Let be the set of all continuous complex-valued functions defined on a subset in For an open subset let be the set of all functions which have continuous partial derivatives of order on , and for , the closure of , we denote by the set of all functions which have continuous partial derivatives of order on
Recall that a spline of degree defined on an interval with nodes is a function which on each subinterval is identical with some polynomial of degree for If we call a cubic spline. For we obtain the definition of a linear spline. Note that is a linear spline if and only if a continuous function on such that on each subinterval the function is linear, so a solution of the differential operator
The aim of the present paper is to provide error estimates for interpolation by special types of multivariate splines, namely harmonic and biharmonic splines. Harmonic splines occur in a natural fashion in mathematical problems, see e.g. the discussion in . Harmonic splines for block partitions have been discussed by various authors in the literature, see , , , . Here a set of the form is called a block in , and a harmonic spline is continuous function on which is harmonic on open and disjoint subdomains (called subblocks) of for such that the closure of is equal to Recall that a function is called harmonic if and
and is the Laplace operator. It is convenient to introduce the following natural generalization: we say that a function is a harmonic spline with respect to the partition if for
and is a harmonic function on each subdomain for Thus a harmonic spline for a partition is a multivariate generalization of a linear spline.
For the partition we define harmonic spline interpolation in the following way: Given a function , we say is a harmonic spline interpolating the data function if is a harmonic spline for and it satisfies the interpolation condition
Note that we require in (3) an interpolation condition for infinitely many points, and in the literature this is often called transfinite interpolation. Transfinite interpolation was already considered by Gordon and Hall in  (cf. ) and this concept has found many applications in mesh generation, geometric modelling, finite element methods and spline analysis, see , , , , .
The existence of a harmonic spline interpolant is easy to establish when we assume that the Dirichlet problem is solvable111The Dirichlet problem is solvable for a domain if for each continuous function defined on the boundary of there exists a continuous function defined on the closure of which is harmonic in and interpolates on the boundary, i.e. for all for each domain for The uniqueness of the harmonic interpolant is a simple consequence of the maximum principle. The main results in , , ,  are optimal estimates for the error
for a given twice differentiable function in the case of block partitions with respect to the supremum norm defined by
for . In the present paper we shall present an explicit error estimate for harmonic splines for a partition given by annular subdomains of the form
for given positive radii . Indeed, we shall proof the following:
Let be positive numbers, and let Assume that is the harmonic spline interpolating for all with and Then
where is the dimension of the space and
Note that this result is very similar to the error estimate for linear splines (see e.g. [13, p. 31]): Assume that is the (unique) linear spline interpolating a twice differentiable function at the points Then
The most difficult part in Theorem 1 is to establish the explicit nature of the constant in the estimate (4). Let us also emphasize that Theorem 1 is an important ingredient to establish an error -estimate for interpolation with biharmonic splines – the next topic we want to discuss.
The definition of a harmonic spline can be traced back in old sources (see.g. ), and the concept is very intuitive. For the definition of a biharmonic spline we use the approach given in  where the explicit description as a piecewise polyharmonic function is used and which emphasizes the analogy to the case of univariate cubic splines. For our purposes it is convenient to use the following definition:
Let be open disjoint sets in and define A function is a biharmonic spline for the partition if and the restriction of to each is biharmonic, i.e. and
for all and for .
In the definition of a biharmonic spline the matching of the boundary behaviour of the biharmonic functions defined on is simply expressed by the requirement that is a -function on . This corresponds to the definition of a cubic spline on the interval with nodes : it is a function which on each subinterval is identical with a solution of the differential equation for
The existence of an interpolating biharmonic spline requires additional assumptions on the smoothness of the domains and higher regularity of the data function on the boundary of which might be expressed in terms of Hölder or Sobolev spaces. On the other hand, we are interested only in error estimates, so at this place we do not need to dwell in the more difficult question of the existence of an interpolation biharmonic spline for a partition (see also Section 4 for some comments in the case of annuli, and for the details on the existence of interpolation polysplines consult the monograph , chapter ).
Our main result is the following error estimate where we recall that the -norm of a measurable function is defined by
Let and let . Assume that is a biharmonic spline for the partition which satisfies the (transfinite) interpolation conditions
and the boundary conditions for the normal derivative
Then, with as above, the following estimate holds:
The above result should be compared with the -error estimate of a one-dimensional cubic spline: Assume that and let be a cubic spline interpolating at the points and satisfying the additional boundary condition
see e.g. . Let us also mention that for the supremum norm the following error estimate
holds, see [13, p. 55]. We leave the question open whether in Theorem 3 one may replace the -norm by the supremum norm. In passing we mention that in  the inequality (9) has been generalized to -splines where is a differential operator with constant coefficients of order
Let us briefly describe the structure of the paper: in Section 2 we shall discuss error estimate for harmonic interpolation splines with respect to a general partition . This problem is closely related to the problem of finding the smallest constant among all constants which satisfy the inequality
for all vanishing on the boundary In Section 2 we will characterize the constant :
Let be a bounded regular domain and let be the solution of the Dirichlet problem for the data function Then
The function is the unique function which vanishes on the boundary of and satisfies
The function plays an eminent role in various area of mathematics and is called the torsion function, see e.g. the fundamental work of G. Pólya, G. Szegö about isoperimetric inequalities in , or the monography . There is a vast literature on this subject with many ramifications and it would take too much space to survey the results, so we only mention a very incomplete list of new references , , , , . In Section 2 we shall provide a self-contained proof of Theorem 4 which is based on a Green function approach.
In Section 3 we provide the proof of Theorem 3. It is remarkable that the -error estimate in the biharmonic case can be performed by an iterative argument where the -error estimate for harmonic splines is used twice.
In Section 4 we present a proof of an orthogonality relation for biharmonic interpolation splines which is used in Section 3.
In Section 5 we provide a computation of the best constant for the annular domain and we prove the following inequalities:
Finally, let us mention that some of the presented concepts can be generalized. Recall that a function is called polyharmonic of order if and
where is the -th iterate of see , , . Polyharmonic functions are often used in applied mathematics, see e.g. , , , , , , , . Slightly more general than in  we define a function to be polyspline of order for a partition if for and for all and for . Cardinal polysplines of order on strips or annuli have been discussed by the first two authors in a series of papers , , .
In this paper we have dealt with transfinite interpolation and it might be of interest to compare our results with the thin plate splines of order (in introduced by J. Duchon in  for the interpolation at a finite number of scattered points . Thin plate splines are polyharmonic functions of order on the set since they are a finite linear combination of translates of the fundamental solution of in In contrast to a polyspline a thin plate spline is only a function in which is the reason for the requirement By definition, an interpolating thin plate spline is defined as the unique minimizer of the integral functional
among all functions having all partial derivative of total order in and interpolating the data. Here we used multi-index notation with and and . In 
one can find the error estimates which served as model example for error estimates for interpolation with radial basis functions, see e.g.,  , . Unlike our results, in these references the constants for the error estimates are not explicit, and only Sobolev norms are used; only in two dimensions some explicit constants are found, cf. , . It is our expectation that the error estimates for polysplines can be used to improve the error estimates for thin plates for data which is structured along curves, a subject we hope to address in a future paper.
2 Harmonic interpolation splines
We say that an open set is regular in if each boundary point is regular, see [1, p. 179] for definition, and [1, Theorem 6.5.5] for a characterization. It is known that for a regular bounded domain the Dirichlet problem is solvable, see [1, Theorem 6.5.5 and 6.5.4].
At first we discuss the error estimate for harmonic interpolation splines:
Assume that are pairwise disjoint bounded regular domains and define If and is the harmonic spline interpolating on then
where is the smallest constant such that (10) holds for all function vanishing on the boundary for .
Proof. We consider Then for all Further since is harmonic on and continuous on Hence for
The statement is now obvious.
A fundamental theorem in potential theory states that for any open set in with or for any bounded open set in the Green function exists, see [1, p. 90]. Further we denote by the volume of the unit ball and define
If is a bounded domain in then
Equality holds when is a regular domain.
Proof. Assume that vanishes on the boundary If is unbounded on the inequality (10) is trivial. If is bounded it follows that
for some Then it can be shown that the following representation formula holds
for all , and all which vanish on the boundary Clearly (15) implies that
for any The first result follows since is the smallest number satisfying (10).
Now assume that is regular, so the Dirichlet problem is solvable for . Then there exists a harmonic function in such that for all Clearly the function
is in and vanishes on Further Since is the smallest constant satisfying (10) we infer that
The representation formula (15) shows that
The proof is complete.
Next we turn to -estimates of harmonic splines:
Let be a bounded domain in . Assume that vanishes on the boundary and that is bounded. Then for any and its conjugate exponent defined by we have
Proof. The representation formula (15) and the Hölder inequality show that
Write take the -th power on both sides and integrate with respect to then
Then Fubini’s theorem shows that
Further we see that
Now take the -th square root in (21) and we arrive at
We apply now the results to the case of annular domains.
Let be real numbers, and let Assume that is a harmonic spline interpolating for all with and then for the supremum norm the estimate
holds, and for the -norm
Now apply Theorem 7 for to the domain and we have
By summing up over and taking the square root we arrive at the second statement.
3 Biharmonic interpolation splines
In this section we want to provide the error estimate for biharmonic interpolation splines on annular domains, see Theorem 3 in the introduction. Note that a simple consequence of the error estimate is the uniqueness of the biharmonic interpolation spline for a given data function
We emphasize that the assumption for the data function, namely in Theorem 3 is weaker than the usual assumption for proving the existence of a biharmonic interpolation spline. Indeed, the first author has proved the existence of a biharmonic spline interpolating a function in [31, p. 446–453] with Dirichlet boundary conditions (25) using a priori estimates for elliptic boundary values problems. For dimension it is required that the data functions are from the fractional Sobolev space . For dimension , A. Bejancu has shown in  the existence of a biharmonic spline interpolating a function with Beppo-Levi boundary conditions and a data function in the weighted Wiener algebra (which is less restrictive) using Fourier series techniques, see also .
Our approach to the proof for the error estimate of biharmonic interpolation splines is inspired by the exposition of Carl de Boor in  for the error estimate of cubic interpolation splines which is deduced via error estimates with piecewise linear functions on in an iterative way. The main observation in the intermediate step of the proof in  is that the second derivative of interpolation cubic spline is the best approximation to the second derivative of the interpolated function. This fact depends on an orthogonality relation between the error and linear splines. We shall use a similar argument in our context of biharmonic and harmonic splines and for this we use the following well known fact from best approximation in Hilbert spaces. For convenience of the reader we include the short proof:
Let be a subspace of a Hilbert space and . Assume that has the property that
Then for all
Proof. Put Due to the orthogonality condition (24) we have
It follows that Since is a subspace we can replace by , and the proof is finished.
Let us recall the main result of the paper stated in the introduction:
Let and let . Assume that is a biharmonic spline for the partition which satisfies the transfinite interpolation conditions
Then the following error estimate holds:
Proof. At first we note that
We apply (20) to the function on the annular domain : note that vanishes for any with for hence
By summing up and taking the square root we see that
Now we want to estimate the right hand side: let us put and Note that is a harmonic spline – but unfortunately it does not interpolate the function so we cannot repeat the error estimate for interpolating harmonic splines. In Theorem 11 below we prove that the equality
holds for all harmonic splines for the partition . Then Proposition 9 implies that
holds for all harmonic splines for the partition Let us take the harmonic spline Hence,
is the error for the harmonic spline interpolating and by Theorem 8 we obtain
It follows that
This ends the proof.
4 Proof of the orthogonality relation
In this section we want to prove the following result:
Let and let and as in Theorem 3. Then for all harmonic splines for the partition for
Let us write . Then and Clearly we have
A natural approach is to prove (26) by appling Green’s formula (see [1, p. 307]) to each summand on the right hand side. In order to apply Green’s formula, let us take in the interval . Then and are twice differentiable in a neighborhood of and we apply Green’s formula. Since is harmonic we obtain that
where is defined by
and denotes the exterior normal derivative, and is the surface measure on the sphere for Now we want to take limits and in (27). The limit for the left hand side clearly exists, so