Multidimensional Chebyshev Systems - just a definition
aa r X i v : . [ m a t h . F A ] A p r Multidimensional Chebyshev Systems (Haarsystems) - just a definition
Ognyan KounchevInstitute of Mathematics and InformaticsBulgarian Academy of SciencesandIZKS, University of BonnMay 22, 2018
Abstract
The notion of Chebyshev system was coined by S. Bernstein al-though many resutls were proved already by A. Markov. Chebyshevsystems play important role in the one-dimensional Moment problemand Approximation and Spline theory. We generalize the notion ofChebyshev system for several dimensions and define the Multidimen-sional Chebyshev Systems of order N . We prove that this definition issatisfied by the solutions of a wide class of elliptic equations of order2 N . This definition generalizes a very large class of Extended Com-plete Chebyshev systems in the one-dimensional case. This is the firstof a series of papers in this area, which solves the longstanding prob-lem of finding a satisfactory multidimensional generalization of theclassical Chebyshev systems introduced already by A. Markov morethan hundered years ago. History of Chebyshev systems
It was namely in the Moment problem where the notion of Chebyshev systemsappeared for the first time on the big stage, and provided a very natural andbeautiful generalizations of the results of Gauss, Jacobi, Chebyshev, Stieltjes,Markov, and others.The classical Moment problem is defined as follows: Find a non-negativemeasure dµ such that Z ba t j dµ ( t ) = c j for j = 0 , , ..., N. The solution of the problem includes conditions on the constants c j providingsolvability. In the case of N = 2 n − Gauss-Jacobi quadrature ; this solution is based on the orthogonalpolynomials P n (of degree n ) which are orthogonal with respect to the innerproduct defined by (cid:10) t j , t k (cid:11) := c j + k . The history is well described in the book of M. Krein and A. Nudelman ”TheMarkov Moment Problem” [5], actually based on a 1951 paper of M. Kreindevoted to the ideas of Chebyshev. There it is said that A. Markov hasrealized that one may consider successfully the Moment problem of the type Z ba u j ( t ) dµ ( t ) = c j for j = 0 , , ..., N, where the system of continuous functions { u j ( t ) } Nj =0 represent a Cheby-shev system in the interval [ a, b ], i.e. any linear combination u ( t ) = N X j =0 α j u j ( t )has no more than N zeros in [ a, b ].Further, by U N we will denote the subspace of the space of continuousfunctions C ([ a, b ]) generated by the Chebyshev system, i.e. U N := ( u ( t ) : u ( t ) = N X j =0 α j u j ( t ) ) . (1)2n general, in areas other than Approximation theory and Moment prob-lem, people have tried to find those properties of the one-dimensional al-gebraic polynomials which make them so nice. Apparently, the Chebyshevproperty seems to be such. Let us remind also the famous Chebyshev alternance theorem which hasbeen proved for
Chebyshev systems , and which one would like to see in amultivariate setting (cf. [5], chapter 9 , Theorem 4 . Theorem 1 ( Chebyshev-Markov ) Let f ∈ C ([ a, b ]) . A necessary andsufficient condition for the element u ∈ U N to solve problem inf u ∈ U N k f − u k C = inf u ∈ U N (cid:18) max x ∈ [ a,b ] | f ( x ) − u ( x ) | (cid:19) =: δ is the existence of N + 2 points t < · · · < t N +2 such that δε ( − j = f ( t j ) − u ( t j ) for j = 1 , ..., N + 2 where ε = 1 or ε = − . What concerns other areas where Chebyshev systems have found numer-ous applications, one has to mention the book of [11] which contains an ex-haustive consideration of spline theory where splines are piecewise functionsbelonging to a Chebyshev system.
Let us provide some basic definitions.Consider the system of functions { u j ( t ) } Nj =0 defined on some interval [ a, b ]in R and the linear space defined in (1).3 efinition 2 We call the system of functions { u j ( t ) } Nj =0 Chebyshev (or T − system)iff for every set of constants { c j } Nj =0 and every choice of the points t j ∈ [ a, b ] with t < t < · · · < t N there is a unique solution u ∈ U of the equations u ( t j ) = c j for j = 0 , , ..., N. It is equivalent to say that u ( t j ) = 0 for j = 0 , , ..., N implies u ≡ . Proposition 3
Assume that the space U ⊂ C [ a, b ] is given. Then if forsome set of knots t j ∈ [ a, b ] with t < t < · · · < t N , and for arbitrary constants { c j } we have unique solution u ∈ U of the equa-tions u ( t j ) = c j for j = 0 , , ..., N it follows that dim U = N + 1 . One may formulate the above in an equivalent way:
Proposition 4
The following are equivalent1. the system { u j ( t ) } Nj =0 is T − system2. for every u ∈ U N the number of zeros in the interval [ a, b ] is ≤ N.
3. the following determinants satisfy D ( u ; t , t , ..., t N ) := det u ( t ) u ( t ) · · · u ( t N ) u ( t ) u ( t ) · · · u ( t N ) · · · · u N ( t ) u N ( t ) · · · u N ( t N ) = 0 Definition 5 If D ( u ; t , t , ..., t N ) ≥ then { u j ( t ) } Nj =0 is called T + − system. .1 Examples The classical polynomials, the trigonometric polynomials in smaller intervals[0 , π ]!1. the system { u j ( t ) = t α j } Nj =0 on subintervals of [0 , ∞ ]2. the system u j ( t ) = 1 s j + t for 0 < s < s < ··· < s N on closed subint. of (0 , ∞ ) .
3. the system u j ( t ) = e − ( s j − t ) for 0 < s < s < · · · < s N on ( −∞ , ∞ ) .
4. if G ( s, t ) is the Green function associated with the operator Lf = − ddx (cid:18) p dfdx (cid:19) + qf and some boundary conditions on the interval [ a, b ] , then the system u j ( t ) = G ( s j , t ) for 0 < s < s < ··· < s N on closed subint. of [ a, b ]For further examples, see the monograph of M. Krein and A. Nudel’man[5] and of S. Karlin and W. Studden, [6]. We usually work with differentiable systems of functions, and we count themultiplicities of the zeros.
Definition 6
Let { u j ( t ) } Nj =0 ∈ C N [ a, b ] be a T − system. We call it Ex-tended Chebyshev system ( ET − system) if in U N we may uniquely solve the Hermite interpolation problem : u ( k ) ( t j ) = c j,k for k = 0 , , ..., d j ith arbitrary numbers { c j,k } where X ( d j + 1) = N + 1 . It is equivalent to say that if for some u ∈ U N holds u ( k ) ( t j ) = 0 for k = 0 , , ..., d k then u ≡ . There are equivalent formulations with determinants and zeros:
Proposition 7
The following are equivalent:1. the system { u j ( t ) } Nj =0 ∈ C N [ a, b ] is and ET − system2. for every u ∈ U the number of zeros counted with the multiplicities is ≤ N.
3. the modified determinants (see the detailed explanation in [5], end ofsection , chapter , or [6]), det u ( t ) u ′ ( t ) · · · u ( k − ( t ) u ( t ) · · · u ( t N ) u ( t ) u ′ ( t ) · · · u ( k − ( t ) u ( t ) · · · u ( t N ) · · · · · · · u N ( t ) u ′ N ( t ) · · · u ( k − N ( t ) u N ( t ) · · · u N ( t N ) = 0There is a nice characterization of ET − systems. A basic example of ET − system is the following: Let w i ∈ C N − i [ a, b ] be positive functions on[ a, b ] for i = 0 , , ..., N. Then the functions u ( t ) = w ( t ) (2) u ( t ) = w ( t ) Z ta w ( t ) dt (3) u ( t ) = w ( t ) Z ta w ( t ) Z t a w ( t ) dt · dt (4) · ·· (5) u N ( t ) = w ( t ) Z ta w ( t ) Z t a w ( t ) · · · Z t N − a w N ( t N ) dt N · · · dt (6)form an ET − system. 6 efinition 8 An T − system { u k } Nk =0 is called M + − system if every subsystem { u k } mk =0 for m = 0 , , ..., N is a T + − system. There is an important result which is basically due to S. Bernstein (cf.[5], Theorem 4 . . , chapter 2 , and the footnote at the end). Theorem 9
Every T − system { u k } Nk =0 on the interval [ a, b ] may be linearlytransformed to an M + − system { v k } nk =0 on the interval ( a, b ) . A similar result may be proved in the differentiable case.
Corollary 10
Every ET − system in [ a, b ] may be linearly transformed to an ECT − system in ( a, b ) . The role of the
ECT systems becomes clear from the following funda-mental result.
Theorem 11
If the space U N is generated by an ECT − system of order N then it has a basis { v j } Nj =0 which is representable in the above form (2)-(6).Hence, U N is a set of solutions to the following equation L N u = 0 for t ∈ ( a.b ) and L N = N Y j =0 ddt w j ( t ) . (7)The proof is available in [6] (chapter 11 , Theorem 1 .
1) and for M + − systems in [5], Theorems 4 .
1, 4 . .
2, chapter 2.
We have to note that all ”brute force generalizations” of the Chebyshevsystems fail. We will mention some of them.7 .1 Generalization by zero sets – theorem of Mairhu-ber
Apparently, the first attempt has been to generalize the Chebyshev systemsby considering the set of zeros:
Definition 12
Let K be a compact topological space. The system of func-tions { u j } Nj =0 is called Chebyshev of order N iff Z ( u ) ≤ N for every u ∈ U. The following result shows that there are no non-trivial examples of mul-tidimensional systems satisfying Definition 12 (cf. [5], chapter 2 , section 1). Theorem 13 (Mairhuber, ) The only spaces K having a Chebyshev sys-tem satisfy K ⊂ R or K ⊂ S . The result of Theorem 13 is intuitively clear since ”general position”function in C ( K ) has a zero set which is a subset of K of codimension 1 . In particular, if K = R then the zero set is roughly speaking union of somecurves, and it would be more reasonable to speculate about the numberof these components then to consider Definition 12 above. Speculating inthis direction in the multidimensional case, in view of the PolyharmonicParadigm [7], one may try to replace the points on R by closed surfaces, andthe spaces U by solutions of Elliptic PDEs. Going further, one may obtainsome interesting results if one uses spheres in the case of the polyharmonicoperator ∆ N by considering the space U N = (cid:8) u : ∆ N u = 0 in D (cid:9) for some bounded domain D ⊂ R n . In particular, one may prove that ifa polyharmonic function of order N in a domain D (a function satisfying∆ N u = 0 in D ) is zero on a set of n concentric spheres, then u ≡ . Thisresult has been proved apparently a long time ago by means of the Almansitheorem, see e.g. [1]. However these hopes to try to generalize Definition12 are only vain. They have been broken by and example which has beenpublished apparently for a first time in 1982, by Atakhodzhaev, [1]. It showsa non-zero biharmonic function which is zero on two embedded ovals in R . .2 Generalization by Haar property The following result belongs to A. Haar, [5]:
Theorem 14
Let the space U N ⊂ C [ a, b ] be generated by a Chebyshev system { u j } Nj =0 . Then for every f ∈ C [ a, b ] the best approximation problem min u ∈ U N k f − u k C has unique solution. Extending this definition to the multivariate case seems to be very rea-sonable but the work with best approximations is very heavy and until nowhas not led to success.
One needs a new point of view on the Chebyshev systems which would makethem generalizable to several dimensions. We propose the point of view ofboundary value problems: We consider a special class of ET − systems whichare ”generalizable”. Definition 15
We say that the system { u j } N − j =0 ∈ C N − [ a, b ] is a Dirich-let type Chebyshev system, or DT − system, in the interval [ a, b ] if forevery two points α and β in [ a, b ] and for every set of constants c j and d j we are able to solve uniquely the following interpolation problem , with u ∈ U N , u ( k ) ( α ) = c k for k = 0 , , ..., N − u ( k ) ( β ) = d k for k = 0 , , ..., N − Remark 16
Obviously, all ET − systems are DT − systems but not vice versa. Let us state an equivalent formulation which we are going to mimic inthe multivariate case.
Proposition 17
The system { u j } N − j =0 ∈ C ∞ [ a, b ] is a Dirichlet typeChebyshev system in the interval [ a, b ] iff for every two points α and β in [ a, b ] , for every set of constants c j and d j , and for every ε > , we re able to solve the following approximate interpolation problem, with u ∈ U N = span { u j } N − j =0 , (cid:12)(cid:12) u ( k ) ( α ) − c k (cid:12)(cid:12) < ε for k = 0 , , ..., N − (cid:12)(cid:12) u ( k ) ( β ) − d k (cid:12)(cid:12) < ε for k = 0 , , ..., N − U N is finite-dimensional. In the present section we will provide a multivariate generalization to the DT -systems of Definition 15.One might use the properties of the Dirichlet type Chebyshev system pro-vided in Proposition 17 as possible way to make a multivariate generalization.However we would like to have also the properties exposed by Theorem 11as well. Remark 18
It is expected that a further research would prove equivalencebetween the solvability of problem (8)-(9) and the representation of the space U N as a set of solutions to an equation L N u = 0 . Whatever the definition of
Multidimensional Chebyshev sys-tems , we would like to retain the properties in Theorem 11 and Proposi-tion 17. In general, it would need in the future to make a proper refinementof these properties which would make then into two equivalent sets of condi-tions. However at the present moment we will restrict ourselves with somespecial though sufficiently wide generalization.We consider a subspace of functions U with U ⊂ C ∞ ( D ) for somebounded domain D ⊂ R n such that its boundary ∂D is infinitely smooth,and assume that D locally ”lies on one side of the boundary”. These are theusual conditions for the solvability of Elliptic Boundary Value problems, seee.g. [9]. For simplicity assume that D is connected and simply connected aswell.In the following definition we will mimic the properties of the Chebyshevsystems provided in Theorem 11. Definition 19
We will say that the elliptic operator P N of order N definedin the domain D, is factorizable if there exist N uniformly strongly elliptic perators Q ( j )2 of second order, defined in the domain D, and satisfying thefollowing properties:1. Every operator Q ( j )2 satisfies the maximum principle in D.
2. Every operator Q ( j )2 satisfies condition ( U ) s for uniqueness in theCauchy problem in the small.
3. The following equality holds P N = N Y j =1 Q ( j )2 = Q (1)2 Q (2)2 · · · Q ( N )2 . However in the next definition we will mimic the interpolation propertiesof the Chebyshev systems exposed by Proposition 17.
Definition 20
We say that the space U satisfies the Multivariate BVPInterpolation of order N iff the following conditions hold:1. The approximate solvability of BVP on subdomains holds inthe following sense: Let the ”boundary differential operators” B j ( x ; D x ) ,j = 1 , , ..., N, with smooth coefficients and of orders ≤ N, defined in D begiven. Let D be an arbitrary subdomain of D with D ⊂ D, and such that D satisfies the above conditions as D , and also D \ D has only non-compactconnected components. Let c j ∈ C ∞ ( ∂D ) for j = 0 , , ..., N − . Then forevery ε > there exists an element u ∈ U such that | B j u − c j ( x ) | ≤ ε for all x ∈ ∂D , and j = 1 , ..., N. (10) In the case of the whole domain, i.e. D = D, inequality (10) holds with ε = 0 .
2. The unique solvability of BVP on subdomains holds: If for some u ∈ U holds B j u ( x ) = 0 for all x ∈ ∂D , for j = 1 , ..., N, (11) then u ≡ . The differential operator P satisfies condition ( U ) s for uniqueness in the Cauchy prob-lem in the small in G provided that if G is a connected open subset of G and u ∈ C r ( G )is a solution to P ∗ u = 0 and u is zero on a non-emplty subset of G then u is identicallyzero. Elliptic operators with analytic coefficients satisfy this property (cf. [ ? ], part II, chapter 1 .
4; [3], p. 402). emark 21 In the case of the domain D we have unique solvability of theElliptic BVP since we may take ε = 0 ! Remark 22
For Theorem 23 below it is important to note that if an op-erator B j is non-characteristic in every direction then it is elliptic, see [9],(Definition . in chapter , section . ). We would like for a
Multidimensional Chebyshev space U to sat-isfy analogs to both Theorem 11 and Proposition 17. In this respect wemay prove the following theorem which shows that it is better to define the Multidimensional Chebyshev spaces by means of Definition 19 than bymeans of Definition 20.
Theorem 23
Let the elliptic operator P N in the domain D be factorizable by Definition 19. Then the space U N = (cid:8) u ∈ H N ( D ) : P N u = 0 in D (cid:9) satisfies the Multivariate BVP interpolation of Definition 20.
Proof.
We choose the following boundary operators B j = N Y i = N − j +2 Q ( i )2 for j ≥ , and in particular, B = id. The uniqueness (11) follows by induction in N, from item 1 in Definition19. Item 1 in Definition 20 follows inductively in N. Let us consider for sim-plicity the case N = 2 . Let u ∈ H ( D ) satisfy P u = Q (1) Q (2) u = 0 in D . Let us put w = Q (2) u By the ( U ) s property of the operator Q (1) , it follows that a Runge typetheorem holds, namely, for every ε > w ε ∈ H ( D )of the equation Q (1) w ε = 0 in D and k w ε − w k ≤ ε, (cf. [8], [10], [3]). Now we want to find a solution u ε ∈ H ( D ) such that Q (2) u ε = w ε in D and k u ε − u k D ≤ ε. But here we use the ( U ) s property of12he operator Q (2) since we compare Q (2) u = w and Q (2) u ε = w ε and we knowthat k w ε − w k ≤ ε. By the ( U ) s property as above we may find a solution u ε to the non-homogeneous equation which satisfies k u ε − u k D ≤ ε. This ends the proof.
Remark 24
An alternative reference for the Runge-Lax-Malgrange type the-orem is [4] (Theorem . . ), [12] ( Section . , Theorems . , . , and Theo-rem . for which it is mentioned there, that it was formulated by F. Browderwith error). For some integer N ≥ U := (cid:8) u : ∆ N u ( x ) = 0 in D (cid:9) . Proposition 25
The space U satisfies Definition 19. The proof follows directly after we define the operators B j = ∆ j − for j = 1 , , ..., N. We still have to check that the DT − systems defined in Definition 15 satsfythe multivariate counterpart in Definition 20. Proposition 26
Let us assume that in Definition 20 the space dimension is n = 1 . Then the set U coincides with a Dirichlet type Chebyshev system DT of order N from Definition 15. Proof.
Indeed, let us take the set D = [ a, b ] and apply the interpolationpropety to the case D = D. Then we know that for all constants c j and d j we have unique solvability of the problem u ( k ) ( a ) = c k for k = 0 , , ..., N − u ( k ) ( b ) = d k for k = 0 , , ..., N − . U is 2 N dimensional; we may take the solution v j to the problem u ( k ) ( a ) = δ j,k for k = 0 , , ..., N − u ( k ) ( b ) = 0 for k = 0 , , ..., N − w j to the problem u ( k ) ( a ) = 0 for k = 0 , , ..., N − u ( k ) ( b ) = δ j,k for k = 0 , , ..., N − U. The approximate solvability of the Dirichletproblem for D ⊂ [ a, b ] implies now the exact solvability since U is finite-dimensional. Indeed, we will take a sequence of solutions u ε ( t ) and the limit.
1. In the case of Multidimensional Chebyshev Systems we need the ap-proximate solvability of the Dirichlet problem since we have infinite-dimensional spaces, and there is no equivalence between the uniquenessand the existence but we have a substitute which is the Fredholm prop-erty of the regular Elliptic BVPs. The one-dimensional Proposition 17traces the smooth path for the multivariate generalization.
Acknowledgement 27
The present research has been partially sponsoredby project DO − − , . . with the Bulgarian NSF. References [1] M. Atakhodzaev, Ill-posed internal boundary value problems for thebiharmonic equation, VSP, 2002 . [2] L. Bers, F. John and M. Schechter, Partial Differential Equations, In-terscience, New York, 1964.[3] Felix Browder, Approximation by Solutions of Partial Differential Equa-tions, American Journal of Mathematics, vol. 84, no. 1, p. 134, 1962;Functional analysis and partial differential equations. II, Mathematis-che Annalen, vol. 145, no. 2, pp. 81-226, 1962.144] Hormander, L., The Analysis of Linear Partial Differential Operators I,Springer-Verlag, 1990 . [5] Krein, M. , Nudelman A., The Markov Moment problem and Extremalproblems, AMS translation from the Russian edition of 1973 . [6] S. Karlin, W. Studden, Tchebycheff Systems: with applications in anal-ysis and statistics, Intersci. Publ., 1968 . [7] Kounchev, O., Multivariate Polysplines, Academic Press, San Diego,2001 . [8] P. Lax, A stability theorem for solutions of abstract differential equationsand its application to the study of local behavior of solutions of ellipticequations, Comm. Pure Appl. Math. 9 (1956) 747–766.[9] J.-L. Lions and E. Magenes, Non-homogeneour Boundary Value Prob-lems and Applications, Springer, Berlin-Heidelberg, 1970 . [10] B. Malgrange, Existence et approximation des solutions des equationsaux derivees partielles et des equations de convolution, Ann. Inst.Fourier (Grenoble) 6 (1955–1956) 271–355.[11] L. Schumaker, Spline Functions: basic theory , Academic Press, NY,1983 ..