# Periodic polynomial spline histopolation.

1. INTRODUCTION

The given histopolation problem may in general be reduced to an equivalent interpolation problem and the derivative of the interpolant is the histopolant. On the contrary, a certain integral of the histopolant is the solution of a corresponding interpolation problem. This correspondence keeps the periodicity only in one direction, namely, the derivative of a periodic interpolant is periodic but not vice versa. This means that, at periodic histopolation, some problems like, e.g., convergence or error estimates cannot be reduced to similar problems at periodic interpolation. Fortunately, when asking about the existence and uniqueness of the solution in spline spaces we are successful because the uniqueness problem could be solved for the corresponding homogeneous problems in finite dimensional spaces and the periodicity would be preserved in both directions. The existence and uniqueness of the solution at periodic polynomial spline histopolation is the main problem in this paper. Several cases are treated and the reader can see that different tools are needed in the proofs of assertions.

2. THE HISTOPOLATION PROBLEM FOR PERIODICITY

For a given grid [[DELTA].sub.n] of points a = [x.sub.0] <[x.sub.1] <...<[x.sub.n] = b define the spline space

[mathematical expression not reproducible] (the set of all polynomials ofdegreeatmostm)fori=1,... ,n, S [member of][C.sup.m-1][a,b]}.

It is known that dim [X.sub.m]([[DELTA].sub.n]) = n + m. The space [X.sub.p,m]([[DELTA].sub.n]) of periodic splines is

[mathematical expression not reproducible].

Then dim [X.sub.p,m]([[DELTA].sub.n]) = n and this could be shown, e.g., in following way.

Lemma 1. Let X be a vector space with dim X = n and [[phi].sub.i],i= 1,... ,k, be linear functionals defined on X that are linearly independent. Then dim[mathematical expression not reproducible].

Touse this result we take functionals [[phi].sub.i](S) = [S.sup.(i)](b)-[S.sup.(i)](a),i = 0,... ,m-1, defined on [X.sub.m]([[DELTA].sub.n]). Their linear independence could be verified on polynomials [p.sub.i](x) =[x.sup.i+1],i = 0,... ,m-1.

Denote the sizes of the intervals [h.sub.i] = [x.sub.i] - [x.sub.i-1], i = 1,... , n. In the periodic histopolation problem we have to find S [member of] [X.sub.p,m]([[DELTA].sub.n]) such that

[mathematical expression not reproducible] (1)

for given numbers [z.sub.i]. Conditions (1) are called histopolation conditions.

Our main task in this paper is to answer the question: When has the formulated periodic histopolation problem aunique solution for any given values [z.sub.i], i = 1,... ,n?

As our problem is linear, this question could be reformulated equivalently as follows: When the corresponding homogeneous problem has only trivial solution, i.e., when

[mathematical expression not reproducible]

3. EXISTENCE AND UNIQUENESS

In this section we first indicate the cases where the solution exists and is unique.

Proposition 2. For m even the periodic histopolation problem has a unique solution.

Proof. Let m = 2k. Consider in [X.sub.m]([[DELTA].sub.n]) the seminorm [mathematical expression not reproducible]. Suppose that S [member of] [X.sub.p,m]([[DELTA].sub.n]). Then we get by using integration by parts and periodicity properties of the spline S

[mathematical expression not reproducible]

Let now, in addition, [mathematical expression not reproducible]. Then ||S|| = 0 or S [member of] [P.sub.k-1]. It is immediate to check that a periodic polynomial S is constant. The homogeneous histopolation conditions then yield S = 0, which completes the proof. []

Recall that the sign change zero of a function f is a number z such that f(z) = 0 and there exists [[epsilon].sub.0] > 0 such that f(z - [epsilon])f(z + [epsilon]) < 0 for all [epsilon] [member of] (0, [[epsilon].sub.0]). If S [member of] [X.sub.m]([[DELTA].sub.n]), then let Z(S) be the number of sign change zeros of S in the interval [[x.sub.0],[x.sub.n]). In the case m = 0 we talk here about sign change point z requiring only f(z - [epsilon]) f (z + [epsilon]) < 0 for all [epsilon] [member of] (0,[[epsilon].sub.0]).

Lemma 3 (see, e.g., ). For S [member of] [X.sub.p,m]([[DELTA].sub.n]) it holds

[mathematical expression not reproducible].

Lemma4. If S [mathematical expression not reproducible], and S(x)=0, where x [member of][ [x.sub.i-1],[x.sub.i]] for some i, then S(x)=0,x [member of] [a,b].

Proof If S(x) = 0,x [member of] [[x.sub.i-1],[x.sub.i]], then for x [member of] [[x.sub.i],[x.sub.i+1]] use the Taylor expansion

[mathematical expression not reproducible]

[mathematical expression not reproducible], it holds [S.sup.(m)]([x.sub.i] + 0) = 0 and S (x) =0, x[member of] [[x.sub.i],[x.sub.i+1]].

We may continue going from [x.sub.i+1] to the right or similarly from [x.sub.i-1] to the left and establish S(x) = 0, x [member of] [a,b]. []

Proposition 5. For m odd and n odd the periodic histopolation problem has a unique solution.

Proof Let S [member of] [X.sub.p,m]([[DELTA].sub.n]) and [mathematical expression not reproducible]. If S(x)=0, x [member of] [[x.sub.i-1],[x.sub.i]], then by Lemma 4 it holds , S = 0, whichis already a contradiction. The condition S(x) [greater than or equal to] 0 for all x [member of] [[x.sub.i-1],[x.sub.i]] and S([xi]) > 0 for some [xi] [member of] [[x.sub.i-1],[x.sub.i]] gives [mathematical expression not reproducible], which is not the case. Similarly, S(x) [less than or equal to] 0 for all x [member of] [[x.sub.i-1],[x.sub.i]] and S([xi]) < 0 for some [xi] [member of] [[x.sub.i-1],[x.sub.i]] do not take place. Thus, there are sign change zeros [[eta].sub.i] [member of] ([x.sub.i-1],[x.sub.i]), i = 1,... ,n, of S and Z(S) [greater than or equal to] n. But by Lemma 3 it holds Z(S) [less than or equal to]n-1, which is a contradiction. This means that the homogeneous problem has only a trivial solution. []

Let us remark that the proof of Proposition 5 is valid for arbitrary m [greater than or equal to] 1 and n odd.

Proposition 6. For m = 1 and n even the homogeneous periodic histopolation problem has a non-trivial solution.

Proof. Take [[eta].sub.i] = ([x.sub.i-1] +[x.sub.i])/2, i=1,... ,n. Let c [not equal to] 0. Consider the function S(x) = [c.sub.i] ( x - [[eta].sub.i]),x [member of] [[x.sub.i-1],[x.sub.i]],i=1,... ,n. It holds [mathematical expression not reproducible], for any choice of numbers [c.sub.i]. The choice of [c.sub.i] = (-2c)/[h.sub.i] for i = 1,3,... and [c.sub.i] = (2c)/[h.sub.i]for i = 2,4,... ensures that S [member of] [X.sub.p,1]([[DELTA].sub.n]), S [not equal to] 0, with

S([x.sub.0])=S([x.sub.2]) = ...=S(xn)=c

and

S([x.sub.1])=S([x.sub.3]) = ...=S([x.sub.n-1]) = -c. []

Proposition 7. For m odd and n = 2 the homogeneous periodic histopolation problem has a non-trivial solution.

Proof. For m = 1 the assertion is already proved by Proposition 6. We prove the general case by induction. Denote [[eta].sub.i] = ([x.sub.i-1]+[x.sub.i])/2, i = 1,2. Let m = 2k-1 and S [member of] [X.sub.p,m]([[DELTA].sub.2]) be such that S [not equal to] 0 and

[mathematical expression not reproducible]. (2)

Clearly, this holds for the spline S from the proof of Proposition 6 in the case m = 1. Define

[mathematical expression not reproducible]

or

[mathematical expression not reproducible]. (3)

Then [mathematical expression not reproducible] and (3) implies that, for any numbers [c.sub.0,i],

[S.sub.1]([x.sub.i-1]+0) = [S.sub.1]([x.sub.i]-0), i=1,2. (4)

If [c.sub.0,1] and [c.sub.0,2] are such that

[S.sub.1]([x.sub.1]-0) = [S.sub.1]([x.sub.1]+0), (5)

then [S.sub.1] [member of] [X.sub.p,m+1]([[DELTA].sub.2]). Next, define S by

[mathematical expression not reproducible]

or

[mathematical expression not reproducible]

WeseethatShastheform(2), S' = [S.sub.1] ,and

S([x.sub.i-1]+0) =-S([x.sub.i]-0), i = 1,2. (6)

If, in addition to (5), we have

S([x.sub.1]-0) = S([x.sub.1+0]), (7)

then S [member of] [X.sub.p,m+2]([[DELTA].sub.2])dueto(4)-(7).

It remains to show that by (5) and (7) we can determine suitable numbers [c.sub.0,1] and [c.sub.0,2]. Equation (5) is, in fact,

[mathematical expression not reproducible]

and (7) is

[mathematical expression not reproducible]

But this system has the non-zero determinant ([h.sub.1] + [h.sub.2])/2. However, as S [not equal to] 0 then S [not equal to] 0. []

We say that the grid [x.sub.0] < [x.sub.1] <... < [x.sub.n] is pairwise uniform if n is even and for any i even it holds that [x.sub.i+1] -[x.sub.i] = h, [x.sub.i+2]-[x.sub.i+1] = [h.sub.2] or [x.sub.i+1] -[x.sub.i] = [h.sub.2], [x.sub.i+2]-[x.sub.i+1] = [h.sub.1].

Corollary 8. The homogeneous periodic histopolation problem has a non-trivial solution for m odd and the pairwise uniform grid.

In particular, the case of the uniform grid for m odd and n even is included in Corollary 8. This result could be found in [10,13].

In general, we state as an open problem the following.

Conjecture. For m odd and n even the homogeneous periodic histopolation problem has a non-trivial solution.

Define the subspace of [X.sub.p,m]([[DELTA].sub.n]) as

[mathematical expression not reproducible]

For m odd and n even it may be that [X.sub.0,p,m]([[DELTA].sub.n]) [not equal to] {0} (if the Conjecture is true, then always). It is natural to ask what dim [X.sub.0,p,m]([[DELTA].sub.n]) is in this case.

Remove from the grid [[DELTA].sub.n] : a = [x.sub.0] <[x.sub.1] <... <[x.sub.n] = b a knot [x.sub.i]. We get the grid [mathematical expression not reproducible] with the number of subintervals n--1, which is odd. By Proposition 5 it holds [mathematical expression not reproducible]. Clearly, [mathematical expression not reproducible]. The sum [mathematical expression not reproducible] is a direct sum that follows from the relation

[mathematical expression not reproducible].

Thus, the equality

[mathematical expression not reproducible]

implies that dim [X.sub.0,p,m]([[DELTA].sub.n]) = 1.

The obtained results about the existence of non-trivial solutions for the homogeneous problem yield the following.

Theorem 9. For m even or m and n odd the periodic histopolation problem has for each [z.sub.i],i=1,... ,n, the unique solution. For m odd and n even there may exist (if the Conjecture is true, then always exist) [z.sub.i],i = 1,... ,n, such that the periodic histopolation problem does not have a solution.

4. BIBLIOGRAPHICAL NOTES

In this section we acquaint the reader with a subjective list of works on periodic spline interpolation and histopolation. The results about the existence and uniqueness of a solution for periodic polynomial spline interpolation could be found in . A short overview of existence results by several authors are presented in , which contains also convergence estimates for problems on a uniform grid with interpolation knots not necessarily in grid points. The paper  contains results about properties of periodic interpolating polynomial splines on subintervals. The existence and uniqueness results of periodic solutions for the uniform grid case in several papers are based on the theory of circulant matrices, see, e.g. [4,5]. The general non-uniform grid is considered in  for low degree periodic splines with convergence estimates. The work  gives error estimates for the periodic quadratic spline interpolation problem arising from the histopolation problem with these splines. In  the existence and uniqueness problem of solution in periodic quartic polynomial spline histopolation (m = 4) is stated generally but solved only for the uniform grid. Unlike in the other studies, the spline representation via moments is used. Our Proposition 2 gives here the answer for the general grid case. The periodic interpolation problem on a uniform grid with certain non-polynomial functions is studied in , and histopolation in . Interpolation with periodic polynomial splines of the defect greater than minimal is studied in [11,14,15]. Cubic spline histopolation on a general grid is treated in  from several aspects, including methods of the practical construction of the histopolant.

ACKNOWLEDGEMENTS

This work was supported by institutional research funding IUT20-57 of the Estonian Ministry of Education and Research. We are thankful to the referee for careful reading of the manuscript. The publication costs of this article were covered by the Estonian Academy of Sciences.

REFERENCES

[1.] Ahlberg, J. H., Nilson, E. N., and Walsh, J. L. The Theory of Splines and Their Applications. Academic Press, New York--London, 1967.

[2.] Delvos, F-J. Periodic area matching interpolation. In Numerical Methods of Approximation Theory, Vol. 8. Oberwolfach, 1986; Internationale Schriftenreihe zur Numerischen Mathematik, Vol 81. Birkhauser, Basel, 1987, 54-66.

[3.] Delvos, F-J. Periodic interpolation on uniform meshes. J. Approx. Theory, 1989, 51, 71-80.

[4.] Dubeau, F. On band circulant matrices in the periodic spline interpolation theory. Linear Algebra and its Applications, 1985, 72, 177-182.

[5.] Dubeau, F. and Savoie, J. On circulant matrices for certain periodic spline and histospline projections. Bull. Austral. Math. Soc., 1987, 36, 49-59.

[6.] Dubeau, F. and Savoie, J. De l'interpolation a l'aide d'une fonction spline definie sur une partition quelconque. Ann. Sci. Math. Quebec, 1992, 16, 25-33.

[7.] Kirsiaed, E., Oja, P., and Shah, G. W. Cubic spline histopolation. Math. Model. Anal., 2017, 22, 514-527.

[8.] Kobza, J. and Zencak, P. Some algorithms for quartic smoothing splines. Acta Univ. Palacki. Olomuc, Mathematica, 1997, 36, 79-94.

[9.] Meinardus, G. and Merz, G. Zur periodischen Spline-Interpolation. In Spline-Funktionen. Bibliographisches Institut, Mannheim, 1974, 177-195.

[10.] Ter Morsche, H. On the existence and convergence of interpolating periodic spline functions of arbitrary degree. In Spline-Funktionen. Bibliographisches Institut Mannheim, 1974, 197-214.

[11.] Plonka, G. Periodic spline interpolation with shifted nodes. J. Approx. Theory, 1994, 76, 1-20.

[12.] Rana, S. S. Quadratic spline interpolation. J. Approx. Theory, 1989, 57, 300-305.

[13.] Schumaker, L. Spline Functions: Basic Theory. Wiley, NY, 1981.

[14.] Szyszka, U. Periodic spline interpolation on uniform meshes. Math. Nachr., 1991, 153, 109-121.

[15.] Zeilfelder, F. Hermite interpolation by periodic splines with equidistant knots. Commun. Appl. Anal., 1998, 2, 183-195.

Peeter Oja (*) and Gul Wali Shah

Institute of Mathematics and Statistics, University of Tartu, J. Liivi 2, 50409 Tartu, Estonia; gulwali@ut.ee

Received 19 September 2017, accepted 11 December 2017, available online 28 June 2018

(*) Corresponding author, peeter.oja@ut.ee

Perioodiliste polunomiaalsete splainidega histopoleerimine

Peeter Oja ja Gul Wali Shah

Antud vorgul a = [x.sub.0] < [x.sub.1] <... < [x.sub.n] = b vaadeldakse polunomiaalseid splaine S, mis on igas osaloigus [[x.sub.i-1],[x.sub.i]], i = 1,... ,n, ulimalt m astme polunoomid ja kuuluvad ruumi [C.sup.m-1][a,b]. Splain S on perioodiline, kui [S.sup.(j)](a)=[S.sup.(j)](b), j = 0,... ,m-1. Histopoleerimisulesandes noutakse, et [mathematical expression not reproducible], kus zi on antud arvud. Artiklis on pohiprobleemiks vastuse otsimine kusimusele, millal selline histopoleerimisulesanne on igasuguste arvude [z.sub.i] korral uheselt lahenduv. Probleemile oli lahendus varem teada uhtlase vorgu korral, kus solmed on voetud [x.sub.i] = a + ih,i = 0,... ,n,h = (b - a)/n. Selles artiklis naitame suvalise vorgu korral, et lahend on uhene, kui m on paaris, samuti on lahend uhene, kui m ja n on molemad paaritud. Toestame, et lahend ei ole uhene ehk vastaval homogeensel ulesandel on mittetriviaalne lahend, kui m = 1 ja n on paaris, samuti kui m on paaritu ning n = 2. Uldine juht, kus m on paaritu ja n paaris, naib olevat raske probleem ning selle oletatav lahendus on meil sonastatud hupoteesina. On tahelepanuvaarne, et kui uhtlase vorgu korral on voimalik anda koikide juhtude jaoks uhtse meetodiga vastus, siis uldise vorgu korral kasutame erinevate juhtude puhul lahenduse saamiseks erinevaid toestusvotteid.

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