of matrix tuples $(x,y)$ as follows. Require an injective choice $\sigma :\{1,\dots ,N\}\to \mathbb{N}$ with $\sigma (j)\in F_j$ andand independently require an injective choice $\tau :\{1,\dots ,N\}\to \mathbb{N}$ with $\tau (j)\in F_j$ andThere are finitely many possible pairs of choices $(\sigma ,\tau )$ because every $F_j$ is finite. For each choice these are finitely many polynomial equations. Therefore $E_N$ is a finite union of algebraic sets, hence an algebraic set. Restricting an injective matching to its first $N$ entries provesThe Hilbert basis theorem now supplies $N_0$ such that $E_N=E_{N_0}$ for all $N\ge N_0$. This step takes place in a fixed affine space and uses polynomial equations, so it gives stabilization of the actual sets, not just stabilization of germs at individual parameters.

For $(x,y)\in E_{N_0}\cap (K\times K)$, stabilization gives injective partial length matchings in both directions for every $N$. Fix any length $a>0$. Only finitely many labels have length $a$ at $x$ or at $y$, by local finiteness. Take $N$ large enough to contain every such label at $x$. Its injective matching into the labels at $y$ preserves $q_j$, and hence preserves length, so the multiplicity of $a$ at $x$ is at most its multiplicity at $y$. The reverse matching gives the opposite inequality. Thus the two multiplicities are equal for every $a$, proving equality of the entire unmarked length spectra. No assumed global matching or unjustified exchange of infinitely many neighborhoods is needed.

Finally choose a numberSuppose the two length multisets agree through $L$. For each of the first $N_0$ labels at $x$, select a distinct equal-length label at $y$, which is possible precisely because the truncated multisets agree with multiplicity. Each selected label belongs to the corresponding $F_j$. Do the same in the reverse direction. Then $(x,y)\in E_{N_0}$, and the previous paragraph proves equality of the full length spectra. Since $K$ was selected using only the fixed thick genus-$g$ parameter space, the resulting finite $L$ depends only on $g,\epsilon$. This proveswith each intersection interpreted as a multiset. By the Huber spectral equivalence theorem, the corresponding entire Laplacian spectra also agree. This is the finite matching reduction needed before the analytic and geometric rigidity steps in the Wolpert generic spectral rigidity theorem.

For ordinary homoscedastic fixed-design nonparametric regression, take observations $Y_i=g(x_i)+{ϵ}_i$ at known distinct sites, with independent mean-zero errors of common variance. A degree-$p$ local polynomial estimator at a target $u$ solves the finite-dimensional weighted least squares problemHere $K$ is a nonnegative regression kernel, $h>0$ is the smoothing bandwidth, and sufficient local design rank ensures uniqueness. The estimate of the regression function is $\hat{g}(u)={\hat{\beta }}_0(u)$. Each target has its own locally fitted polynomial; this is not one global polynomial fit. The bandwidth controls the localization and the bias-variance tradeoff.

A natural cubic smoothing spline instead solves the global penalized problemThe Sobolev space here consists of functions with square-integrable weak derivatives through order two. The minimizer is a natural cubic spline with knots at the observation sites and linear tails outside the extreme sites. Equivalently one may perform the minimization on that finite-dimensional spline space: the minimum roughness property of the natural cubic spline interpolant shows that replacing a candidate by the natural spline with the same site values preserves the data loss and decreases the second-derivative penalty. The smoothing parameter $\lambda$ controls curvature; it differs from the local bandwidth $h$.

For the interval-average observation problem, the printed upper limit in the loss is an indexing error: there are only $n-1$ responses and intervals, so a term with $i=n$ would involve the undefined $Y_n$ and $x_{n+1}$. The coherent loss uses the upper limit $n-1$. Put ${\Delta }_i=x_{i+1}-x_i$ and define the linear observation functionalsThe intended criterion on $C^1[a,b]$ is thereforeUnlike the ordinary cubic smoothing spline, this uses interval averages as data and a first-derivative penalty; the resulting spline degree will be two.

Fix any candidate $v\in C^1[a,b]$. By the supplied interpolation property, there is a unique quadratic spline $q\in C^1[a,b]$ with knots $x_1,\dots ,x_n$, constant on both exterior intervals, such that $L_{i}q=L_{i}v$ for every $i$. Put $d=v-q$. ThenOn each interior knot interval $q^{\prime \prime }$ is a constant $c_i$, and on the two exterior intervals $q^{\prime }=0$. Integrate $q^{\prime }{d}^{\prime }$ by parts separately on the intervals. At every knot the boundary contributions cancel because both $q^{\prime }$ and $d$ are continuous; the contributions at $a$ and $b$ vanish because $q^{\prime }$ is zero there. ThusThis orthogonality is the key to the quadratic smoothing spline for interval averages. Expanding the roughness penalty givesThe data-fit terms are equal because the interval averages are equal. Consequently,Equality can occur only if $d^{\prime }=0$ everywhere: the derivative is continuous and its squared integral is zero. Then $d$ is constant, and any of its zero interval averages forces that constant to be zero. Therefore every minimizer must equal its matching quadratic spline $q$.

We conclude that the minimizer is a continuously differentiable quadratic spline with knots at the specified sites and constant tails on both exterior intervals. Existence and uniqueness also follow from the supplied interpolation property. The interval-average map identifies the finite-dimensional space of these quadratic splines with ${\mathbb{R}}^{n-1}$, and the criterion becomes a strictly convex quadratic function of that average vector: its data-fit term is $\Vert Y-y{\Vert }^2$, while its penalty is nonnegative and quadratic. It has a unique minimizer, and the strict equality argument above proves uniqueness in the full function class.

A renormalizable quantum field theory requires only finitely many independent field and parameter redefinitions to remove regulator dependence from its perturbative correlation functions, order by order. The allowed local counterterms must close within that finite set; it is not necessary that the unrenormalized integrals be finite.

In four spacetime dimensions the kinetic term fixes the mass dimension of $\varphi$ to one. A coefficient multiplying ${\varphi }^k$ therefore has dimension $4-k$. Power counting excludes couplings of negative mass dimension for perturbative renormalizability. For a single scalar with a conventional nonderivative polynomial potential, the allowed form isThe constant controls vacuum energy and the linear term a possible tadpole. A field-reflection symmetry can remove the odd terms but is not necessary for renormalizability. Terms of degree above four generally require an infinite tower of counterterms when treated as fundamental interactions; they can instead be used in an effective field theory. Stability of the potential is a separate requirement, for example a positive quartic leading term, and should not be confused with the power-counting criterion.

For a graph with $E$ external scalar lines, $I$ internal lines, $L$ loops and $V_k$ vertices of degree $k$, the superficial ultraviolet divergence degree is $\omega =4L-2I$. The identities $L=I-{\sum }_k{V}_k+1$ and $2I+E={\sum }_{k}k{V}_k$ giveWhen every $k\le 4$, only the finitely many low-point local divergences, including the kinetic term, can require subtraction. Subdivergences are removed recursively by the same local counterterms. A vertex with $k>4$ instead permits arbitrarily high divergence degrees as more such vertices are added. This explains the restriction rather than merely asserting it.

The bare Lagrangian density is written in terms of regulator-dependent bare quantities, before taking the regulator away:The coefficients in $V_B$ are related to the renormalized parameters by regulator-dependent redefinitions. The wave-function renormalization supplies a kinetic counterterm. In dimensional regularization $d=4-ϵ$, a quartic coupling has a bare relation of the form ${\lambda }_B={\mu }^{ϵ}[\lambda +\delta \lambda (\lambda ,ϵ)]$, with field factors incorporated according to the chosen definition. The bare parameters and bare correlation functions are held fixed when changing the renormalization scale.

Even without a physical mass, interacting loop subtractions require a reference scale $\mu$ to define the renormalized dimensionless coupling and field normalization. Finite logarithms then involve dimensionless combinations such as $\mu |x_r-x_s|$ or $p^2/{\mu }^2$. This scale is a subtraction convention, not a new physical mass parameter. Its explicit dependence is compensated by the running coupling and field normalization.

Let $G_n=⟨\varphi (x_1)\cdots \varphi (x_n)⟩$ at separated points. Its bare counterpart satisfies $G_{B,n}=Z_{\varphi }^{n/2}{G}_n$. DefineDifferentiating $G_{B,n}$ at fixed bare theory givesThis Callan-Symanzik equation expresses independence of the arbitrary subtraction scale. The renormalization-group beta function describes the compensating coupling change; the field-renormalization anomalous dimension describes the compensating rescaling of each insertion. Coincident composite insertions would require their own additional renormalization, rather than automatically obeying this separated-point equation.

For the two-point function, set $z=p^2/{\mu }^2$ and $t=\frac{1}{2}\log z$. We consider large positive spacelike $p^2$, with continuation to other momenta as required. The dimensionless propagator factor obeysDefine $\bar{g}(0)=g$ and $d\bar{g}/dt=\beta (\bar{g})$. The renormalization-group characteristic solution for a two-point function isTo verify it, the running-coupling flow has $\beta (g){\partial }_g\bar{g}(t)=\beta (\bar{g}(t))$. Also, $\beta (g){\partial }_g{\int }_0^t\gamma (\bar{g}(s))ds=\gamma (\bar{g}(t))-\gamma (g)$. Applying ${\partial }_t-\beta (g){\partial }_g$ to the displayed expression therefore gives $2\gamma (g)d$, exactly the required differential equation. The boundary value $d(1,g)$ records the renormalization condition. This shows how the ultraviolet behaviour is determined by the coupling trajectory and its accumulated anomalous dimension: it can approach zero coupling, a nonzero fixed point, or a finite-scale singularity, depending on $\beta$.

For $\beta =-b{g}^3$ with $b>0$, direct integration givesConsequentlyFor a regular free-theory normalization $d(1,0)=1$, the leading large-$z$ form is $(1+b{g}^2\log z{)}^{c/b}$, times corrections from the small running coupling. This is asymptotic freedom with a logarithmic, rather than a fixed-power, propagator correction. The hypothetical beta and gamma functions are being used as given; this calculation does not assert that they are the actual beta and gamma functions of an arbitrary quartic scalar theory.

For the second beta function put $B=-b>0$, so $\beta (g)=B{g}^3-a{g}^5$. Its positive nonzero fixed point and linearized slope areFor $0<g<g_{\ast }$ the coupling increases with momentum, while for $g>g_{\ast }$ it decreases; each positive trajectory approaches $g_{\ast }$ in the ultraviolet. Negative initial couplings approach $-g_{\ast }$ by the odd symmetry of the beta function, with the same limiting squared coupling. The exactly zero trajectory remains zero. Thus the nonzero ultraviolet fixed point of a cubic-quintic beta function replaces asymptotic freedom by a finite-coupling ultraviolet limit, with $\bar{g}(t)-g_{\ast }=O(e^{-2B^{2}t/a})$. If the anomalous dimension is regular there, the propagator factor has fixed-point behaviour $d(z,g)\sim A(g)z^{\gamma (g_{\ast })}$ when its fixed-point boundary value is nonzero. If $\gamma (g)=c{g}^2$ is retained, the exponent is $\gamma (g_{\ast })=-cb/a$. Trusting this perturbative truncation as a statement about the underlying theory requires $g_{\ast }$ to be sufficiently small; the flow result itself follows from the specified beta function.

A nontrivial example is $f_n=T_{n+1}$, the Chebyshev polynomial characterized by $T_{n+1}(\cos \theta )=\cos ((n+1)\theta )$. Its supremum norm is one. At the $n+2$ points $\cos (j\pi /(n+1))$, $0\le j\le n+1$, its values alternate between $1$ and $-1$; reversing their order gives increasing nodes with the same alternation.

The zero polynomial therefore satisfies the Chebyshev alternation theorem for degree at most $n$, so $E_n(f_n)=1$. Since the smaller polynomial space is contained in the larger one, $E_{n-1}(f_n)\ge 1$; the zero polynomial is feasible in the smaller space as well, giving the reverse inequality. HenceA constant function would also give equality with both errors zero, but this example shows that equality can occur at a positive approximation error.

Let $p$ be the best uniform approximation to the exponential function from the degree-at-most-$n-1$ polynomial space. Suppose that $E_{n-1}(e^x)=E_n(e^x)$. Then the same $p$ is best in the larger space. Its error $g(x)=e^x-p(x)$ is nonzero, since the exponential is not a polynomial. The Chebyshev alternation theorem supplies $n+2$ ordered points where $g$ has alternating signs. By the intermediate value theorem, $g$ has at least one zero in each of the $n+1$ intervening intervals, hence at least $n+1$ distinct zeros.

Applying Rolle's theorem successively $n$ times forces a zero of $g^{(n)}$. But $\mathrm{deg}p\le n-1$, soeverywhere. This is a contradiction. Together with the automatic non-increase of best errors, it provesThe argument is an instance of strict polynomial error decrease from nonvanishing derivatives.

For each $n$, choose the best uniform approximation $p_n^{\ast }$. If its error is nonzero, take its $n+2$ alternating extrema. The intermediate value theorem supplies a point $y_j$ strictly between the $j$th and $(j+1)$st extrema such that $f(y_j)=p_n^{\ast }(y_j)$, for $0\le j\le n$. These $n+1$ points are distinct. The polynomial interpolation problem at them has a unique solution of degree at most $n$, so its solution is exactly $p_n^{\ast }$. This constructs the interpolation nodes from best uniform approximation.

If $E_n(f)=0$, then $f$ itself belongs to the polynomial space, and any $n+1$ distinct nodes give the same conclusion. In either case, take $x_{n,j}=y_j$ or the arbitrary nodes just described. ThenThe Weierstrass approximation theorem gives $E_n(f)\to 0$; alternatively, the bound derived in part 4(b), combined with uniform continuity, gives this convergence directly. ThereforeThe nodes may depend on $f$ and on $n$. The conclusion does not assert a universal sequence of interpolation grids working for every continuous target.

For a real continuous function on $[-1,1]$, the Chebyshev alternation theorem gives a unique best uniform approximation $p_n^{\ast }$ from the space of polynomials of degree at most $n$. If $f$ is not already in that space, a candidate $p$ is best if and only if there exist ordered points $x_0<\cdots <x_{n+1}$ and a sign $\sigma \in \{1,-1\}$ withThus a nonzero best error has at least $n+2$ alternating extrema of equal magnitude. If $f$ is already a polynomial of degree at most $n$, the unique best approximant is $f$ itself with zero error.

Existence does not require assuming that the infimum is attained: a minimizing sequence has uniformly bounded polynomial norms, and evaluation at $n+1$ fixed distinct nodes bounds its coefficients through an invertible interpolation matrix. A convergent coefficient subsequence then attains the infimum. The alternation criterion characterizes that minimum and gives uniqueness. In the following parts, $n\ge 1$ whenever $E_{n-1}$ occurs.

The first Jackson theorem for periodic approximation states that an absolute constant $C_J$ exists such that, for every continuous $2\pi$-periodic function $g$ and integer $n\ge 1$, there is a trigonometric polynomial $q_n$ of degree at most $n$ satisfyingApply it to the continuous periodic function $g(\theta )=f(\cos \theta )$, which is even. Replace the approximant by its even partSince $g(-\theta )=g(\theta )$, the triangle inequality gives $\Vert g-q_n^{\mathrm{ev}}{\Vert }_{\infty }\le \Vert g-q_n{\Vert }_{\infty }$. This symmetrization cancels all sine terms, so $q_n^{\mathrm{ev}}={\sum }_{j=0}^n{\alpha }_j\cos (j\theta )$. The Chebyshev polynomials satisfy $T_j(\cos \theta )=\cos (j\theta )$ and have degree $j$; the latter follows inductively from $T_0=1$, $T_1=x$ and $T_{j+1}=2x{T}_j-T_{j-1}$. Thereforeis an algebraic polynomial of degree at most $n$, and $p_n(\cos \theta )=q_n^{\mathrm{ev}}(\theta )$. Because cosine maps onto $[-1,1]$, the supremum norms agree exactly:Finally, the definition of the best error, periodic Jackson approximation, and part (a) give the successive boundsThusNeither the constant nor the choice of approximation space depends on $f$. The cosine substitution for polynomial approximation and even symmetrization are what preserve the required degree bound.

Set $r=k-1$, with $k\ge 2$ for the final linear-reproduction assertion, and use the normalized knot polynomial ${\psi }_i={\omega }_i/r!$. Differentiate the Marsden identity $r-j$ times with respect to the auxiliary variable $x$, for $0\le j\le r$. This givesThe Taylor polynomial expansion of an arbitrary polynomial $p$ of degree at most $r$ about $x$ is exact:Substitute the differentiated identity into each term and collect the B-spline coefficients. The result isThis is the normalized Marsden dual functional. There is no extra factorial outside the sum, because it is already included in ${\psi }_i$.

To prove independence of $x$ directly, differentiate the finite sum. The contribution from differentiating ${\psi }_i$ in term $j$ cancels the contribution from differentiating $p$ in term $j-1$. Explicitly,The two uncancelled endpoint terms vanish because $p^{(r+1)}={\psi }_i^{(r+1)}=0$. Thus the coefficient functional is constant in the auxiliary variable, as required.

The leading two terms of the knot polynomial giveFor a linear polynomial $p(t)=A+Bt$, only $j=0,1$ survive, whenceThe coefficient points $t_i^{\ast }$ are the Greville abscissae. Inserting these coefficients proves exact linear reproduction:All identities hold on the basic knot interval $[t_k,t_{n+1}]$, including the endpoints with the standard continuous endpoint convention. Repeated knots cause no difficulty in these polynomial differentiations.