Periodic polynomial spline histopolation

Periodic polynomial spline histopolation with arbitrary placement of histogram knots is studied. Spline knots are considered coinciding with histogram knots. The main problem is the existence and uniqueness of the histopolant for any degree of spline and for any number of partition points. The results for arbitrary grid give as particular cases known assertions for the uniform grid but different techniques are used.


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.

THE HISTOPOLATION PROBLEM FOR PERIODICITY
For a given grid ∆ n of points a = x 0 < x 1 < . . . < x n = b define the spline space It is known that dim X m (∆ n ) = n + m. The space X p,m (∆ n ) of periodic splines is Then dim X p,m (∆ 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 ϕ i , i = 1, . . . , k, be linear functionals defined on X that are linearly independent. Then dim(∩ k i=1 ker ϕ i ) = n − k.
To use this result we take functionals Denote the sizes of the intervals In the periodic histopolation problem we have to find S ∈ X p,m (∆ n ) such that for given numbers z 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 a unique solution for any given values z 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

EXISTENCE AND UNIQUENESS
In this section we first indicate the cases where the solution exists and is unique.
Then we get by using integration by parts and periodicity properties of the spline S Let now, in addition, 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 ε 0 > 0 such that f (z − ε) f (z + ε) < 0 for all ε ∈ (0, ε 0 ). If S ∈ X m (∆ n ), then let Z(S) be the number of sign change zeros of S in the interval [x 0 , x n ). In the case m = 0 we talk here about sign change point z requiring only Lemma 3 (see, e.g., [13]). For S ∈ X p,m (∆ n ) it holds This holds for all m ∈ N ∪ {0}.
We may continue going from x i+1 to the right or similarly from x i−1 to the left and establish S( Proposition 5. For m odd and n odd the periodic histopolation problem has a unique solution. Proof. Let S ∈ X p,m (∆ n ) and  Proof. Proof. For m = 1 the assertion is already proved by Proposition 6. We prove the general case by induction. Denote η i = (x i−1 + x i )/2, i = 1, 2. Let m = 2k − 1 and S ∈ X p,m (∆ 2 ) be such that S ̸ = 0 and Clearly, this holds for the spline S from the proof of Proposition 6 in the case m = 1. Define Then S ′ 1 = S and (3) implies that, for any numbers c 0,i , If c 0,1 and c 0,2 are such that then S 1 ∈ X p,m+1 (∆ 2 ). Next, defineS byS We see thatS has the form (2),S ′ = S 1 , and If, in addition to (5), we haveS thenS ∈ X p,m+2 (∆ 2 ) due to (4)- ( 7). It remains to show that by (5) and (7) we can determine suitable numbers c 0,1 and c 0,2 . Equation (5) is, in fact, and (7) is .
We say that the grid x 0 < x 1 < . . . < x n is pairwise uniform if n is even and for any i even it holds that 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 p,m (∆ n ) as For m odd and n even it may be that X 0,p,m (∆ n ) ̸ = {0} (if the Conjecture is true, then always). It is natural to ask what dim X 0,p,m (∆ n ) is in this case. Remove from the grid ∆ n : The obtained results about the existence of non-trivial solutions for the homogeneous problem yield the following.

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 [1]. A short overview of existence results by several authors are presented in [10], which contains also convergence estimates for problems on a uniform grid with interpolation knots not necessarily in grid points. The paper [9] 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 [6] for low degree periodic splines with convergence estimates. The work [12] gives error estimates for the periodic quadratic spline interpolation problem arising from the histopolation problem with these splines. In [8] 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 [3], and histopolation in [2]. 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 [7] from several aspects, including methods of the practical construction of the histopolant.