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Use the quartic treble-knot representation of a quadratic B-spline. Specify the indexing as well as the controls: for the interior of the long curve, take the quartic spline knot sequenceWith the standard B-spline convention that control $Q_j$ multiplies $N_{j,4}$ supported on $[{\tau }_j,{\tau }_{j+5}]$, the answer isIn other words, keep the three interior quartic Bézier curve controls $E_{1,i},E_{2,i},E_{3,i}$ for each span in that order. The offset $3i-1$ matters for the stated knot indexing; shifting all indices together gives an equivalent convention.

To prove the representation, insert each integer interior knot once, changing multiplicity three to four. A degree-four knot of multiplicity four separates Bézier curve spans while leaving their shared endpoint common. The three interior controls $E_{1,i},E_{2,i},E_{3,i}$ remain, and knot insertion constructs the shared endpoint from its neighboring controls with weights $1/2,1/2$, since adjacent knot intervals have equal length. In fact,The right endpoint is obtained in the same way. Hence every extracted quartic span has exactly the five controls proved in part (b). Knot insertion leaves the curve unchanged, so the assembled quartic B-spline is identical to the original quadratic one on every span.

There is also a derivative check at each join:This is why treble knots are appropriate: degree four with interior knot multiplicity three permits $C^{4-3}=C^1$ continuity, exactly the generic continuity of the uniform quadratic B-spline. For a finite clamped curve, include its endpoint controls and endpoint knot multiplicities using the same degree elevation of Bernstein coefficients and knot insertion; the displayed formulas describe the long interior requested here, without imposing an unstated end condition.

This addresses the general interpolation bound in part (I); the subsequent (a)–(c) address the cubic case in part (II). Write the interpolant in the B-spline basis asThe stipulated uniquely defined spline interpolation operator presupposes invertibility of the B-spline collocation matrix. For the standard increasing distinct sites, the conditions $N_i(x_i)>0$ give it by the Schoenberg–Whitney theorem. Thus $c=A_{\mathbf{x}}^{-1}Rf$.

Use the standard partition-normalized B-splines of the question: they are nonnegative and satisfy the subpartition of unity for B-splines, ${\sum }_j{N}_j(t)\le 1$. This also holds outside the basic knot interval; one may extend the knots and regard the finite family as a subfamily of a full partition of unity. The recurrence proof of the full partition is given in Question 6(c). Repeated knots retain the same nonnegativity and subpartition properties, with zero-denominator recurrence terms interpreted as zero or by knot limits. ConsequentlySampling has norm at most one, $\Vert Rf{\Vert }_{{\ell }^{\infty }}\le \Vert f{\Vert }_{\infty }$, and soHere the induced matrix norm is the maximum absolute row sum. The label $L^{\infty }$-normalized means the usual partition normalization, not rescaling each spline to have maximum value one. Such a rescaling would invalidate the specific coefficient bound and the cardinal values used below.

For this part relabel one consecutive block of $k+1$ distinct knots as $t_0,\dots ,t_k$. This is a local reindexing; no extra global knot is being introduced. Let $g_t(u)=(u-t{)}_{+}^{k-1}$. Its Lagrange interpolation polynomial through the knot values isEach quotient $\omega (u)/(u-t_i)$ is monic of degree $k$. Comparing leading coefficients givesThe same leading coefficient is the order-$k$ divided difference $[t_0,\dots ,t_k]g_t$, by the Newton interpolation polynomial representation. Multiplying by $k$ therefore gives the explicit B-spline formulaDistinct knots make every denominator nonzero. For $k=1$ the degree-zero truncated power is interpreted as an interval-indicator convention; values at individual endpoints do not affect its integral.

The order-$k$ divided difference defining $M_i$ is a finite linear combination of values at its knots. Hence integration can be interchanged with it directly. For a knot value $u\in [a,b]$,It follows thatThe final equality holds because the order-$k$ divided difference of a degree-$k$ monic polynomial is its leading coefficient, one. ThusThis proves the unit-integral normalization of a B-spline. Its support is contained in $[t_i,t_{i+k}]$: below all its knots, the sampled truncated power agrees with a degree-$k-1$ polynomial, annihilated by the order-$k$ divided difference; above all its knots, every sample is zero. No mass is omitted by integration over $[a,b]$.

Since $N_i=(t_{i+k}-t_i)M_i/k$, the corresponding partition-normalized B-spline has integral $(t_{i+k}-t_i)/k$. Its normalization is different from that of $M_i$.

The Korovkin theorem states that if positive linear operators on continuous functions $L_n:C[0,1]\to C[0,1]$ satisfythen $\Vert L_{n}f-f{\Vert }_{\infty }\to 0$ for every real continuous function $f$.

Let $r=k-1\ge 2$ and $h_n=|{\Delta }_n|$. For the B-splines in this question, $N_{i,n}\ge 0$ and their sum is one on the basic interval. Thus the Schoenberg spline operator $V_n$ is linear and positive. To verify the three test functions, suppress the index $n$ and write the Marsden identity with ${\omega }_i(x)={\prod }_{j=1}^r(x-t_{i+j})$. Comparing the coefficients of $x^r,x^{r-1},x^{r-2}$ gives the monomial B-spline coefficientswhereThe first coefficient is the partition of unity; the second is the Greville abscissa. Every internal knot and the sample point ${\tau }_i$ lie in $[t_i,t_{i+k}]$, of width at most $k{h}_n$. Consequently $|{\tau }_i-{\xi }_i|\le k{h}_n$. Since all these numbers lie in $[0,1]$, for any two internal knots $u,v$ we also haveAveraging gives $|{\tau }_i^2-q_i|\le 2k{h}_n$. Nonnegative weights of sum one now yieldHere the spline order $k$ is fixed, so all three errors tend to zero. The Korovkin theorem therefore provesNo special selection such as the Greville abscissae is required for the sample points. In fact the local-support error bound for a Schoenberg spline operator also gives $\Vert V_{n}f-f{\Vert }_{\infty }\le \omega (f,k{h}_n)$. Under the usual continuous-spline convention the operators have the codomain required by the stated theorem. If the permissive knot notation is taken to allow full interior multiplicities and hence discontinuous splines, the same conclusion in the uniform norm still follows from this direct bound, although each approximant need not then be continuous.

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.

Use the standard nonnegative partition-normalized B-splines $N_j$; their sum is at most one, and is exactly one for the complete clamped basis on $[0,1]$. The printed question introduces $M_i$ only in its last matrix formula without defining it. The intended normalization is the unit-integral normalization of a B-spline:For supports wholly inside the interval, $c_i=(t_{i+k}-t_i)/k$. The positivity and partition convention, sometimes called $L_{\infty }$ normalization, should not be confused with individually rescaling every spline to have maximum exactly one.

The characterization in part 1(a) gives the normal equations for the orthogonal projection:Dividing each equation by $c_i$ yieldsThis is the mixed-normalization spline Gram matrix. It need not be symmetric, but $G=D^{-1}H$ with $D=\mathrm{diag}(c_i)$ and $H_{ij}=(N_i,N_j{)}_{L_2}$. The ordinary Gram matrix $H$ is positive definite because the splines are linearly independent, so $G$ is invertible and $a=G^{-1}d$.

Since each $M_i$ is nonnegative and has integral one,The induced matrix operator norm is the maximum absolute row sum, soFinally, nonnegativity and the subpartition of unity for B-splines giveTaking the supremum over $t$ and then over $\Vert f{\Vert }_{\infty }\le 1$ proves the maximum-norm bound for spline projection:The formulas also establish linearity and boundedness of the projection on continuous inputs. Its range is exactly the finite-dimensional spline space $\mathcal{S}$, since it fixes every spline in that space. Thus, when the splines are continuous, it is an operator $C[0,1]\to C[0,1]$ with range $\mathcal{S}$; it is not onto all of $C[0,1]$ as the printed wording says. For example, $e^t$ cannot belong to a fixed finite-degree spline space because it is not a polynomial on any nonempty knot interval. The missing $M_i$ normalization and this range correction are needed for a precise reading of the result.

Use unit knot spacing and the centered Cardinal cubic B-spline convention $C(t)={\sum }_i{P}_i{M}_3(t-i)$, where $M_3$ is supported on $[-2,2]$. On the span $[i,i+1]$, with $s=t-i$, the active control points are $P_{i-1},P_i,P_{i+1},P_{i+2}$, and their B-spline weights areFor $t=11/4$, take $i=2$, $s=3/4$. The weights and their derivatives areThe PDF has $P_3=(0,0,0)$ and $P_4=(0,1,0)$, so the point and tangent vector areThus, under this centered convention,The original PDF specifies neither the knot origin nor its indexing relative to $P_i$. If instead the convention associates span $[i,i+1]$ with $P_i,P_{i+1},P_{i+2},P_{i+3}$, the identical local weights multiply $P_2,P_3,P_4,P_5$ at the requested parameter, givingThese are parameter translations of the same uniform construction, rather than contradictory evaluations with the same knots. A specified knot convention is necessary to select the numerical pair uniquely; changing the knot spacing would also rescale the derivative.

The Cox-de Boor recurrence and De Boor's algorithm handle nonuniform knots and repeated knots without converting the whole curve. With a tabulated recurrence they require $O(p^2)$ arithmetic after locating the span and use only local control points; a naive recursive basis evaluation wastes work by recomputing the same terms. De Boor's algorithm evaluates the vector directly using affine combinations. Inside the span these are convex combinations, which makes the geometric construction transparent and avoids the cancellation that can occur in a power-basis representation. Basis evaluation is especially useful when the same knots and parameter are shared by many coordinate functions or many curves, since the weights can be reused.

Precomputed power coefficients followed by Horner's method need only $O(p)$ arithmetic per coordinate and are attractive for many evaluations within an unchanged span, particularly for fixed-degree uniform B-splines. Coefficient construction has an initial cost, control-point edits require updates to affected spans, and poor scaling of high-degree power coefficients can damage accuracy. Rescaling each span to $[0,1]$ helps. Bézier curve extraction has a similar initial conversion cost; De Casteljau's algorithm then gives stable local evaluation, subdivision, a convex hull bound and geometric flatness tests useful for rendering and intersection.

Subdivision is convenient when an entire polygonal approximation is wanted: refine only where needed and reuse the hierarchy. It is less economical for a single prescribed parameter and requires a geometric error bound before its approximate point can replace exact evaluation. Span lookup itself costs $O(\log n)$ for sorted arbitrary knots, constant work for a uniform interior span, or amortized constant work for an ordered sequence of sample parameters. Use de Boor for robust general knots, shared basis weights for batches, and precomputed local polynomials or Bézier pieces for repeated interrogation.

Write a degree-$p$ B-spline as $C(t)={\sum }_i{P}_i{N}_{i,p}(t)$ and locate its spline knot span. There are several equivalent exact evaluation methods.

Evaluate the at most $p+1$ nonzero basis functions by the Cox-de Boor recurrence, then form their weighted sum of control points. Alternatively apply De Boor's algorithm directly to the $p+1$ active control points. In a span $[t_k,t_{k+1})$, initialize $d_j^{(0)}=P_{k-p+j}$ and repeatedly interpolate:for $r=1,\dots ,p$ and $j=p,p-1,\dots ,r$. The last point $d_p^{(p)}$ is $C(t)$; zero knot intervals are handled by the standard limiting convention or an appropriate neighboring nonempty span.

A third method precomputes the local polynomial coefficients on each span and evaluates them by Horner's method. For a uniform cubic B-spline, a fixed four-by-four basis matrix gives these coefficients. A fourth method uses knot insertion to express the relevant span as a Bézier curve, followed by De Casteljau's algorithm. Both interpolation algorithms also split the curve at the requested parameter. Repeated subdivision supplies arbitrarily accurate polygonal evaluation, although a finite subdivision mesh is generally an approximation rather than the exact requested point. Basis summation, de Boor interpolation, local polynomial evaluation and Bézier extraction are equivalent exact representations.

A practical family consists of constant-distance parallel surfaces and smooth variable normal offsetsConstants cover wall thickness, cutter radius and clearance. A scalar B-spline or piecewise polynomial displacement field permits thickness variation, local bumps and smooth blending; compact support makes local edits economical. Use the base parameterization or compatible data on its subdivision hierarchy so that evaluating the displacement and its derivatives requires only a small local stencil. Choose displacement size and gradients to avoid unintended folds, and trim global self-intersections where the physical application requires an envelope. A constant displacement smaller than the local curvature radii prevents the corresponding local focal singularities, although global clearance still needs a separate check.

When remaining within a polynomial or rational representation is more useful than prescribing an exact distance, a related practical form is $P+\lambda (u,v)(P_u\times P_v)$ with a low-degree scalar field $\lambda$. Its displacement is normal but has magnitude $|\lambda |\Vert P_u\times P_v\Vert$, so it is not a constant-distance parallel surface. For polynomial $P$ and $\lambda$, it stays polynomial; for rational input it stays rational. More general low-degree vector displacement fields similarly preserve efficient evaluation, while relinquishing the normal-offset condition. These forms make explicit the tradeoff between an exact metric offset, which involves normalization, and an easily represented deformation. Simple scalar normal-displacement fields are useful and directly evaluable; algebraic displacement fields are useful when representation closure is required.

The transverse row sequence in the extrusion invariance of Loop subdivision obeysIn the convention $g_k^{\prime }={\sum }_j{a}_{k-2j}{g}_j$, the centered univariate subdivision mask isThe even coefficients sum to one and the odd coefficients sum to one; the whole binary mask sums to two. Its generating expression is $a(z)=z^{-2}(1+z{)}^4/8$, identifying the Cardinal cubic B-spline refinement rule. Its limit is $G(y)={\sum }_j{g}_j{M}_3(y-j)$ for the centered unit-grid cubic basis $M_3$. The curve across rows is this cubic B-spline; extrusion generates the whole limit surface. The along-extrusion direction itself has constant height, so its profile is a straight line rather than an additional nonconstant cubic curve.

Use the standard partition normalization of the B-splines, so $N_j\ge 0$ and the subpartition of unity for B-splines gives ${\sum }_{j=1}^n{N}_j(t)\le 1$. Write the interpolating spline as $P_{x}f={\sum }_j{c}_j{N}_j$ and the sampled values as $y_i=f(x_i)$. The interpolation equations are $A_{x}c=y$, hence $c=A_x^{-1}y$ whenever the stated interpolation map is well defined. For every $t$,The sampling map also satisfies $\Vert y{\Vert }_{{\ell }^{\infty }}\le \Vert f{\Vert }_{\infty }$. Taking supremum norms and then the operator norm therefore provesThis is the B-spline interpolation operator norm bound, and it applies on any interval containing the sampling sites on which these normalized B-splines are used.

There is a small qualification to the printed hypotheses: diagonal positivity alone does not ensure uniqueness unless the usual site assumptions are supplied. For example, with two quadratic B-splines on knots $1,2,3,4,5$ and $x_1=x_2=3$, both diagonal values equal $1/2$, but $A_x$ has two identical rows and is singular. The intended increasing sites $x_1<\cdots <x_n$ with $N_i(x_i)>0$ satisfy the Schoenberg–Whitney theorem, making $A_x$ invertible; alternatively invertibility is implicit in the premise that $P_x$ is the unique interpolation map. The displayed bound is proved under that premise, rather than inferring invertibility from diagonal positivity alone.

For unit-spaced knots, the quadratic cardinal B-spline is a translate ofFor example, the Cox-de Boor recurrence gives $N_{j,3}(t)=(t-j)N_{j,2}(t)/2+(j+3-t)N_{j+1,2}(t)/2$ from the two linear hats. Substituting their linear pieces gives the three expressions above. At $x_i=i+3/2$, only the B-splines with indices $j=i-1,i,i+1$ have nonzero values, and their values are respectivelyOnly indices $1\le j\le n$ belong to the given basis, so entries outside that range are omitted at the boundary.

Put $s_j=t_{i+j}$ for $0\le j\le k$, with the knots in increasing order. The explicit formula for a divided difference at distinct nodes givesEach truncated power function is a piecewise polynomial function of degree $k-1$ and is $C^{k-2}$ for $k\ge 2$. Therefore their sum has the same global smoothness. The jump in the $(k-1)$st derivative at $s_j$ equalswhich is nonzero. Thus each $s_j$ really is a knot, and generally there is no additional smoothness.

If $t<s_0$, the values being differenced are the values of the degree-$k-1$ polynomial $(s-t{)}^{k-1}$, so their $k$th divided difference vanishes. If $t>s_k$, every sampled truncated power function is zero. To check that both support endpoints occur, for $s_0<t<s_1$ subtract the vanished full polynomial divided difference to obtainFor $s_{k-1}<t<s_k$, only the last term survives and is positive. The nonnegativity from the recurrence in (b) actually gives positivity throughout the interior: for each $s_0<t<s_k$, at least one lower-order B-spline covering $t$ has a positive recurrence coefficient, and induction starts with positive order-one indicators. Hence $\mathrm{supp}{M}_i=[t_i,t_{i+k}]$ and $M_i\in C^{k-2}$ for $k\ge 2$, with degree $k-1$ on its knot intervals. For $k=1$, the B-spline is the step function $1_{[s_0,s_1)}(t)/(s_1-s_0)$, so no continuity is asserted.

The unit-integral normalization of a B-spline also follows directly: integrating $k(s_j-t{)}_{+}^{k-1}$ over $s_0\le t\le s_k$ gives $(s_j-s_0{)}^k$. The $k$th divided difference of this monic degree-$k$ polynomial is one. Thus $\int M_i(t)dt=1$, explaining its normalization.