An upper bound on the number of zeros of a piecewise polinomial function
aa r X i v : . [ m a t h . NA ] O c t AN UPPER BOUND ON THE NUMBER OF ZEROS OFA PIECEWISE POLINOMIAL FUNCTION
MARCO B. CAMINATI
Abstract.
A precise tie between a univariate spline’s knots andits zeros abundance and dissemination is formulated.As an application, a conjecture formulated by De Concini and Pro-cesi is shown to be true in the special univariate, unimodular case.As a supplement, the same conjecture is shown, through comput-ing a counterexample, to be false when unimodularity hypothesisis dropped. Introduction
A piecewise polynomial function, or univariate spline, here just spline ,of degree m is a function s ∈ C m − ( R , R ) which admits a partition ofthe real axis into a finite or numerable family of bound intervals suchthat on each of them s is the restriction of a polynomial of maximumdegree m . The bounds of the intervals are called knots . More strongly,the knots of s are the points in which s is not C ∞ , and each interval be-tween consecutive knots is a polynomiality domain . If the set of knotscoincides with Z one speaks of cardinal splines, a variation handy fortheoretical and formal elaborations. We refer the reader to [2] for amore extensive treatment of splines.One could think of s as the adjoining of a finite or numerable set ofpolynomials done so as to keep up the best possible smoothness overthe junction points we just defined as knots. This endeavour must bringsome costraints on the choice of polynomials, and particularly on thezeros of s . Our goal is to elaborate on that, and draw an adaptationof the fundamental theorem of algebra which fully takes into accountthe cited constraints.Firstly, we will restrict ourselves to the finite knots case. There’s obvi-ously no loss of generality in doing that, since one can always think afinite knots spline as the restriction on a bound interval of a numerably-knotted spline: what happens outside the outermost pair of nodes isof no interest to us at the moment. Once done that, one could try toapply the fundamental theorem of algebra to each of the polynomial-ity domain to get a first gross estimation of the upper bound over the Date : 2008/03/16.2000
Mathematics Subject Classification.
Primary 41A15, Secondary 65D07.
Key words and phrases. spline, B-spline, approximation, knot, zero.
1N UPPER BOUND ON THE NUMBER OF ZEROS OF A SPLINE 2
Figure 1. number of zeros of s , which is our goal. We must hold a moment andrealize a couple of thing before doing that: one, that the constraintswe invoked will allow a much better approach to that, and two, thatthere can be polynomiality domains on which s is identically zero, acase which in the case of polynomials does not pose much problem, andwhich here must be taken care of.So, we will specialize the definition of zeros to the one of separatedzeros (see 2.1), which is necessary to work around the annoying eventof a piecewise constant spline. Such a case for m = 1, suitable to de-piction (for the knots are identifiable at a glance with the “spikes”) isportrayed in figure 1. In its right part the result of inserting a flat zonebetween two adjacent polynomiality domains of its left part is shown.For our purpose of counting the zeros this insertion is not to be takeninto account, and that’s the motivation for the new definition.2. Main results
Definition 2.1. • a, b ∈ R are said to be f -separated iff f is a function definedand non-constant on [ a, b ], i.e. ∃ c ∈ [ a, b ] /f ( c ) = f ( a ) • Broadening the above definition: a countable subset of R { . . . < a i < a i +1 < . . . } i ∈ Z is said to be f -separated iff ∀ i ∈ Z , a i and a i +1 are f -separated. m∀ i = j ∈ Z , a i and a j are f -separated.When there will be no possibility of confusion we shall dispense our-selves from writing the prefix f -. in front of “separated”. Remark . We can now draw our first gross estimation of an upperbound on the number of separated zeros of s as already sketched in theintroduction. It’s plainly given by m ( N + 1), where N is the number ofknots, and is obtained by applying the fundamental theorem of algebra N UPPER BOUND ON THE NUMBER OF ZEROS OF A SPLINE 3 to each polynomiality domain and taking advantage of the definitionof separated zeros.This remark assures that the following definition is correct.
Definition 2.3. Z ( s ) ∈ N is the number of zeros of the finite-knottedspline s .2.5 can be viewed as the basic result of this paper from the technicalpoint of view, expressing the ties we were seeking between the numberof knots, the degree and the number of separated zeros of a splinehaving compact support. It’s preceded by a simple analytic lemma. Lemma 2.4 (strong Rolle) . Hypothesis: (1) f is derivable in [ a, b ] , ( a < b )(2) f ( a ) = f ( b )(3) a, b are f -separatedThesis: ∃ z ∈ ] a, b [ such that: (1) f ′ ( z ) = 0(2) a, z, b are f ′ -separatedProof. f , being continuous on [ a, b ], admits maximum and minimumon [ a, b ]. Employing the separation hypothesis we can suppose, forexample, the existence of z ∈ [ a, b ] /f ( z ) = max [ a,b ] f > f ( a ) = f ( b ).Being f derivable, f ′ ( z ) = 0.Moreover, f ′ can’t be constant on [ a, z ], otherwise f ′ would be thereidentically zero and f ( z ) = f ( a ) against what just stated. Then a and z are f ′ -separated. Similarly so are z and b . (cid:3) Now our informal idea is to apply repeated derivation to lower thedegree of the spline, and employ 2.4 to control the number of separatedzeros thus elicited. We thus end up with a spline of degree one (apolygonal path), for which the connection between zeros and knotsnumber is trivial and is formally expressed in 2.11.At first we proceed with our idea on a particular kind of spline, havingas outermost knots a couple of zeros of maximum order, which givesthem the quality of being conserved through derivation. This is aquality because simplifies our reasoning and permits to deduce ourfirst result. This special kind of splines, subject of 2.5, encompasses,notably, the ones with compact support, but it’s wider, containing allthe splines which can be smoothly flattened outside a compact interval.
Proposition 2.5.
Be given a spline s : R −→ R of degree m , being { α , . . . , α n } its ordered knots ( m, n ∈ Z + ). N UPPER BOUND ON THE NUMBER OF ZEROS OF A SPLINE 4
What’s more, suppose that α and α n are separated zeros of order † ≥ m ‡ Then • n ≥ m + 1 • s has at most n − m − separated zeros in ] α , α n [ .Proof. It suffices to show that Z ( s ) ≤ n − m + 1 , (2.1)which immediately implies the second statement, but also the first,because (cfr. 2.2) Z is well defined and non negative, so in this casewe get Z ( s ) ≥ m . m = 1 means that s gives a polygonal pathcomposed of n line segments; this path can’t, quite obviously, admitmore than n separated zeros (by the way, that’s true even dropping thelast request on α and α n ). For a formal proof of this last elementaryassertion see 2.11.Now let m >
1. Suppose we found k + 2 separated zeros for s ; wedenote them the following way: α =: z < z < . . . < z k < z k +1 := α n (2.2)Now we apply 2.4 (explicitly noting that we are entitled to do so,being s derivable, even more being it in C m − ) to each [ z i − , z i ] , i =1 , , . . . , k + 1 and work out z ′ < . . . < z ′ k < z ′ k +1 ( z ′ i ∈ ] z i − , z i [)(2.3) k + 1 separated zeros for s ′ .Now, again, s ′ is a spline of degree m − s Moreover, α and α n are zeros of order ≥ m − s ′ , we canconclude from the inductive hypothesis that k + 1 ≤ n − ( m − − ⇔ k ≤ n − m − (cid:3) The two statements proved in 2.5 give respectively the followingcouple of corollaries.
Corollary 2.6.
The compact support of a spline of degree m ≥ con-tains at least m + 1 consecutive polynomiality domains. † Here we mean order of a zero z according the following analytic definition,where f ∈ C m − :ord( z, f ) := min (cid:16) { j ∈ { , . . . , m − } /f ( j ) ( z ) = 0 } ∪ { m } (cid:17) Obviously this reduces to the algebraic definition given in terms of multiplicity ofa zero in case f is a polynomial. ‡ This implies the order is either m or + ∞ N UPPER BOUND ON THE NUMBER OF ZEROS OF A SPLINE 5
Equivalently † : each connected component of the support of s has “length”at least m + 1 in term of polynomiality domains.Proof. Just note that the nodes delimiting the support are forcibly alsozeros of order m , and that they are obviously mutually separated.The second formulation is obtained by choosing a connected compo-nent of the support and smoothly zeroing the spline outside of it, thenapplying what just showed. (cid:3) Corollary 2.7.
A smallest-support spline (see 2.6) never attains zerovalue inside the support itself. In particular, it’s either always strictlypositive or negative there.
Now we wish to write the general spline as a linear combinationof translations of compact-supported splines, so as to employ 2.5 anddesume the relations between m, n and the number of knots we arelooking for. In other words, given a generic spline with finitely manyknots, we would like to extend it outside its outermost knots to acompact-supported spline, which is subject to 2.5. To this end weintroduce a simple technical lemma.
Lemma 2.8.
Given a spline s of degree m and knots α < . . . < α n ,there is a compact-supported spline ¯ s extending s | [ α ,α n ] and such thatthe set of its knots is included in { α − m, α − m + 1 , . . . , α , . . . , α n , α n + 1 , . . . , α n + m } Proof.
We need to introduce B m , the B -spline of degree m ‡ . It’s a par-ticularly well-behaved cardinal spline supported on [0 , m + 1] (cfr. 2.6)whose prominent property, and the one we will need, is that translatingit at steps of length one we obtain a basis of the whole space of cardinalsplines. Again, refer to [2] for full treatment. We can limit ourselves toshow the thesis on the left side, and there’s no loss assuming α = 0.Consider § s − ( x ) := X j = − m λ j T j B m ( x ) . As just said, we can find a set of coefficients { λ j } j = − m,..., such that s − and s coincide on [0 , min { α , } ]. This implies that the “gluing” (cid:0) s − || s (cid:1) ( x ) := (cid:26) s − ( x ) , se x ≤ s ( x ) , se x ≥ m having all of its knots included in the set { α − m, α − m + 1 , . . . , α , . . . , α n } † This second formulation is suited for splines with countably many nodes as well. ‡ To avoid confusion the reader is warned that in [2] M m denotes the B-spline ofdegree m − § T λ denotes the unidimensional translation operator: T λ f : x f ( x − λ ) N UPPER BOUND ON THE NUMBER OF ZEROS OF A SPLINE 6 and is obviously identically zero at the left of α − m . (cid:3) We finally get to the main result of this paper.
Theorem 2.9.
A spline of degree m and knots α < . . . < α n has atmost n + m − separated zeros in [ α , α n ] .Proof. Called s the given spline, let us build, as granted by 2.8, itscompact-supported extension ¯ s . The latter has at most N = n +1+2 m knots, so by (2.5) will also have at most N − m − n + m − α − m, α n + m [. So its zeros in [ α , α n ] are at most n + m − s and ¯ s coincide by construction. (cid:3) The following corollary give a precise meaning to the idea that 2.9permits to render a spline zero everywhere by scattering a sufficientnumber of zeros inside its nodes, and in the right places. This way weget a result not resorting to the notion of separated zeros.
Corollary 2.10. If s is a spline of degree m and knots α < . . . < α n such that (1) card ( s − ( { } ) ∩ [ α , α n ]) ≥ n + m (2) ∀ j = 1 , . . . , n : ( | s ( α j − ) | + | s ( α j ) | ) · s (] α j − , α j [) ⊃ { } Then s is identically zero on [ α , α n ] . The formulae (1) and (2) are a compact symbolic way to render therequest that s is zero in at least n + m points (hypothesis (1)) whichmoreover are “scattered enough” along [ α , α n ] (hypothesis (2)): indeed(2) succinctly expresses the condition that s is zero either in one pointof the interior or in both the ends of any given polynomiality domain. Proof.
The m + n zeros, can’t be separated as by 2.9, therefore s isidentically zero on at least one polynomiality domain. We shall seethat it is zero on the whole [ α , α n ].By contradiction. There would be, for example, j ∈ { , , . . . , n } / (cid:26) p := s | [ α j − ,α j ] = 0 s | [ α j ,α j +1 ] = 0To preserve the regularity of s , p should have a zero of order m in α j .But according to (2), p has an additional zero in [ α j − , α j [, whilst itsdegree is just m . (cid:3) m = 1, which in turn has been employed in the proof of 2.5 itself. Lemma 2.11.
A spline of degree and knots α < . . . < α n , n ∈ Z + admits at most n separated zeros in [ α , α n ] . N UPPER BOUND ON THE NUMBER OF ZEROS OF A SPLINE 7
Proof.
By induction on n . If n = 1, and [ α , α ] contains two distinctzeros, they oviously cannot be separated. Let us now suppose that s is a spline of knots α < . . . < α n , and let k be the maximumnumber of separated knots of s restricted to [ α , α n − ]. By the inductivehypothesis k ≤ n −
1. Now we reason distinguishing two separate cases: • s ( α n − ) = 0In this case the zeros of s in ] α n − , α n ] can’t be separated from α n − , because s | [ α n − ,α n ] is a first degree polynomial. • s ( α n − ) = 0In this case s | [ α n − ,α n ] is a non-identically-null polynomial, hav-ing therefore at most one zero, so s has at most one zero in[ α n − , α n ].In any case the total number of separated zeros in [ α , α n ] is ≤ k + 1 ≤ n . (cid:3) Validating the conjecture
We briefly introduce a framework for multivariate splines which gen-eralizes the unidimensional case treated until now; see [1] for details.In the unidimensional case we had polynomiality domains given bybounded intervals and the boundaries between them given by points(knots). In the case of R s they become respectively polyhedra andhyperplanes. An interesting fact is that under unimodular hypothesisone can still build a family, given by the so-called Box splines, havingmany analogies with that of the B-splines. One just has to slightlymodify the way of describing the polynomiality domains to adequateit to the increased degrees of freedom. Instead of explicitly enumeratethe knots, one gives a finite m -ple X := ( a , . . . , a m ) of vectors of R s ,and builds the lattice it generates. The cells thus obtained are thepolynomiality domains we wanted to describe. Given X , a unique boxspline B X is determined upon requests of completeness and minimumsupport. Its degree is m − s and its support is † the convex hull P [0 , X of P { , } X . At this point we have the means to restate the conjecturestated in [1], section entitled “Explicit Projections”. Claim 3.1.
Given a m -ple X := ( a , . . . , a m ) ⊆ R s and defined as { ω , ω , . . . , ω n } = Ω := ◦ Σ [0 , X ∩
12 Σ Z X the set of the semi-integral points of the interior ‡ of the support P [0 , X of B X , the matrix ( A X ) i,j := B X (Σ { } X + ω i − ω j ) † The notation P BA , where V ⊇ A is a vector space over the field K ⊇ B , denotesall the possible linear combinations of vectors in A with coefficients in B. ‡ If X does not span R s one must resort to the notion of relative interior, cfr [1]. N UPPER BOUND ON THE NUMBER OF ZEROS OF A SPLINE 8 is invertible.
Not without unimodularity.
The symbolic calculation soft-ware [3] has been exploited to show that in case X is not unimodularthere exists a simple counterexample to 3.1. A symbolic manipulationhad to be utilized since the determinant to be computed happens to bevery near to zero even in simple cases. The counterexample is obtainedby taking X = B := ((1 , , (1 , , (0 , , ( − , A B =
18 18
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12 18
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00 0 0 0 0 0 0 0 0
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38 116 which has symbolically-computed determinant 0. On the other hand,the unimodular case X = A := ((1 , , (1 , , (0 , A A =
12 12
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00 0 det ( A A ) = which leads to our last result.3.2. True in unidimensional, unimodular case.
In this case X must be made up of (repeated) 1’s and/or − X = X m := (1 , , . . . , | {z } m +1 times (2) B X = B m (3) (cid:26) , , , , . . . , m + 12 (cid:27) , | Ω | = 2 m + 1 N UPPER BOUND ON THE NUMBER OF ZEROS OF A SPLINE 9
Theorem 3.2. A X m is nonsingular.Proof. Let us regard the j -th column c j of A X m as the image throughthe translated B-spline T ω j − Σ { } X B X m =: T λ j B X m of the set Ω = (cid:26) , , , , . . . , m + 12 (cid:27) Now let P m +1 j =1 d j · c j be a null linear combination of the c j ’s. In ourcurrent view this means that the cardinal spline P m +1 j =1 d j · T λ j B m iszero in 2 m + 1 distinct points over [0 , m + 1] satisfying also request (2)of 2.10, which then allows to conclude that m +1 X j =1 d j · T λ j B m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [0 ,m +1] ≡ T λ B m , . . . , T λ m +1 B m is a linear independent system and, moreover, neither of its elementsis identically zero on [0 , m + 1], so the only possibility is that d j =0 ∀ j = 1 , . . . , m + 1. (cid:3) References [1] C. De Concini, C. Procesi:
Topics in hyperplane arrangements, polytopes andbox-splines , preprint March 13, 2008.[2] I. J. Schoenberg:
Cardinal Spline Interpolation , Soc. for Industrial and AppliedMath, 1973[3] M. B. Caminati, boxspline2d, a a hybrid Matlab-Mathematica package forthe numerical/pointwise and symbolic/explicit (as a piecewise polynomial)calculation of the B-Spline supported on a given 2D domain available athttp://sourceforge.net/projects/boxspline2d/
Dipartimento di Matematica ”Guido Castelnuovo”, Sapienza - Uni-versit`a di Roma
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