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<title>A comment on Ewald Quak’s “About B-splines”: A comment on Ewald Quak’s “About B-splines”</title>
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<h1>A comment on Ewald Quak’s “About B-splines”</h1>
<p class="authors">
<span class="author">Carl de Boor\(^\ast \)</span>
</p>
<p class="date">June 6, 2016.</p>
</div>
<p>\(^\ast \)University of Wisconsin, Department of Computer Sciences, e-mail <span class="tt">deboor@cs.wisc.edu, crdeboor@gmail.com</span>. </p>

<div class="abstract"><p> The early contributions to B-spline theory by Tiberiu Popoviciu and by Liubomir Chakalov are recalled. </p>
<p><b class="bf">MSC.</b> 41-XX, 65-XX, 60-XX. </p>
<p><b class="bf">Keywords.</b> B-spline, B-spline recurrence, Marsden’s identity, knot insertion, Popoviciu, Chakalov. </p>
</div>
<h1 id="a0000000002"></h1>
<p>The publication of a paper on the many aspects of B-splines in this journal suggests adding a comment on the contributions to B-splines made by this journal’s founder, Tiberiu Popoviciu. </p>
<p>Briefly, in his paper <span class="cite">
	[
	<a href="#P34b" >P34b</a>
	]
</span> on \(n\)-convex functions (a terminology he introduced), Povoviciu proves the recurrence relation for B-splines (see <span class="cite">
	[
	<a href="#Q" >Q</a>
	, 
	(42)
	]
</span>), Marsden’s identity (see <span class="cite">
	[
	<a href="#Q" >Q</a>
	, 
	p.46
	]
</span>), and mentions without proof that the B-splines form a generating set for what he calls <b class="bf">elementary functions of degree \(n\) with \(m\) vertices</b> by which he means smooth functions whose \((n-1)\)st derivative is a continuous broken line with \(m-2\) interior breaks. In addition, you can find in <span class="cite">
	[
	<a href="#P34a" >P34a</a>
	]
</span> Boehm’s formula for knot insertion (if you know what to look for). </p>
<p>To be a bit more explicit (as laid out in <span class="cite">
	[
	<a href="#BP03" >BP03</a>
	]
</span>), Popoviciu introduces (see <span class="cite">
	[
	<a href="#P34b" >P34b</a>
	, 
	p.89
	]
</span>), for a given strictly increasing sequence \(x_1{\lt}\cdots {\lt}x_m\), certain piecewise polynomial functions \(\Psi _i\) with support in the interval <a class="footnote" href="#a0000000003">
  <sup class="footnotemark">1</sup>
</a> \((x_i\mathbin {\ldotp \ldotp }x_{i+n+1})\), \(i=1,\ldots ,m-n-1\), of degree \(\le n\), and proves (see <span class="cite">
	[
	<a href="#P34b" >P34b</a>
	, 
	p.93
	]
</span>) the relation </p>
<div class="equation" id="recurrence">
<p>
  <div class="equation_content">
    \begin{equation}  (x_{n+i+1}-x_i)\Psi _i(x) = (x-x_i)\Psi '_i(x)+(x_{n+i+1}-x)\Psi _{i+1}'(x), \label{recurrence} \end{equation}
  </div>
  <span class="equation_label">1</span>
</p>
</div>
<p> with, as he writes, the \(\Psi _i'\) defined just like the \(\Psi _i\) except that \(n\) is replaced by \(n-1\). Compare (<a href="#recurrence">1</a>) with the well-known recurrence relation (see <span class="cite">
	[
	<a href="#Q" >Q</a>
	, 
	(42)
	]
</span>) </p>
<div class="displaymath" id="a0000000004">
  \[  B_{i,n+1} = \omega _{i,n+1}B_{in} + (1-\omega _{i+1,n+1})B_{i+1,n}, \qquad \omega _{i,n+1}(x):=\tfrac {x-x_i}{ x_{i+n}-x_i}  \]
</div>
<p> in which \(B_{jk}\) is the B-spline with knots \(x_j,\ldots ,x_{j+k}\) (normalized to be part of a partition of unity), and add to that the fact that Popoviciu’s formula for \(\Psi _i\) readily reduces to \(\Psi _i=\chi _{\lower 3pt\hbox{$\scriptstyle (x_i\mathbin {\ldotp \ldotp }x_{i+1})$}}\) for \(n=0\), to conclude that, for given \(n\), \((x_{i+n+1}-x_i)\Psi _i=B_{i,n+1}\). In fact, since Popoviciu’s formula for \(\Psi _i\) involves ratios of Vandermonde determinants, it is not that hard to derive directly that </p>
<div class="equation" id="bspline">
<p>
  <div class="equation_content">
    \begin{equation}  \Psi _i(x) = \mathord {\setbox 0=\hbox{$\Delta $}\dimen 0=\wd 0\divide \dimen 0 by 2 \kern \dimen 0\vrule height\ht 0 depth\dp 0 width.5pt\kern -.25pt\kern -\dimen 0\box 0(x_i,\ldots ,x_{i+n+1})}(\cdot -x)_+^n \label{bspline} \end{equation}
  </div>
  <span class="equation_label">2</span>
</p>
</div>
<p> with \(\mathord {\setbox 0=\hbox{$\Delta $}\dimen 0=\wd 0\divide \dimen 0 by 2 \kern \dimen 0\vrule height\ht 0 depth\dp 0 width.5pt\kern -.25pt\kern -\dimen 0\box 0(x_i,\ldots ,x_{i+n+1})}=[x_i,\ldots ,x_{i+n+1}]\) the divided difference<a class="footnote" href="#a0000000005">
  <sup class="footnotemark">2</sup>
</a> functional at the nodes \(x_i,\ldots ,x_{i+n+1}\) (provided one knows the handy notation \((\cdot -a)_+^n\) for the \(n\)th truncated power). The simple formula (<a href="#bspline">2</a>) could have been of help in simplifying some of Popoviciu’s arguments concerning \(n\)-convexity. It could also have readily supplied the fact (not proved in the paper) that the \(\Psi _i\) have all derivatives of order \({\lt}n\) continuous. </p>
<p>Popoviciu uses the recurrence relation (<a href="#recurrence">1</a>) to prove the positivity of the \(\Psi _i\) on their support, as well as the following formula (see <span class="cite">
	[
	<a href="#P34b" >P34b</a>
	, 
	p.93
	]
</span>) </p>
<div class="equation" id="marsden">
<p>
  <div class="equation_content">
    \begin{equation}  \sum _{i=1}^{n+1}(x_{i+n+1}-x_i)\Psi _i(\xi )(x-x_{i+1})\cdots (x-x_{i+n})= (x-\xi )^n \label{marsden} \end{equation}
  </div>
  <span class="equation_label">3</span>
</p>
</div>
<p> which we now call Marsden’s identity because of <span class="cite">
	[
	<a href="#Ma70" >Ma70</a>
	]
</span>. </p>
<p>While the sequence \(x_1,\ldots ,x_m\) starts out strictly increasing, in the last section of <span class="cite">
	[
	<a href="#P34b" >P34b</a>
	]
</span> all is specialized to the case </p>
<div class="displaymath" id="a0000000006">
  \[ x_1=\cdots =x_{n+1}=a {\lt} b= x_{n+2}=\cdots =x_{2n+2} \]
</div>
<p> which we now associate with the names Bernstein and Bézier. </p>
<p>Finally, on page 7 of <span class="cite">
	[
	<a href="#P34a" >P34a</a>
	]
</span>, Popoviciu uses induction to derive from the formula </p>
<div class="equation" id="knotinsert">
<p>
  <div class="equation_content">
    \begin{equation}  (t_n-t_0)\mathord {\setbox 0=\hbox{$\Delta $}\dimen 0=\wd 0\divide \dimen 0 by 2 \kern \dimen 0\vrule height\ht 0 depth\dp 0 width.5pt\kern -.25pt\kern -\dimen 0\box 0(t_0,\ldots ,t_n)} = (t_n -\xi )\mathord {\setbox 0=\hbox{$\Delta $}\dimen 0=\wd 0\divide \dimen 0 by 2 \kern \dimen 0\vrule height\ht 0 depth\dp 0 width.5pt\kern -.25pt\kern -\dimen 0\box 0(\xi ,t_1,\ldots ,t_n)}+(\xi -t_0)\mathord {\setbox 0=\hbox{$\Delta $}\dimen 0=\wd 0\divide \dimen 0 by 2 \kern \dimen 0\vrule height\ht 0 depth\dp 0 width.5pt\kern -.25pt\kern -\dimen 0\box 0(t_0,\ldots ,t_{n-1},\xi )} \label{knotinsert} \end{equation}
  </div>
  <span class="equation_label">4</span>
</p>
</div>
<p> the fact that, for an increasing refinement \(\sigma \) of the increasing sequence \(\tau \), </p>
<div class="equation" id="oslo">
<p>
  <div class="equation_content">
    \begin{equation}  \mathord {\setbox 0=\hbox{$\Delta $}\dimen 0=\wd 0\divide \dimen 0 by 2 \kern \dimen 0\vrule height\ht 0 depth\dp 0 width.5pt\kern -.25pt\kern -\dimen 0\box 0(\tau _0,\ldots ,\tau _n)} = \sum _j\alpha _j(\tau ,\sigma )\mathord {\setbox 0=\hbox{$\Delta $}\dimen 0=\wd 0\divide \dimen 0 by 2 \kern \dimen 0\vrule height\ht 0 depth\dp 0 width.5pt\kern -.25pt\kern -\dimen 0\box 0(\sigma _j,\ldots ,\sigma _{j+n})}\label{oslo} \end{equation}
  </div>
  <span class="equation_label">5</span>
</p>
</div>
<p> with the coefficients \(\alpha _j(\tau ,\sigma )\) nonnegative and summing to 1. Given formula (<a href="#bspline">2</a>), you will recognize in (<a href="#knotinsert">4</a>) Boehm’s now standard formula for knot insertion, and in (<a href="#oslo">5</a>) a formula for expressing the B-spline with knots \(\tau _0,\ldots ,\tau _n\) in terms of B-splines of the same order on a finer knot sequence \(\sigma \). </p>
<p>Another early contributor to B-splines is Liubomir Chakalov who studied (see <span class="cite">
	[
	<a href="#C38" >C38</a>
	]
</span> or its discussion in <span class="cite">
	[
	<a href="#BP03" >BP03</a>
	]
</span>) B-splines in the sense that he focused on the Peano kernel \(u\) in the integral representation </p>
<div class="displaymath" id="a0000000007">
  \[  \mathord {\setbox 0=\hbox{$\Delta $}\dimen 0=\wd 0\divide \dimen 0 by 2 \kern \dimen 0\vrule height\ht 0 depth\dp 0 width.5pt\kern -.25pt\kern -\dimen 0\box 0(x_1,\ldots ,x_{n+1})}f = \int u(s) D^nf(s)\, {\rm d}s  \]
</div>
<p> of the divided difference. Necessarily, </p>
<div class="displaymath" id="a0000000008">
  \[  u(x) = \mathord {\setbox 0=\hbox{$\Delta $}\dimen 0=\wd 0\divide \dimen 0 by 2 \kern \dimen 0\vrule height\ht 0 depth\dp 0 width.5pt\kern -.25pt\kern -\dimen 0\box 0(x_1,\ldots ,x_{n+1})}(\cdot -x)_+^{n-1}/(n-1)!  \]
</div>
<p> hence, by (<a href="#bspline">2</a>) (see also <span class="cite">
	[
	<a href="#Q" >Q</a>
	, 
	p.22
	]
</span>), \(u\) is a B-spline, </p>
<div class="displaymath" id="a0000000009">
  \[  u=B_{1n}/((x_{n+1}-x_1)(n-1)!).  \]
</div>
<p> Chakalov provides the B-spline recurrence relation in a form that involves a derivative and, more importantly, provides the representation </p>
<div class="displaymath" id="a0000000010">
  \[  u(x) = \tfrac 1{2\pi {\rm i}}\int _C \tfrac {(z-x)^{n-1}\, {\rm d}z}{(n-1)!\prod \limits _{j=1}^{n+1}(z-x_j)}  \]
</div>
<p> of this B-spline in terms of a contour integral (with the contour \(C\) dependent on \(x\) and the \(x_i\)) which was rediscovered many years later by Meinardus and put to good use in <span class="cite">
	[
	<a href="#Me74" >Me74</a>
	]
</span>. </p>
<p><span class="cite">
	[
	<a href="#Q" >Q</a>
	]
</span> mentions Sommerfeld, and could have mentioned Favard, and even Laplace, as people who, like Popoviciu and Chakalov, came across B-splines in their work well before (Curry and) Schoenberg, although Popoviciu and Chakalov are the only ones I am aware of who developed the B-spline recurrence relations. It seems that it is not enough to have a good idea or insight. One needs, like Schoenberg, the appreciation and courage to develop the idea systematically, make its objects mathematically presentable by giving them names, and give them much exposure in many papers. </p>
<p><small class="footnotesize">  </small></p>
<div class="bibliography">
<h1>Bibliography</h1>
<dl class="bibliography">
  <dt><a name="B80">B80</a></dt>
  <dd><p><a href ="http://dx.doi.org/10.1016/0010-4485(80)90154-2">W. <i class="sc">Boehm</i>, <i class="itshape">Inserting new knots into B-spline curves</i>,  Computer-Aided Design , <b class="bf">12</b> (1980) no. 4, pp.199–201. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="BP03">BP03</a></dt>
  <dd><p><a href ="http://dx.doi.org/10.1016/S0021-9045(03)00117-5"> C. <i class="sc">de Boor</i> and A. <i class="sc">Pinkus</i>, <i class="it">The B-spline recurrence relations of Chakalov and of Popoviciu</i>, J. Approx. Theory, <b class="bf">124</b> (2003) no. 1, pp.115–123. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="C38">C38</a></dt>
  <dd><p>L. <i class="sc">Chakalov</i>, <i class="it">On a certain presentation of the Newton divided differences in interpolation theory and it applications</i>, Annuaire Univ. Sofia, Fiz. Mat. Fakultet, <b class="bf">34</b> (1938), pp.353–394 (in Bulgarian). </p>
</dd>
  <dt><a name="Ma70">Ma70</a></dt>
  <dd><p><a href ="http://dx.doi.org/10.1016/0021-9045(70)90058-4"> M.J. <i class="sc">Marsden</i>, <i class="it">An identity for spline functions with applications to variation-diminishing spline approximation</i>, J. Approx. Theory, <b class="bf">3</b> (1970), pp.7–49. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="Me74">Me74</a></dt>
  <dd><p>G. <i class="sc">Meinardus</i> , <i class="it">Bemerkungen zur Theorie der B-Splines</i>, in Spline-Funktionen (K. Böhmer, G. Meinardus, and W. Schempp Eds.), Bibliographisches Institut (Mannheim), 1974, pp.165–175. </p>
</dd>
  <dt><a name="P34a">P34a</a></dt>
  <dd><p><a href ="http://ictp.acad.ro/popoviciu"> T. <i class="sc">Popoviciu</i>, <i class="it">Sur quelques propriétés des fonctions d’une ou de deux variables réelles</i>, Mathematica, <b class="bf">8</b> (1934), pp.1–85. Retrieved on October 3rd, 2016, from <span class="tt">http://ictp.acad.ro/popoviciu</span> <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
 <img src="img-0002.png" alt="\includegraphics[scale=0.06]{auth-homepage-link.png}" style="width:13.5px; height:11.879999999999999px" />
</a> </p>
</dd>
  <dt><a name="P34b">P34b</a></dt>
  <dd><p><a href ="http://ictp.acad.ro/popoviciu"> T. <i class="sc">Popoviciu</i>, <i class="it">Sur le prolongement des fonctions convexes d’ordre superieur</i>, Bull. Math. Soc. Roumaine des Sc., <b class="bf">36</b> (1934), pp.75–108. Retrieved on October 3rd, 2016, from <span class="tt">http://ictp.acad.ro/popoviciu</span> <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
 <img src="img-0002.png" alt="\includegraphics[scale=0.06]{auth-homepage-link.png}" style="width:13.5px; height:11.879999999999999px" />
</a> </p>
</dd>
  <dt><a name="Q">Q</a></dt>
  <dd><p><a href ="http://ictp.acad.ro/jnaat/journal/issue/view/2016-vol45-no1"> E. <i class="sc">Quak</i>, <i class="it">About B-splines. Twenty answers to one question: What is the cubic B-spline for the knots -2,-1,0,1,2? </i>, J. Numer. Anal. Approx. Theory, <b class="bf">45</b> (2016) no. 1, pp.37–83. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="S64">S64</a></dt>
  <dd><p><a href ="http://dx.doi.org/10.1073/pnas.52.4.947"> I.J. <i class="sc">Schoenberg</i>, <i class="it">Spline functions and the problem of graduation</i>, Proc. Amer. Math. Soc., <b class="bf">52</b> (1964), pp.947–950. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
</dl>


</div>
</div> <!--main-text -->
<footer id="footnotes">
<ol>
  <li id="a0000000003">I was pleased to see already in Popoviciu’s article this handy use of \(\mathbin {\ldotp \ldotp }\) in the description of an interval.</li>
  <li id="a0000000005">I was pleased to see this literal notation for divided differences already in <span class="cite">
	[
	<a href="#S64" >S64</a>
	]
</span>.</li>
</ol>
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