Bivariate general Appell interpolation problem

In this paper, the solution to a bivariate Appell interpolation problem proposed in a previous work is given. Bounds of the truncation error are considered. Ten new interpolants for real, regular, bivariate functions are constructed. Numerical examples and comparisons with bivariate Bernstein polynomials are considered.

In this paper we will consider the bivariate interpolation problem proposed in [11]. We will give the unique solution expressed in the basis of the so-called bivariate, general Appell polynomial sequences [21]. It can be so formulated: let X be the linear space of bivariate, real continuous functions having continuous partial derivatives of all necessary orders, defined in a domain D ⊂ IR 2 . Usually, for simplicity, D = [0, 1] × [0, 1]. We look for, if there exists, the bivariate polynomial i n [f ], n ∈ IN, such that, for any f ∈ X where L is a linear functional on X such that L(1) = 0. Bivariate and, in general, multivariate interpolation has widely employed in the literature (see for example [18,19,23,24,27,29,30] and the references therein).
This paper is organized as follows. In order to make the work as autonomous as possible, Section 2 is a preliminary section. In fact, it includes some known definitions and results that we need in the paper. In Section 3, we find the unique solution of the bivariate interpolation problem mentioned above and give, also, a "complementary" polynomial interpolant. The remainder is analyzed in Section 4 by using the well-known Sard's formula. Then, in Section 5, we give some particular examples of interpolants that doesn't appear in the literature. Section 6 contains numerical examples of bivariate real functions approximations. Comparisons of the new interpolants with the bivariate Bernstein approximation is also given. Finally, in Section 7, we provide some concluding remarks.

Preliminaries
Let A(t) and φ (y, t) be two power series such that with α 0 = ϕ 0 (y) = 1, α k ∈ IR, k ≥ 1 and ϕ k (y) are real polynomials of degree k in the variable y.
The sequences (α k ) k∈IN  Remark 1 Differential relation (2a) is known in the literature (see, for example, [1,11,16] and references therein), but in different contexts and with different approaches.
It has been proved [11] that ∀n ∈ IN r n (x, y) = n k=0 n k α n−k p k (x, y) = n k=0 k i=0 n k where {p k } b k∈IN is the so-called elementary bivariate Appell sequence whose elements are defined as p n (x, y) = n k=0 n k x k ϕ n−k (y), It is also known that the elements of the bivariate sequences {r k } b k∈IN and {p k } b k∈IN have as generating functions F (x, y; t) = A(t)e xt φ(y, t) and G (x, y; t) = e xt φ(y, t), (5) respectively, that is, For any k ∈ IN, letp k (x, y) be the following polynomial of degree k Furthermore, let (β k ) k∈IN be the numerical sequence defined by n k=0 n k α n−k β k = δ n0 , that is, if A(t) is as in (1), then The determinant forms for r n (x, y) and p n (x, y) [11], respectively, are fundamental in the sequel: , n > 0. (10) Moreover, the following recurrence relation holds: From (7) we get and, by symmetry, we can obtain β k , k = 0, . . . , n. We define S n := span {p 0 , . . . , p n } , n∈ IN, with p k as in (4) or in (10) for all k ∈ IN. Observe that ∀n ∈ IN, {p 0 , . . . , p n } is a set of n + 1 linear independent bivariate polynomials.
In the sequel of the paper any fundamental notation and hypothesis introduced so far will be used without references, unless otherwise specified.

Bivariate general Appell interpolation
Let X be the linear space of bivariate real functions defined in D ⊂ IR 2 and belonging to C N (D). Note that ∀n ∈ IN, S n ⊂ X.
Let L be a linear functional on X with We consider the numerical sequence (α k ) k∈IN defined in (7), (β k ) k∈IN being as in (13).
In addition, we consider the general bivariate Appell polynomial sequence {r n } b n∈IN defined equivalently as in (2a) or (9). In this case we say that the bivariate polynomial sequence {r n } b n∈IN is associated with the functional L and, when necessary, we denote it by r L n b n∈IN . The generating function of the bivariate Appell polynomial sequence r L n b n∈IN is connected to the linear functional L by means of the following result. , (14) where L x,y means that the linear functional L acts on e xt φ (y, t) with respect to the variables x and y.
For every f ∈ X we look for, if there exists, the bivariate polynomial i n [f ] such that ∀k ∈ IN, k ≤ n, We call this problem the general bivariate Appell interpolation problem. If, ∀n ∈ IN, i n [f ] exists, we call it the bivariate Appell interpolant of f of order n associated with the functional L.
We note that this problem is very closely related to the corresponding univariate problem in [10, p. 101]. Therefore, we say that it is its "natural" bivariate extension.
In the sequel we will adopt the following notation for the derivatives of a bivariate function f : . Theorem 1 (The main theorem) [11] For every f ∈ X the bivariate polynomial of total degree n given by is the unique element of S n such that that is, i n [f ] is the bivariate Appell interpolant of f associated with the functional L.
Proof Let's define the linear functionals We get Then for the sequence r L n b n∈IN , from (9) we have L j r L k = k!δ kj , j = 0, . . . , k, that is the systems {L n } n∈IN and r L n b n∈IN are biorthogonal [14]. Hence the polynomial (15) satisfies the interpolation conditions (16). The uniqueness follows from the linear independence of the linear functionals L j , j = 0, . . . , n.
Remark 2 In order to remove the calculation of L(f ) from the bivariate Appell interpolant of f (15), we consider an arbitrary fixed point (u, v) ∈ D. Then we get the bivariate polynomial The polynomial i * n [f ](x, y) satisfies the interpolation conditions: We note that the interpolant i * n [f ] replaces the calculation of the functional L(f ) by the evaluation of the function in a suitable point. Therefore, we call it complementary Appell interpolant of f related to functional L.

Remark 3
We observe that the Appell interpolant and its complementary interpolant can be considered finite Appell polynomial expansions of bivariate functions. They are the natural extensions to the bivariate polynomial case of the following univariate formulas, respectively, [10, Th. 7.1, p. 101] and [10, p. 103] where a L,i i∈IN is the univariate Appell polynomial sequence associated with the functional L.

Remainder for Appell interpolation
For the Appell bivariate interpolants i n [f ] defined as in (15) and i * n [f ] defined as in (17), we consider the error at any (x, y) ∈ D.
Definition 1 For any f ∈ X, (x, y) ∈ D, the remainder for the interpolants (15) and (17) are, respectively, the linear functionals and In this case we say that E n [f ] and E * n [f ] have order n with respect to ϕ.
and for even n, n = 2k, Theorem 3 Let f ∈ C n+1 D . For the remainder T n [f ] the following representation holds: where T x,y n is the functional T n with respect to the variables x and y.
In order to get error bounds for E n [f ] we can apply Holder's inequality, for example in the case of the sup norm · . Let f ∈ C n+1 D . We set Then we get  (20) and

Example 1 (Evaluating functional)
Assuming the bivariate Appell interpolating polynomial becomes where α k are related to β k by the relation (7) and by substituting in (21), we get From Theorem 3 we get the following estimate: We call polynomial (21), or equivalently (22), general partial Taylor formula for f at the initial point (x 0 , y 0 ).

Example 2 (Integral functional)
Assuming After easy calculations we get f (1, y) − f (0, y). From Theorem 3 we obtain the following estimate: We call the interpolant (24) (or (25)) integral of first forward difference. The complementary interpolant associated with the functional (23) is Assuming we get the bivariate Appell interpolating polynomial After easy calculations we obtain and α i are as in (1).
From Theorem 3 we get the following estimate: The complementary interpolant is A. Let φ (y, t) = e yt . It is known [11] that the related bivariate elementary Appell sequence is

Some particular examples of bivariate Appell interpolation
In the literature [4,5,7,15] p k (x, y) is denoted also by H (1) n (x, y) and called bivariate Hermite polynomial. Now we consider different functionals.

A1. Evaluating functional.
Let L(f ) be defined as In this case ∀k ∈ IN we get Consequently, from (7), Then the general partial Taylor formula, that in this case we denote by t n [f ], becomes We call this polynomial partial-Taylor formula at starting point (0, 0). We observe that this formula is quite different from the generalized Taylor formula in [16].
From Example 1 we get the following estimate: M i,j , with M i,j defined as in (20).

A2. Integral functional.
For any f ∈ X let's consider the integral functional L as in (23). From (13) we have The bivariate Appell sequence associated with L is given by the recurrence formula (see (11)) In order to give an explicit expression of r n , from (28) B (2) n (x) being the Bernoulli polynomial of order 2 [8,13,22,26] (2) n (x) t n n! .
Bernoulli polynomial of order 2 can be written also in terms of Bernoulli numbers of order 2, as follows with B (2) k given by From (31) we get where B j is the j th Bernoulli number. Finally, from (29), (30) and (31), after calculations, we get r n (x, y) = n k=0 n k B (2) k (x)y n−k = n k=0 n k B k (x)B n−k (y).
Other properties of these polynomials will be studied in a future work. The integral Appell interpolant is (34) Formula (34) can be called, also, polynomial expansion of a bivariate real function in natural bivariate Bernoulli polynomials of order 2.
The complementary integral Appell interpolant is

A3. Arithmetic mean functional.
For any f ∈ X let's consider the functional M 0 (f ) defined in (26). In particular, for (x 0 , y 0 ) = (0, 0), we get Therefore, β 0 = 1 and ∀k ∈ IN, k ≥ 1, Then, the bivariate Appell sequence associated with the functional M 0 is given by the recurrence formula In order to give an explicit expression of r n , from (37) we get r n (x, y) = n k=0 n k E (2) k (x)y n−k , (38) E (2) n (x) being the Euler polynomial of order 2 [2,9] defined by the generating function 2 e t + 1 2 e xt = ∞ n=0 E (2) n (x) t n n! or by with E (2) k (0) given by From Corollary 1.9 in [2] we get E (2) where E j (0) is the value of the j th univariate Euler polynomial E j (x) at x = 0. Finally, from (38), (39) and (40), after calculations, we get r n (x, y) = n k=0 n k E (2) E (2) n n∈IN is a univariate Appell sequence; therefore, from (38), according to Theorem 6.13 in [10, p. 90], we get r n (x, y) = E (2) n (x + y).
To the authors knowledge the bivariate sequence {r k } b k∈IN defined in (38) or, equivalently in (41), does not appear in the literature. We call it bivariate natural Euler polynomial of order 2 and denote it by E (2) n b k∈IN : E (2) n (x, y) = n k=0 n k E k (x)E n−k (y) = E (2) n (x + y).
The first bivariate natural Euler polynomials of order 2 are E (2) 0 (x, y) = 1, E 1 (x, y) = x + y − 1, It is also the polynomial expansion of a bivariate real function in bivariate natural Euler polynomials of order 2. We note that interpolant (43) approximates a functions by only boundary values. In addition, it is the natural extension to the bivariate case of the univariate polynomial [10, p. 133]

The complementary Appell interpolant is
B. Let φ(y, t) = e yt 2 . It is known [11] that in this case the elementary Appell sequence is {p n } b n∈IN , with p n (x, y) = H (2) n (x, y) = n! n 2 k=0 x n−2k y k k!(n − 2k)! .
We note that the numbers α j , j = 0, . . . , n, and the sequence {H n } b n∈IN appear in [13], but in a different context.

B3. Arithmetic mean functional
Let's consider the functional as in (36). In this case

Finally, the bivariate Appell interpolant is
The complementary interpolant is Remark 8 All the bivariate HKF-based Appell interpolants satisfy the known heat equation.

Remark 9
All the bivariate Appell interpolants connected to the arithmetic mean linear functional use only boundary values. Table 1 contains the list of the considered polynomial sequences and the related Appell interpolants.

Numerical results
In order to verify the previous theoretical results we consider the comparison between some functions and the related bivariate Appell interpolant. Particularly, we consider the following functions and their interpolants. For every function we calculate the maximum error with n = 1, . . . , 10, k = 1, 2, 3 and Q∈ A1, A2, A3, B1, B2, B3 . In order to compare the numerical results of our interpolant with other approximants, we consider the well-known bivariate Bernstein polynomial [25]. Tables 2, 3, and 4 show the results for f 1 , f 2 and f 3 respectively. The last column of each table contains the maximum error in the case of approximation by means of bivariate Bernstein polynomials.
From the previous tables we can observe that the results obtained by the interpolants based on H (1) n (cases A1, A2, A3) are satisfactory and comparable favorably with those obtained by Bernstein approximations. The interpolants based on H (2) n (cases B1, B2, B3) need a more in depth study from a computational point of view, particularly taking into account stability and accuracy.

Conclusions
In this paper we proposed a new type of linear interpolation for bivariate functions, called bivariate Appell interpolation. The interpolant conditions are not usual, but they are expressed in terms of a linear functional L, with L(1) = 0, on the space C N (X), N > 1, where X is a linear space of bivariate real functions defined in D ⊂ IR 2 . We proved that for every f ∈ C N (X) there exists a unique bivariate polynomial i  f (u, v), being (u, v) an arbitrary fixed point. The truncation error for the bivariate interpolants are defined and bounds are given by Sard's Theorem. As examples we considered the bivariate Appell polynomials based on H (i) n (x, y), i = 1, 2, and, for every family, three different linear functionals. So we obtained ten new bivariate interpolants for real, regular bivariate functions. We gave also numerical examples and comparisons with the bivariate Berstein polynomial. The comparison is advantageous except in the case of H (2) n (x, y), for which further investigations are needed. Further developments are possible. Beside the aforementioned computational aspects, the study of interpolant series for analytic functions with particular properties seems to be of interest. Other developments can be applications of interpolants, such as numerical cubature and numerical solution of boundary value problems for partial differential equations. Furthermore, theoretical attention can be given to the role of two variables in the definition of bivariate Appell extension.
Data availability Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
Funding Open access funding provided by Università della Calabria within the CRUI-CARE Agreement.

Conflict of interest The authors declare no competing interests.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.