Abstract
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.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
Interpolation theory for real functions is a classic problem both in mathematical and numerical analysis. In fact, on the one hand, it is connected to representability of an analytic function f(x) as a series \(\displaystyle \sum\limits_{n=0}^{\infty }c_{n} \phi _{n}(x)\), where \(\left \{\phi _{n}\right \}_{n\in \text {I N}}\) is a prescribed sequence of functions, called basis functions, and cn are real constants related to the function f [3, 6]. On the other hand, interpolation is fundamental in numerical approximation of functions, numerical quadrature and cubature, boundary value methods, etc. [12, 16, 17, 28]. In an interpolation problem the choice of basic functions, that is the system \(\left \{\phi _{n}\right \}_{n\in \text {I N}}\), is very important.
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 ⊂IR2. Usually, for simplicity, D = [0,1] × [0,1]. We look for, if there exists, the bivariate polynomial in[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.
2 Preliminaries
Let A(t) and \(\phi \left (y,t\right )\) be two power series such that
with α0 = φ0(y) = 1,αk ∈IR,k ≥ 1 and φk(y) are real polynomials of degree k in the variable y.
The sequences \(\left (\alpha _{k}\right )_{k\in \text {I N}}\) and \(\left (\varphi _{k}\right )_{k\in \text {I N}}\) generate the elements of the bivariate polynomial sequence \(\left \{r_{n}\right \}^{b}_{n\in \text {I N}}\) (the superscript b stands for bivariate) satisfying [11]
Remark 1
Differential relation (2b) 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
where \(\left \{p_{k}\right \}^{b}_{k\in \text {I N}}\) is the so-called elementary bivariate Appell sequence whose elements are defined as
It is also known that the elements of the bivariate sequences \(\left \{r_{k}\right \}^{b}_{k\in \text {I N}}\) and \(\left \{p_{k}\right \}^{b}_{k\in \text {I N}}\) have as generating functions
respectively, that is,
For any k ∈IN, let \(\hat p_{k}(x,y)\) be the following polynomial of degree k
Furthermore, let \(\left (\beta _{k}\right )_{k\in \text {I N}}\) be the numerical sequence defined by
that is, if A(t) is as in (1), then
The determinant forms for rn(x,y) and pn(x,y) [11], respectively, are fundamental in the sequel:
Moreover, the following recurrence relation holds:
From (7) we get
and, by symmetry, we can obtain βk, \(k=0,\dots , n\).
We define
with pk as in (4) or in (10) for all k ∈IN. Observe that ∀n ∈IN, \(\left \{p_{0},\dots , p_{n}\right \}\) 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.
3 Bivariate general Appell interpolation
Let X be the linear space of bivariate real functions defined in D ⊂IR2 and belonging to CN(D). Note that ∀n ∈IN, \(\mathcal {S}_{n}\subset X\).
Let L be a linear functional on X with L(1)≠ 0. \(\forall p_{k} \in \left \{p_{\nu }\right \}^{b}_{\nu \in \text {I N}}\), we set
We consider the numerical sequence (αk)k∈IN defined in (7), (βk)k∈IN being as in (13). In addition, we consider the general bivariate Appell polynomial sequence \(\left \{r_{n}\right \}^{b}_{n\in \text {I N}}\) defined equivalently as in (2a) or (9).
In this case we say that the bivariate polynomial sequence \(\left \{r_{n}\right \}^{b}_{n\in \text {I N}}\) is associated with the functional L and, when necessary, we denote it by \(\left \{{r_{n}^{L}}\right \}^{b}_{n\in \text {I N}}\).
The generating function of the bivariate Appell polynomial sequence \(\left \{{r_{n}^{L}}\right \}^{b}_{n\in \text {I N}}\) is connected to the linear functional L by means of the following result.
Proposition 1
For the generating function \(F\left (x,y;t\right )\) of the general bivariate Appell sequence \(\left \{{r_{n}^{L}}\right \}^{b}_{n\in \text {I N}}\) the following identity holds:
where Lx,y means that the linear functional L acts on \(e^{xt}\phi \left (y,t\right )\) with respect to the variables x and y.
Proof
Relation (14) follows from (5), (8), (13), the second equality in (6) and the linearity of the functional:
For every f ∈ X we look for, if there exists, the bivariate polynomial in[f] such that ∀k ∈IN, k ≤ n,
We call this problem the general bivariate Appell interpolation problem. If, ∀n ∈IN, in[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 \(\mathcal {S}_{n}\) such that
that is, in[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 \(\left \{{r_{n}^{L}}\right \}^{b}_{n\in \text {I N}}\), from (9) we have \(\displaystyle L_{j}\left ({r_{k}^{L}} \right )=k!\delta _{kj}\), \(j=0,\dots , k\), that is the systems \(\left \{L_{n}\right \}_{n\in \text {I N}}\) and \(\left \{{r_{n}^{L}}\right \}^{b}_{n\in \text {I N}}\) are biorthogonal [14]. Hence the polynomial (15) satisfies the interpolation conditions (16). The uniqueness follows from the linear independence of the linear functionals Lj, \(j=0,\dots , n\). □
Corollary 1
For any \(f \in \mathcal {S}_{n}\), in[f](x,y) = f(x,y), ∀(x,y) ∈ D.
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 \(\left \{a_{L,i}\right \}_{i\in \text {I N}}\) is the univariate Appell polynomial sequence associated with the functional L.
4 Remainder for Appell interpolation
For the Appell bivariate interpolants \(i_{n}\left [f\right ]\) defined as in (15) and \(i_{n}^{*}\left [f\right ]\) defined as in (17), we consider the error at any \(\left (x,y\right )\in D\).
Definition 1
For any \(f\in X,\ \left (x,y\right )\in D\), the remainder for the interpolants (15) and (17) are, respectively, the linear functionals
and
Remark 4
We observe that for any \(q\in \mathcal {S}_{n}\), \(\displaystyle E_{n}\left [q\right ]\left (x,y\right )=0\) and \(\displaystyle E_{n}^{*}\left [q\right ]\left (x,y\right )=0\), \(\forall \left (x,y\right )\in D\). In this case we say that \(E_{n}\left [f\right ]\) and \(E_{n}^{*}\left [f\right ]\) have order n with respect to φ.
In order to estimate errors (18) and (19), in the following theorem we remember Sard’s formula [17, 29].
Theorem 2 (Sard’s formula for bivariate functions)
Let \(f\in C^{n+1}\left (\overline D\right )\). Then for odd n, n = 2k + 1,
and for even n, n = 2k,
where \(\sum _{\mu =0}^{k}{~}^{\prime }a_{\mu }=\frac {1}{2}{a}_{0}+{\cdots }+ a_{k-1}+a_{k}\).
Now, let \(T_{n}\left [f\right ]=E_{n}\left [f\right ]\) or \(E_{n}^{*}\left [f\right ]\).
Theorem 3
Let \(f\in C^{n+1}\left (\overline D\right )\). For the remainder \(T_{n}\left [f\right ]\) the following representation holds:
for n = 2ρ + 1;
for n = 2ρ, where
τ = τ(y) > 0, σ = σ(x) > 0 are numbers such that (τ,y),(x,σ) ∈ ∂D. \(K_{1}^{\mu }(u)\), \(K_{2}^{\mu }(v)\), Ki,j(u,v) are the Sard kernel functions of \(T_{n}\left [f\right ]\) [17] defined as
where \(T_{n}^{x,y}\) is the functional Tn with respect to the variables x and y.
Proof
The error functional \(T_{n}\left [f\right ]\) vanishes at each element of \(\mathcal {S}_{n}\), according to Remark 4. The result follows from Th. 3.2.2 in [17, p. 105].□
If all the Sard kernels have constant sign, from the general mean value theorem, there exist points \(\left (\xi _{i},0\right ), \left (0, \zeta _{i}\right )\) and \(\left (\eta , \theta \right ), \left (\eta _{1}, \theta _{1}\right ), \left (\eta _{2}, \theta _{2}\right )\) in D such that, for \(f\in C^{n+1}\left (\overline D \right )\),
In order to get error bounds for \(E_{n}\left [f\right ]\) we can apply Holder’s inequality, for example in the case of the sup norm \(\left \| \cdot \right \|\). Let \(f\in C^{n+1}\left (\overline D\right )\). We set
Then we get
Now we consider some important examples of functionals L. For each functional we determine the two interpolants in[f] and \(i_{n}^{*}[f]\), and an error bound in the case of in[f] (for \(i_{n}^{*}[f]\) an analogous bound can be obtained). In the expressions of the error bound we set \(\displaystyle M=\max \limits _{\stackrel {i,j\ge 0}{i+j\le n+1}} M_{i,j},\) with Mi,j defined as in (20) and \(\displaystyle R=\max \limits _{\stackrel {1\le k\le n}{(x,y)\in \overline D} } \left |{r_{k}^{L}}(x,y)\right |\).
Example 1 (Evaluating functional)
Assuming
the bivariate Appell interpolating polynomial becomes
Remembering that \(\displaystyle {r_{k}^{L}}(x,y)={\sum }_{i=0}^{k} \binom ki \alpha _{k-i} p_{i}(x,y)\), where αk are related to βk by the relation (7) and by substituting in (21), we get
From Theorem 3 we get the following estimate:
and
We call polynomial (21), or equivalently (22), general partial Taylor formula for f at the initial point (x0,y0).
Example 2 (Integral functional)
Assuming
the bivariate Appell interpolating polynomial related to the integral functional (23) becomes
After easy calculations we get
being \({\Delta } f(y)=f\left (1,y\right )-f\left (0,y\right )\).
From Theorem 3 we obtain the following estimate:
and
We call the interpolant (24) (or (25)) integral of first forward difference.
The complementary interpolant associated with the functional (23) is
Example 3 (Arithmetic mean functional)
Let \(\left (x_{0}, y_{0}\right )\) be an arbitrary fixed point of D. Assuming
we get the bivariate Appell interpolating polynomial
After easy calculations we obtain
where
and αi are as in (1).
From Theorem 3 we get the following estimate:
and
The complementary interpolant is
5 Some particular examples of bivariate Appell interpolation
Now we will give some particular examples of interpolants in the case D = [0,1] × [0,1].
-
A. Let \(\phi \left (y,t\right )=e^{yt}\).
It is known [11] that the related bivariate elementary Appell sequence is \(\left \{p_{k}\right \}_{k\in \text {I N}}^{b}\), with
$$ p_{k}(x,y)=(x+y)^{k},\qquad \forall k\in \text{I N}. $$(27)In the literature [4, 5, 7, 15] pk(x,y) is denoted also by \(H_{n}^{(1)}(x,y)\) and called bivariate Hermite polynomial.
Now we consider different functionals.
-
A1.
Evaluating functional.
Let L(f) be defined as
$$ L(f)=f(0,0),\qquad \forall f\in X. $$In this case ∀k ∈IN we get
$$ \beta_{k}=L\left( p_{k}\right) =L\left( (x+y)^{k}\right)= \left\{ \begin{array}{lll} 1 & & k=0\\[0.3em] 0 & & k>0. \end{array} \right. $$Consequently, from (7),
$$ \alpha_{k}=\left\{ \begin{array}{lll} 1 & & k=0\\[0.3em] 0 & & k>0. \end{array} \right. $$Then the general partial Taylor formula, that in this case we denote by tn[f], becomes
$$ t_{n}[f](x,y) =f\left( 0,0\right)+ \sum\limits_{k=1}^{n} f^{(k,0)}(0,0) \frac{\left( x+y\right)^{k}}{k!}. $$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:
$$ \begin{array}{lllllll} \displaystyle \left| E_{n}[f] \right| &\displaystyle \le \frac{2M}{(n + 1)!}\ \left( 2^{n} + 2 + { \sum\limits_{\mu=1}^{\rho}}\binom{n + 1}{\mu} \left( 2^{\mu-1} + 1\right) \right) \\[0.2in] &\displaystyle \quad + \frac{2M}{(\rho+1)!(\rho+1)!} \left( 2^{\rho}+1\right), \end{array}\qquad n=2\rho+1, $$and
$$ \begin{array}{lllllll} \displaystyle \left| E_{n}[f] \right| &\displaystyle \le \frac{2M}{(n + 1)!}\ \left( 2^{n-1} + 1 + { \sum\limits_{\mu=1}^{\rho}}\binom{n + 1}{\mu} \left( 2^{\mu-1} + 1\right)\right) \\[0.2in] &\displaystyle \quad + \frac{2M}{(\rho+1)!\rho!}\left( \ 2^{\rho-1} +2^{\rho-2}+1\right), \end{array}\qquad n=2\rho, $$where \(\displaystyle M=\max \limits _{i,j\ge 0, \\ i+j\le n+1} M_{i,j},\) with Mi,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
$$ \forall k \in \text{I N},\qquad \beta_{k}= L\left( p_{k}\right) ={{\int}_{0}^{1}} {{\int}_{0}^{1}} \left( x+y\right)^{k}dx dy=\frac{2\left( 2^{k+1}-1\right)}{(k+1)(k+2)}.$$The bivariate Appell sequence associated with L is given by the recurrence formula (see (11))
$$ r_{0}\left( x,y\right)=1, \quad r_{k}(x,y)=\left( x + y\right)^{k} - 2 \sum\limits_{j=0}^{k-1} \frac{k!\left( 2^{k-j+1} - 1\right) }{j! (k - j + 2)!}r_{j}(x,y),\quad k>0, $$or, equivalently, by the determinant form
$$ r_{0}\left( x,y\right) = 1, \quad r_{k}\left( x,y\right) = \left( - 1\right)^{n} \left| \begin{array}{ccccccccccc} 1 &x+y &(x+y)^{2} &{\cdots} &(x+y)^{k} \\ 1 &1 &\frac{7}{6} &{\cdots} &\frac{2\left( 2^{k+1}-1\right)}{\left( k+1\right)\left( k+2\right)} \\[0.5em] 0 &1 &2 &{\cdots} &\frac{2\left( 2^{k}-1\right)}{k+1} \\ {\vdots} & {\ddots} &{\ddots} &{\ddots} &{\vdots} \\ {\vdots} & &{\ddots} &{\ddots} &{\vdots} \\ 0 &{\cdots} &0 &1 &k \end{array} \right| ,\quad k>0. $$From Proposition 1, the generating function of the sequence \(\left \{r_{n}\right \}_{n\in \text {I N}}^{b}\) is
$$ F\left( x,y;t\right)= \frac{e^{xt} e^{yt}}{\displaystyle {{\int}_{0}^{1}} {{\int}_{0}^{1}} e^{(x+y)t} dx\ dy} = \frac{t^{2}}{\displaystyle \left( e^{t}-1\right)^{2}}e^{(x+y)t}. $$(28)In order to give an explicit expression of rn, from (28) we can write
$$ \left( \frac t{e^{t}-1}\right)^{2}e^{(x+y)t}= \sum\limits_{n=0}^{\infty} r_{n}(x,y) \frac{t^{n}}{n!}. $$Hence
$$ r_{n}(x,y)= \sum\limits_{k=0}^{n} \binom nk B_{k}^{(2)}(x)y^{n-k}, $$(29)\(B_{n}^{(2)}(x)\) being the Bernoulli polynomial of order 2 [8, 13, 22, 26] defined by the generating function
$$ \left( \frac t{e^{t}-1}\right)^{2}e^{xt}= \sum\limits_{n=0}^{\infty} B^{(2)}_{n}(x) \frac{t^{n}}{n!}. $$Bernoulli polynomial of order 2 can be written also in terms of Bernoulli numbers of order 2, as follows
$$ B^{(2)}_{n}(x) = \sum\limits_{k=0}^{n} \binom nk B_{k}^{(2)}x^{n-k}, $$(30)with \(B_{k}^{(2)}\) given by
$$ \left( \frac t{e^{t}-1}\right)^{2}= \sum\limits_{k=0}^{\infty} B^{(2)}_{k} \frac{t^{k}}{k!}. $$(31)From (31) we get
$$ B^{(2)}_{k}= \sum\limits_{i=0}^{k} \binom ki B_{i} B_{k-i}, $$where Bj is the j th Bernoulli number. Finally, from (29), (30) and (31), after calculations, we get
$$ r_{n}(x,y) = \sum\limits_{k=0}^{n} \binom nk B_{k}^{(2)}(x)y^{n-k}= \sum\limits_{k=0}^{n} \binom nk B_{k}(x) B_{n-k}(y). $$(32)We note that \(\left \{B_{n}^{(2)}\right \}_{n\in \text {I N}}\) is a univariate Appell sequence; therefore, from (29), according to Theorem 6.13 in [10, p. 90], we get
$$ r_{n}(x,y)=B_{n}^{(2)}(x+y). $$The bivariate Appell sequence \(\left \{r_{k}\right \}_{k\in \text {I N}}^{b}\) defined in (29) or, equivalently in (32), does not appear in the literature. We call it natural bivariate Bernoulli polynomial sequence of order 2 and we denote it by \(\left \{ {\mathscr{B}}_{k}^{(2)}\right \}_{k\in \text {I N}}^{b}\). Hence, we have
$$ \mathcal{B}_{n}^{(2)}(x,y)= \sum\limits_{k=0}^{n} \binom nk B_{k}(x) B_{n-k}(y)= B_{n}^{(2)}(x+y). $$(33)The first natural bivariate Bernoulli polynomials of order 2 are
$$ \begin{array}{l} \displaystyle \mathcal{B}_{0}^{(2)}(x,y)= 1,\qquad \mathcal{B}_{1}^{(2)}(x,y)= x+y-1,\\[0.7em] \displaystyle \mathcal{B}_{2}^{(2)}(x,y)= (x+y)^{2} - 2 (x+y)+\frac{5}{6}, \\[0.7em] \displaystyle \mathcal{B}_{3}^{(2)}(x,y)= (x+y)^{3}- 3 (x+y)^{2}+ \frac{5}{2}(x+y) -\frac{1}{2},\\[0.7em] \displaystyle \mathcal{B}_{4}^{(2)}(x,y)= (x+y)^{4}- 4 (x+y)^{3}+ 5 (x+y)^{2}- 2 (x+y)+\frac{1}{10}. \end{array} $$Other properties of these polynomials will be studied in a future work.
The integral Appell interpolant is
$$ \begin{array}{llll} i_{n}[f](x,y) & = {{\int}_{0}^{1}} {{\int}_{0}^{1}} f(x,y) dx dy\\ &\hskip 0.5cm + \sum_{k=1}^{n} \frac{\mathcal{B}_{k}^{(2) }(x,y)}{k!}{{\int}_{0}^{1}} \left[f^{(k-1)}\left( 1,y\right)-f^{(k-1)}\left( 0,y\right)\right] dy. \end{array} $$(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
$$ \begin{array}{llll} i_{n}^{*}[f](x,y) &= f(0,0) \\ &\hskip 0.1cm + \sum_{k=1}^{n} \frac{\mathcal{B}^{(2)}_{k}(x,y)-\mathcal{B}^{(2)}_{k}(0,0)}{k!}{{\int}_{0}^{1 }} \left[f^{(k - 1) }\left( 1,y\right) - f^{(k - 1)} \left( 0,y \right)\right] dy. \end{array} $$(35)Remark 5
Formulas (34) and (35) are the bivariate extensions of univariate formulas (8.68) and (8.77) in [10] that are, respectively,
$$ P_{n}(x)={{\int}_{0}^{1}} P_{n}(x) + \sum\limits_{k=1}^{n}\frac{P_{n}^{(k-1)}(1) - P_{n}^{(k-1)}(0)}{k!}B_{k}(x), $$and
$$ f(x) = f(0) + \sum\limits_{k=1}^{n} \frac{B_{k}(x)-B_{k}}{k!}\left[f^{(k-1)}(1) - f^{(k-1)}(0)\right]+R_{n}[f](x). $$ -
A3.
Arithmetic mean functional.
For any f ∈ X let’s consider the functional \({\mathscr{M}}_{0} \left (f\right )\) defined in (26).
In particular, for \(\left (x_{0},y_{0}\right )=\left (0,0\right )\), we get
$$ \mathcal{M}_{0}\left( f\right) = \frac{f\left( 1,1\right)+f\left( 1,0\right)+f\left( 0,1\right)+f\left( 0,0\right)}{4}. $$(36)Therefore, β0 = 1 and ∀k ∈IN, k ≥ 1,
$$ \beta_{k}=\mathcal{M}_{0}\left( (x+y)^{k}\right)= \frac{1+2^{k-1}}2. $$Then, the bivariate Appell sequence associated with the functional \({\mathscr{M}}_{0}\) is given by the recurrence formula
$$ r_{0}\left( x,y\right)=1, \quad r_{n}(x,y) = (x +y)^{n}- \sum\limits_{j=0}^{n-1} \binom{n}{j}\frac{ 1 + 2^{n-j - 1}}{2} r_{j} (x,y ),\quad n \ge 1, $$or, equivalently, by
$$ r_{n}\left( x,y\right) = \left( -1\right)^{n} \left| \begin{array}{ccccccccccc} 1 &x+y &(x+y)^{2} &(x+y)^{3} &{\cdots} &(x+y)^{n} \\ 1 &1 &\frac 32&\frac 52 &{\cdots} &\frac{2^{n-1}+1}2 \\[0.5em] 0 &1 &2 &3 &{\cdots} &n \frac{2^{n-2}+1}2 \\ 0 &0 &1 &1 & &\binom n2\frac{2^{n-3}+1}2 \\ {\vdots} &{\ddots} &{\ddots} &{\ddots} &{\ddots} &{\vdots} \\ 0 &{\cdots} &{\cdots} &0 &1 &\binom n{n-1} \end{array} \right| ,\quad n>0. $$To give the series expansion of \(r_{k}^{{\mathscr{M}}}\left (x,y\right )\), from Proposition 1 we have
$$ F\left( x,y;t\right)= \frac{e^{(x+y)t}}{\displaystyle \mathcal{M}_{0}\left( e^{(x+y)t}\right)} = \frac{4}{\displaystyle \left( e^{t}+1\right)^{2}}e^{(x+y)t}. $$Hence
$$ \left( \frac{2}{e^{t}+1}\right)^{2} e^{(x+y)t}= \sum\limits_{n=0}^{\infty} r_{n}(x,y) \frac{t^{n}}{n!} . $$(37)In order to give an explicit expression of rn, from (37) we get
$$ r_{n}(x,y)= \sum\limits_{k=0}^{n} \binom nk E_{k}^{(2)}(x)y^{n-k}, $$(38)\(E_{n}^{(2)}(x)\) being the Euler polynomial of order 2 [2, 9] defined by the generating function
$$ \left( \frac 2{e^{t}+1}\right)^{2}e^{xt}= \sum\limits_{n=0}^{\infty} E^{(2)}_{n}(x) \frac{t^{n}}{n!} $$or by
$$ E^{(2)}_{n}(x) = \sum\limits_{k=0}^{n} \binom nk E_{k}^{(2)}(0)x^{n-k}, $$(39)with \(E_{k}^{(2)}(0)\) given by
$$ \left( \frac 2{e^{t}+1}\right)^{2}= \sum\limits_{k=0}^{\infty} E^{(2)}_{k} (0)\frac{t^{k}}{k!}. $$(40)From Corollary 1.9 in [2] we get
$$ E^{(2)}_{k}(0)= \sum\limits_{i=0}^{k} \binom ki E_{i}(0) E_{k-i}(0), $$where Ej(0) is the value of the j th univariate Euler polynomial Ej(x) at x = 0.
Finally, from (38), (39) and (40), after calculations, we get
$$ r_{n}(x,y) = \sum\limits_{k=0}^{n} \binom nk E_{k}^{(2)}(x)y^{n-k}= \sum\limits_{k=0}^{n} \binom nk E_{k}(x) E_{n-k}(y). $$(41)\(\left \{E_{n}^{(2)}\right \}_{n\in \text {I N}}\) is a univariate Appell sequence; therefore, from (38), according to Theorem 6.13 in [10, p. 90], we get
$$ r_{n}(x,y)=E_{n}^{(2)}(x+y). $$To the authors knowledge the bivariate sequence \(\left \{r_{k}\right \}_{k\in \text {I N}}^{b}\) 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 \(\left \{ \mathcal {E}_{n}^{(2)}\right \}_{k\in \text {I N}}^{b}\):
$$ \mathcal{E}_{n}^{(2)}(x,y)= \sum\limits_{k=0}^{n} \binom nk E_{k}(x)E_{n-k}(y)= E_{n}^{(2)}(x+y). $$(42)The first bivariate natural Euler polynomials of order 2 are
$$ \begin{array}{l} \displaystyle \mathcal{E}_{0}^{(2)}(x,y)=1,\qquad \mathcal{E}_{1}^{(2)}(x,y)= x+y-1,\\[0.7em] \displaystyle \mathcal{E}_{2}^{(2)}(x,y)= (x+y)^{2} - 2 (x+y)+\frac{1}{2}, \\[0.7em] \displaystyle \mathcal{E}_{3}^{(2)}(x,y)= (x+y)^{3}- 3 (x+y)^{2}+ \frac{3}{2}(x+y) +\frac{1}{2},\\[0.7em] \displaystyle \mathcal{E}_{4}^{(2)}(x,y)= (x+y)^{4}- 4 (x+y)^{3}+ 3 (x+y)^{2}+ 2 (x+y)-1. \end{array} $$The mean Appell interpolant is
$$ \begin{array}{lll} i_{n}\left[f\right]\left( x,y\right)= \displaystyle \mathcal{M}_{0}\left( f\right) + \sum\limits_{k=1}^{n} \mathcal{M}_{0}\left( f^{(k,0)}\right) \frac{\mathcal{E}_{k}^{(2)}\left( x,y\right)}{k!}\\[1em] \quad = \displaystyle \frac{f\left( 1,1\right)+f\left( 1,0\right)+f\left( 0,1\right)+f\left( 0,0\right)}{4}\\ \qquad\displaystyle + \sum\limits_{k=1}^{n} \frac{f^{(k,0)} \left( 1,1\right) + f^{(k,0)} \left( 1,0\right) + f^{(k,0)} \left( 0,1\right) + f^{(k,0)} \left( 0,0\right)}{4}\ \frac{\mathcal{E}_{k}^{(2)} \left( x,y\right)}{k!}. \end{array} $$(43)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]
$$ P_{n}[f](x)=\frac{f(1)+f(0)}2+ \sum\limits_{i=1}^{n} \frac{f^{(i)}(1)+f^{(i)}(0)}2 \frac{E_{i}(x)}2. $$The complementary Appell interpolant is
$$ i_{n}^{*}[f](x,y) = f\left( 0,0\right)+ \sum\limits_{k=1}^{n} \mathcal{M}_{0}\left( f^{(k,0)}\right)\frac{\mathcal{E}_{k}^{(2)}\left( x,y\right)-\mathcal{E}_{k}^{(2)}\left( 0,0\right)}{k!}. $$
-
A1.
-
B. Let \(\displaystyle \phi (y,t)=e^{yt^{2}}\).
It is known [11] that in this case the elementary Appell sequence is \(\left \{p_{n}\right \}_{n\in \text {I N}}^{b}\), with
$$ p_{n}(x,y)=H_{n}^{(2)}(x,y)=n! \sum\limits_{k=0}^{\lfloor\frac n2\rfloor}\frac{x^{n-2k}y^{k}}{k!(n-2k)!}. $$(44)The polynomials \(H_{n}^{(2)}(x,y)\) are called Hermite-Kampé de Fériet (HKF) polynomials [4, 5, 7, 13, 20].
-
B1.
Evaluating functional
Assuming \(L\left (f\right )=f\left (0,0\right )\), ∀k ∈IN we have
$$ \beta_{k}=L\left( H_{k}^{(2)}\right)= \left\{ \begin{array}{llllll} 1 & & & & k=0\\ 0 & & & & k\ne 0. \end{array}\right. $$The Appell interpolant is
$$ i_{n}[f](x,y) =f\left( 0,0\right)+ \sum\limits_{k=1}^{n} f^{(k,0)}(0,0) \frac{H_{k}^{(2)}\left( x,y\right)}{k!}. $$This means that the partial Taylor HKF-based polynomial provides also an expansion for a bivariate function in terms of HKF polynomials.
-
B2.
Integral functional.
∀f ∈ X let’s consider the integral functional as in (23). In order to determine the bivariate Appell sequence associated with functional (23), we get
$$ \forall k \in \text{I N}, \qquad \beta_{k}=L\left( H_{n}^{(2)}\right) = k! \sum\limits_{j=0}^{\lfloor\frac k2\rfloor}\frac{1}{(j+1)!(k-2j+1)!}. $$(45)Thus we obtain the bivariate Appell sequence \(\left \{r_{n}\right \}^{b}_{n\in \text {I N}}\) such that
$$ r_{0}\left( x,y\right)=1, \quad r_{n}(x,y)=H_{n}^{(2)}\left( x,y\right)- \sum\limits_{j=0}^{n-1} \binom{n}{j}\beta_{n-j} r_{j}(x,y),\quad n\ge 1. $$For the generating function of \(\left \{r_{n}\right \}_{n\in \text {I N}}^{b}\), from Proposition 1 we get
$$ F\left( x,y;t\right)= \frac{e^{xt+yt^{2}}}{\displaystyle {{\int}_{0}^{1}} {{\int}_{0}^{1}} e^{xt+yt^{2}} dx dy}= \frac{t^{3} e^{xt+yt^{2}}}{\left( e^{t}-1\right)\left( e^{t^{2}}-1\right)}. $$(46)Hence
$$ \frac{t^{3} e^{xt+yt^{2}}}{\left( e^{t}-1\right)\left( e^{t^{2}}-1\right)}= \sum\limits_{n=0}^{\infty} r_{n}(x,y) \frac{t^{n}}{n!}. $$(47)We call \(\left \{r_{n}\right \}^{b}_{n\in \text {I N}}\) bivariate Bernoulli HKF-based polynomial sequence associated with functional (23) and denote it by \(\left \{{\mathscr{H}}_{k}\right \}_{k\in \text {I N}}^{b}\). The first polynomials of this sequence are
$$ \begin{array}{l} \displaystyle \mathcal{H}_{0}(x,y)=1 ,\quad \mathcal{H}_{1}(x,y)=x-\frac 12 ,\quad \mathcal{H}_{2}(x,y)= x^{2}-x+2y-\frac 56, \\ \displaystyle \mathcal{H}_{3}(x,y)= x^{3} - \frac 32 x^{2} - \frac 52 x - 3 y + 6 x y +\frac 32,\\ \displaystyle \mathcal{H}_{4}(x,y)=x^{4}- 2x^{3}-5x^{2}+6x+12 y^{2}+12 x^{2} y - 12xy - 10y+\frac{29}{30} . \end{array} $$From (47),
$$ \mathcal{H}_{n}(x,y)= \sum\limits_{j=0}^{n} \binom{n}{j}\alpha_{n-j} H_{j}^{(2)}(x,y),\qquad n\ge 1, $$(48)where
$$ \frac{t^{3}}{\left( e^{t}-1\right)\left( e^{t^{2}}-1\right)}= \sum\limits_{k=0}^{\infty} \alpha_{k} \frac{t^{k}}{k!} $$and
$$ \alpha_{k}=k! \sum\limits_{j=0}^{\lfloor\frac k2\rfloor} \frac{B_{k-2j}B_{j}}{j!(k-2j)!}, $$with Bs the s th Bernoulli number.
Remark 6
From (7) and (12), for \(k=1,\dots ,n\), we get the following identity:
$$ k! \sum\limits_{j=0}^{\lfloor\frac k2\rfloor} \frac{B_{k-2j}B_{j}}{j!(k-2j)!}= (-1)^{k} \left| \begin{array}{ccccccccccccc} \beta_{1} &1 &0 &{\cdots} &{\cdots} &0 \\ \beta_{2} &\binom 21 \beta_{1} &1 & 0 &{\cdots} &0\\ {\vdots} & {\vdots} & {\ddots} &{\ddots} & &{\vdots} \\ {\vdots} & {\vdots} & {\vdots} & {\ddots} &{\ddots} &{\vdots} \\ \beta_{k-1} &\binom{k-1}{k-2} \beta_{k-2} &\binom{k-1}{k-3} \beta_{k-3} &{\cdots} &{\ddots} &1 \\ \beta_{k} &\binom{k}{k-1} \beta_{k-1} &\binom{k}{k-2} \beta_{k-2} &{\cdots} &{\cdots} &\binom{k}{1} \beta_{1} \end{array} \right| , $$where βj, \(j=1,\dots , k\), are defined as in (45).
We note that the numbers αj, \(j=0,\dots , n\), and the sequence \(\left \{{\mathscr{H}}_{n}\right \}^{b}_{n\in \text {I N}}\) appear in [13], but in a different context.
The bivariate Appell interpolant related to the sequence \(\left \{{\mathscr{H}}_{k}\right \}_{k\in \text {I N}}^{b}\) is
$$ i_{n}[f](x,y) = {{\int}_{0}^{1}} {{\int}_{0}^{1}} f \left( x ,y\right)dx dy+ \sum\limits_{k=1}^{n} \frac{\mathcal{H}_{k}\left( x ,y\right)}{k!}{ {\int}_{0}^{1}} {{\int}_{0}^{1}} f^{(k,0)} (x,y) dx dy, $$which provides an expansion of a bivariate function in terms of HKF polynomials.
The complementary Appell interpolant is
$$ i_{n}^{*}[f](x ,y) = f \left( x_{0},y_{0}\right)+ \sum\limits_{k=1}^{n} \frac{\mathcal{H}_{k} \left( x,y\right) - \mathcal{H}_{k}\left( x_{0},y_{0}\right)}{k!}{ {\int}_{0}^{1}} {{\int}_{0}^{1}} f^{(k,0)}(x ,y) dx dy . $$ -
B3.
Arithmetic mean functional
Let’s consider the functional as in (36). In this case
$$ \beta_{0}=1, \quad \beta_{k}=\mathcal{M}_{0}\left( H_{k}^{(2)}\right) =\frac 14\left( 1+ \sum\limits_{j=0}^{\left[ \frac k2\right]} \frac{k!}{j! (k-2j)!}+ \left\{ \begin{array}{cl} \frac{n!}{\left( \frac n2\right)!} & n\ even\\ 0 & n\ odd \end{array} \right. \right),\ \ k\ge 1. $$Then the bivariate Appell sequence associated with the functional \({\mathscr{M}}_{0}\) is given by the recurrence formula
$$ r_{n}(x,y)=H_{n}^{(2)}(x,y)- \sum\limits_{k=0}^{n-1} \binom nk \beta_{n-k}r_{k}(x,y). $$From Proposition 1 the generating function is
$$ F(x,y;t)=\frac{e^{xt+yt^{2}}} {\mathcal{M}_{0} \left( e^{xt+yt^{2}}\right)}. $$Since \(\displaystyle {\mathscr{M}}_{0} \left (e^{xt+yt^{2}}\right ) =\frac {\left (e^{t}+1\right ) \left (e^{t^{2}}+1\right )}4, \) then
$$ \frac {4 e^{xt+yt^{2}}}{\left( e^{t}+1\right) \left( e^{t^{2}}+1\right) }= \sum\limits_{n=0}^{\infty} r_{n}(x,y) \frac{t^{n}}{n!}. $$Setting
$$ \frac 4{\left( e^{t}+1\right) \left( e^{t^{2}}+1\right) }= \sum\limits_{k=0}^{\infty} \alpha_{k} \frac{t^{k}}{k!}, $$we have
$$ \alpha_{k}= k! \sum\limits_{s=0}^{\left[ \frac k2\right]} \frac{E_{s}(0) E_{k-2s}(0)}{s! (k-2s)!}, $$where En(x) is the classic Euler polynomial of degree n. Then, from (3) we obtain
$$ r_{n}(x,y)= \sum\limits_{k=0}^{n} \binom nk \alpha_{n-k} H_{k}^{(2)}(x,y). $$(49)Remark 7
The bivariate Appell sequence \(\left \{r_{n}\right \}^{b}_{n\in \text {I N}}\) appears in [13], but in a different context. We call it bivariate Euler HKF-based polynomial sequence of order 2 and denote it by \(\left \{\mathcal {K}_{n}^{(2)}\right \}^{b}_{n\in \text {I N}}\).
The first bivariate Euler HKF-based polynomials of order 2 are:
$$ \begin{array}{l} \displaystyle \mathcal{K}_{0}^{(2)}(x,y) = 1,\quad \mathcal{K}_{1}^{(2)}(x,y) = x - \frac 12 ,\quad \mathcal{K}_{2}^{(2)}(x,y)= x^{2} - x + 2y-1, \\ \displaystyle \mathcal{K}_{3}^{(2)}(x,y)= x^{3} - \frac 32 x^{2} -3 x - 3 y + 6 x y +\frac 74,\\ \displaystyle \mathcal{K}_{4}^{(2)}(x,y)=x^{4}- 2x^{3}-6 x^{2}+ 7x+12 y^{2}+12 x^{2} y -12xy-12y . \end{array} $$Finally, the bivariate Appell interpolant is
$$ i_{n}\left[f\right]\left( x,y\right)= \mathcal{M}_{0}\left( f\right)+ \sum\limits_{k=1}^{n} \mathcal{M}_{0}\left( f^{(k,0)}\right)\frac{\mathcal{K}_{k}^{(2)}\left( x,y\right)}{k!}. $$The complementary interpolant is
$$ i_{n}^{*}[f](x,y) = f\left( 0,0\right)+ \sum\limits_{k=1}^{n} \mathcal{M}_{0}\left( f^{(k,0)}\right)\frac{\mathcal{K}_{k}^{(2)}\left( x,y\right)-\mathcal{K}_{k}^{(2)}\left( 0,0\right)}{k!}. $$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.
-
B1.
Table 1 contains the list of the considered polynomial sequences and the related Appell interpolants.
6 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
-
\(f_{1}(x,y)=\sin \limits \left (x+y\right )\)
-
\(f_{2}(x,y)=\ln \left (x+y+5\right )\)
-
\(f_{3}(x,y)=\displaystyle e^{-\frac {x+2}{4y+9}}\)
and their interpolants. For every function we calculate the maximum error
with \(n=1,\dots , 10\), k = 1,2,3 and Q\(\in \left \{A1, A2, A3, B1, B2, B3\right \}\).
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 f1, f2 and f3 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 \(H_{n}^{(1)}\) (cases A1, A2, A3) are satisfactory and comparable favorably with those obtained by Bernstein approximations. The interpolants based on \(H_{n}^{(2)}\) (cases B1, B2, B3) need a more in depth study from a computational point of view, particularly taking into account stability and accuracy.
7 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 CN(X), N > 1, where X is a linear space of bivariate real functions defined in D ⊂IR2. We proved that for every f ∈ CN(X) there exists a unique bivariate polynomial in[f](x,y) such that \(L\left (i_{n}[f]^{(j,0)} \right ) =L\left (f^{(j,0)} \right )\), \(j=0,\dots , n\). To the bivariate Appell interpolant in[f], which depends on the functional L, is associated the complementary interpolant \(i_{n}^{*}[f]\), in which L(f) is substituted by 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 \(H_{n}^{(i)}(x,y)\), 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 \(H_{n}^{(2)}(x,y)\), 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.
Change history
18 July 2022
Funding information was added to the article.
References
Anshelevich, M.l.: Appell polynomials and their relatives. Int. Math. Res. Not. 65, 3469–3531 (2004)
Aygunes, A.A.: Higher-order Euler-type polynomials and their applications. Fixed Point Theory Appl. 2013(1), 1–11 (2013)
Boas, R.P., Buck, R.C.: Polynomial Expansions of Analytic Functions, vol. 19. Springer Science & Business Media (2013)
Bretti, G., Cesarano, C.: Ricci, P.E. Laguerre-type exponentials and generalized Appell polynomials. Comput. Math. Appl. 48, 833–839 (2004)
Bretti, G., Natalini, P., Ricci, P.E.: Generalizations of the Bernoulli and Appell polynomials. Abstr. Appl. Anal. 7, 613–623 (2004)
Buhmann, M.D.: Radial basis functions. Acta Numerica 9, 1–38 (2000)
Cesarano, C.: A note on generalized Hermite polynomials. Int. J. Appl. Math. Informat. 8, 1–6 (2014)
Choi, J.: Explicit formulas for Bernoulli polynomials of order n. Indian. J. Pure Appl. Math. 27(7), 667–674 (1996)
Corcino, C.B., Corcino, R.B., Canete, J.A.A.: Some formulae of Genocchi polynomials of higher order. arXiv:2009.03481 (2020)
Costabile, F.A.: Modern Umbral Calculus. An Elementary Introduction with Applications to Linear Interpolation and Operator Approximation Theory, vol. 72. Walter de Gruyter GmbH & Co KG (2019)
Costabile, F.A., Gualtieri, M.I., Napoli, A.: General bivariate Appell polynomials via matrix calculus and related interpolation hints. Mathematics 9(9), 964 (2021)
Costabile, F.A., Gualtieri, M.I., Napoli, A.: Lidstone-Euler interpolation and related high even order boundary value problem. Calcolo 58(25), 1–24 (2021)
Dattoli, G., Cesarano, C., Lorenzutta, S.: Bernoulli numbers and polynomials from a more general point of view. Rend. Math. Appl. 22(7), 193–202 (2002)
Dattoli, G., Germano, B., Ricci, P.E.: Comments on monomiality, ordinary polynomials and associated bi-orthogonal functions. Appl. Math. Comput. 154(1), 219–227 (2004)
Davis, P.J.: Interpolation and Approximation. Courier Corporation (1975)
Dragomir, S.S., Qi, F., Hanna, G.T., Cerone, P.: New Taylor-like expansions for functions of two variables and estimates of their remainders. J. Korean Soc. Ind. Appl. Math. 9(2), 1–15 (2005)
Engels, H.: Numerical Quadrature and Cubature. Academic Press (1980)
Gasca, M., Sauer, T.: Polynomial interpolation in several variables. Adv. Comput. Math. 12(4), 377–410 (2000)
Gasca, M., Sauer, T.: On the history of multivariate polynomial interpolation. Numerical Analysis: Historical Developments in the 20th Century, 135–147 (2001)
Gould, H.W., Hopper, A.T., et al.: Operational formulas connected with two generalizations of Hermite polynomials. Duke Math. J. 29(1), 51–63 (1962)
Khan, S., Raza, N., et al.: General-Appell polynomials within the context of monomiality principle. Int. J. Anal., 328032 (2013)
Kim, D.S., Kim, T.: A note on higher-order Bernoulli polynomials. J. Inequal. Appl. 2013(1), 1–9 (2013)
Lorentz, R.A.: Multivariate Birkhoff Interpolation. Springer (2006)
Lorentz, R.A.: Multivariate Hermite interpolation by algebraic polynomials: A survey. J. Comput. Appl. Math. 122(1-2), 167–201 (2000)
Martinez, F.L.: Some properties of two-dimensional Bernstein polynomials. J. Approx. Theory 59(3), 300–306 (1989)
Nörlund, N. E.: Vorlesungen über Differenzenrechnung, vol. 13 Springer (1924)
Olver, P.J.: On Multivariate Interpolation. Available online. http://www.math.umn.edu/~olver (1916)
Powell, M.J.D., et al.: Approximation Theory and Methods. Cambridge University Press (1981)
Sard, A.: Linear approximation. American Mathematical Soc (1963)
Xu, Y.: Polynomial interpolation in several variables, cubature formulae, and ideals. Adv. Comput. Math. 12(4), 363–376 (2000)
Funding
Open access funding provided by Università della Calabria within the CRUI-CARE Agreement.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare no competing interests.
Additional information
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
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/.
About this article
Cite this article
Costabile, F.A., Gualtieri, M.I. & Napoli, A. Bivariate general Appell interpolation problem. Numer Algor 91, 531–556 (2022). https://doi.org/10.1007/s11075-022-01272-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-022-01272-4