The interpolation of periodic functions at equidistant nodes by trigonometric polynomials is a basic task of approximation theory with far-reaching applications (see, e.g., Chapter 3 in Plonka et al. [2]). The possibility of using FFT algorithms with huge amounts of data has contributed greatly to the popularity of this approximation method. Accordingly, error estimates for such interpolation methods have been intensively studied in the literature. The decisive difference between approximation methods which are based on integral evaluations of the given function f, for example, the Fourier coefficients, and an interpolation method is that information about f must really be available pointwise. This difference becomes particularly important in the case of interpolation of discontinuous functions, where one will focus on the error in L_p norms in particular. As is well-known, the Riemann integrability of a periodic function f is a condition for the L_p error to tend to 0 as the number of nodes n → ∞ (cf. [3]). For a little more smoothness, the approximation order in L_p can be bounded by the best one-sided approximation in L_p using trigonometric polynomials (cf. [4]).

A particularly important class of functions, generally discontinuous functions, for which one would like to obtain error estimates are functions of bounded variation. A first result in this area comes from Zacharias, who proved in [5] with Hilbert space methods that the L₂ error behaves like 1/n. This result was generalized to 1 ≤ p < ∞ in Prestin [6].

To generalize these error estimates to multivariate periodic functions, a suitable concept for multivariate bounded variation is required. The Hardy-Krause definition is appropriate here (see Clarkson and Adams [7] and for more information on these spaces [8], [9] and others). For the dimension d = 2 and interpolation on the tensor product, such results can be found in Prestin and Tasche [10], (see also Kolomoitsev et al. [11]). An essential tool for the proof of the error estimates is the consideration of blending operators, which have been extensively analyzed in the study of Delvos et al. (cf., e.g., [12–15]).

In this study, the results for the approximation error of functions of bounded variation are to be transferred to interpolation methods on sparse grids. Such grids were first introduced in Smolyak [16] and since then have been widely used in interpolation problems, quadrature schemes, and other fields. For more details, see Dũng et al. [17]. These sparse grids are very efficient, especially for large spatial dimensions d, that is, the approximation order is only reduced by a logarithmic factor compared to the tensor product interpolation, although the number of interpolation nodes is only by a log factor bigger than in the univariate case. At this point, it should be noted that error estimates for such interpolation methods of continuous functions are known (see Dũng et al. [17, Chap. 5.3]). Such statements are proved for functions belonging to the spaces Hpr, where r>1/p is assumed, which implies the continuity of the function to be interpolated. Our larger class of functions of bounded variation then provides an order of convergence as in the case r = 1/p. Our approach requires a notation for the definition of bounded variation that is well-suited for large dimensions d. Here, we follow the approach in Aistleitner et al. [1].

Finally, we note that these approximation results for functions of bounded variation are also valid for Fourier sums and the corresponding multivariate hyperbolic cross-variants, where the results can also be obtained using other methods.

Let p∈[1, ∞), d∈ℕ. For 2π-periodic functions f of d variables on the torus 𝕋^d, we consider the space Lp(Td), 1 ≤ p < ∞, supplied by the following norm:

We denote by D = {1, …, d} the set of coordinates with cardinality |D| = d and split it into two domains B⊂D and B¯=D\B, |B|+|B¯|=d. Following Aistleitner et al. [1] by z = y^B:x, where y, x∈𝕋^d, we describe the vector z∈𝕋^d consisting of the components z^j = y^j if j∈B and z^j = x^j otherwise. Such a partition will also be used to represent the vector z∈𝕋^d as a combination of arguments from B and fixed values along coordinates from B̄.

For each coordinate j = 1, ..., d we introduce some arbitrary decomposition Zj, namely

Let ξ=(ξk11,ξk22,…,ξkdd)∈Td be a vector with components ξkjj∈Zj,kj=1,…,uj and ξ+=((ξk11)+,(ξk22)+,…,(ξkdd)+)∈Td, where

Using this notation for a function f:𝕋^d → ℂ, we introduce a d-dimensional difference operator in the following way:

Furthermore, we consider the difference operator and corresponding variation for f:𝕋^d → ℂ with respect to coordinates j∈B and fixed values z^j for j∈B¯:

Then, we define for all B⊆D:

In particular, V^∅f(z) = f(z).

For a function VBf(zB¯)∈Lp(Td-|B|), we have

for 1 ≤ p < ∞ and ||VBf||∞=supzB¯∈Td-|B|VBf(zB¯) for p = ∞.

Let us mention that for B = D, the variation VDf(zB¯) is a constant, which we simply denote as V^Df.

Then, the total variation of a function f:𝕋^d → ℂ is determined by the quantity

A function f:𝕋^d → ℂ for which HV(f) is finite we call function of bounded variation on 𝕋^d in the sense of Hardy-Krause and write f∈HV(𝕋^d).

Remark 2.1. An alternative definition of this kind of bounded variation is discussed in Bakhvalov [18, Lemma 4]. So, f∈HV(𝕋^d) if V^Df < ∞ and for any j∈D there are z0j such that f(z0j:z)∈HV(Td-1), that is, f has bounded variation up to coordinates i∈D\{j}.

Remark 2.2. Let d>1. By definition f∈HV(𝕋^d) iff ||VBf||∞ is finite for all B⊆D. All these 2^d conditions are pairwise independent of each other as can be seen by the following examples [for the case d = 2 cf. ([7], p. 827)].s

Let B₁≠B₂ be arbitrary subsets of D. W.l.o.g. we assume 1∈B₁, 1∉B₂ and we distinguish the 4 possible cases:

Now, we consider functions F:𝕋^d → ℂ of the form

with gk∈HV(T1) and 0<V02π(gk)<∞ for all k = 3, …, d, where V02π denotes the one-dimensional total variation on [0, 2π]. If D⊇A = B∪C with B⊆{1, 2} and C⊆{3, …, d}, then

Hence, ||VAF||∞ is finite, if ||VBf||∞ is finite.

As examples fj:T2→ℂ,j=1,2,3,4 we choose

On the one hand, we conclude for

On the other hand, we conclude for

The main aim of our investigation is to study the approximation order of trigonometric interpolation processes on tensor product and sparse grids for multivariable functions f∈HV(𝕋^d).

In this section, we study an interpolation operator for multivariable functions on tensor product grids. Our approach continues the investigations in Prestin [6] and Prestin and Tasche [10], where the trigonometric interpolation for univariate and bivariate functions and the corresponding approximation bounds were established.

Let Tnd be the space of trigonometric polynomials such that

We define a set of an odd number of equidistant nodes in direction x^j by

Then, the tensor product ⊗j=1dXnj is called a full interpolation grid on 𝕋^d.

For an univariate bounded function f:𝕋 → ℂ, the interpolation operator L_n is of the form

where

is the 2ⁿ-th Dirichlet kernel. For a multivariate function f:𝕋^d → ℂ, the corresponding interpolation operator with respect to the coordinate j takes the form

where I is the identity operator and A⊗B is the algebraic tensor product of A and B.

It is obvious that the operator Lrjj satisfies the interpolation conditions

for each j = 1, …, d.

Let us consider the tensor product of interpolation operators with respect to arguments belonging to the set B⊆D, that is, we define the corresponding interpolation operator for the grid ⊗j∈BXrjj as

Moreover, the interpolation property

holds for any x0B∈⊗j∈BXrjj.

Furthermore, we give the representation for the operator L^B by its Fourier series. Let k={kj}j∈B and |kj|∞≤2rj. So, using Equation 2 we immediately get that

with

We also introduce the intermediate interpolation operator often called blending operator, namely

where A⊕C = A+C−AC is the boolean sum operation. As is known (cf. [14], p. 141), the sum representation for M^B is

and for the remainder operator, we have the product representation

In the next theorem, we establish the approximation property of the blending interpolation operator on a |B|-variate tensor product grid.

Theorem 3.1. Let f∈𝕋^d → ℂ, 1 < p < ∞ and B⊆D be some index set. If ||VUf||p for all U⊆B exists and is a finite number, then it holds true that

where c is some constant depending only on p and |B|.

Proof. For a univariate function f:𝕋 → ℂ in Prestin [6], it was proved that for 1 < p < ∞ the inequality

holds with some constant c depending only on p.

Let B = {j₁, j₂, …j_q}. Thus, using Lemma 2 in Prestin and Tasche [10] and Equation 5 by |B| times, we immediately get that

what has to be proved.

Corollary 3.2. In the case of B = D, Theorem 3.1 states that

and for r_j = n for all j∈D we immediately have

Theorem 3.3. Let f∈HV(𝕋^d) and 1 < p < ∞. Then,

Proof. According to Delvos [14, Proposition 4.1], we can express the remainder as a combination of the remainders of blending operators with lower dimensions:

Then, the proof follows the same estimate as Equation 6.

Corollary 3.4. In the case of B = D for a function f∈HV(𝕋^d), the inequality (Equation 7) takes the form

Furthermore, if r_j = n for all j∈D, then Theorem 3.3 implies that

Remark 3.5. In the case p = 1, the inequality Equation 5 has the form

and Equations 4, 7 read as follows:

and

respectively.

Remark 3.6. For f∈L1(Td), we consider the m-th Fourier coefficients

With B(m)⊆D, we denote the set of indices j such that m_j≠0. Then, according to Fülöp and Móricz [19] for all m∈ℤ^d, the trigonometric Fourier coefficients c_m(f) of f∈HV(𝕋^d) can be estimated by

This estimate is best possible, as demonstrated by the example

where we have equality in Equation 8.

For p = 2, we want to compare the tensor product interpolation with the best approximation. The best approximation in the Hilbert space L2(T)d is given by the Fourier partial sum

By Parseval equation, we estimate

Hence,

Based on the examples provided in Equation 9, it is evident that the order of this estimate cannot be improved.

In the following section, we study an interpolation operator on a sparse grid related to a corresponding Boolean sum operator for the d-dimensional case. Our error estimates for functions of bounded variation complement the results proved in Baszenski and Delvos [12, 13].

To construct a chain of interpolation operators, we consider for each coordinate j∈D the following set of an even number of equidistant nodes:

It is known that for a univariate bounded function f:𝕋 → ℂ, the interpolation operator L~n on the grid (Equation 1) has the form

where

is the 2ⁿ-th modified Dirichlet kernel. In the same way as it was done in Section 3, we will introduce the operators L~B and M~B. Then, the same error estimates are obtained for these approximation methods as in Section 3. The only change is the error estimate for the one-dimensional interpolation. Here, one can refer to Corollary 3.6 in Prestin and Xu [4], where the exact error bound is derived although no explicit constants are given.

Remark 4.1. It is well-known that Kn⋆ is a Lagrange basis function for system of nodes (Equation 1). It is easy to check that for any m≥1 the relation ImL~n⊂ImL~n+m as well as X~n⊂X~n+m are satisfied. Then taking into account Remark 2.2 [13] we have that for operators L~n and L~n+m the ordering L~n<L~n+m and the relation

hold for all n such that 0 ≤ n<n+m.

Now, we introduce a d-dimensional Boolean sum interpolation operator of n-th order in the following way

In an analogous manner as in Section 3, a partial variant GnB with B⊂D can be introduced here and error estimates can be proven. The approach remains the same. To simplify the notation, we therefore restrict ourselves to the case B = D.

To determine the set of interpolation points of the operator Gnd, we note (cf. [15]) that the grid for the operator L~r11L~r22…L~rdd is X~r11×X~r22×…×X~rdd and for L~r11L~r22…L~rdd⊕L~l11L~l22…L~ldd is

Thus, for the operator Gnd, we have the sparse grid of n-th order in the following form

Due to Equation 3, it follows that Gnd interpolates f on each point such that x0∈X~sparsen, that is,

for all x0∈X~sparsen.

Taking into account (Equation 11), we have the sum representation (cf. [13])

Remark 4.2. If we put d = 2, then the operator Gn2 has the form (see for details [13]):

For d = 3, we immediately get the following Boolean sum operator:

Theorem 4.3. If f∈HV(𝕋^d) and 1 < p < ∞, then for all n

where c is some constant depending on d and p.

Proof. Following Baszenski and Delvos [12], we have

Then using Theorem 3.1, we get

Now, the result follows from

Remark 4.4. Let us compare the cardinality of the tensor product grid Xprodn:=⊗j∈DXnj and the sparse grid X~sparsen. The grid Xprodn has 2^dn nodes which is essentially more than n^d−12ⁿ nodes of grid X~sparsen. Nevertheless, the approximation order for f∈HV(T) is only worse by a logarithmic factor n^d−1.