PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 99(113) (2016), 165–175 DOI: 10.2298/PIM1613165C

UMBRAL INTERPOLATION

Francesco Aldo Costabile and Elisabetta Longo

Abstract. A general linear interpolation problem is posed and solved. This problem is called umbral interpolation problem because its solution can be expressed by a basis of Sheffer polynomials. The truncation error and its bounds are considered. Some examples are discussed, in particular generalizations of Abel–Gontscharoff and central interpolation are studied. Numerical examples are given too.

1. Introduction

In [5–7] an application of Appell and Δ_{h}-Appell polynomials to linear interpolation problem for real functions has been given. In this note we will extend this approach to the more general so-called Sheffer polynomials [8,9,12–14,16,24]. For this purpose, let X be the linear space of real functions defined in the interval [a, b] continuous and with continuous derivatives of all necessary orders; let P_{n} ⊂ X, n ∈ ℕ, be the space of polynomials of degree less than or equal to n. Let Q be a δ-operator [20] on P_{n}, and L a linear functional on X with L(1) ≠ 0. Let be XQ = {f ∈ X : Q^{i}f ∈ X, i = 0, …, n, for all n ∈ ℕ}, then for each f ∈ XQ we want a polynomial P_{n}[f], if it exists, of degree less than or equal to n such that

f = P_{n}[f] + R_{n}[f],

with

(1.1)  L(Q^{i}P_{n}[f]) = L(Q^{i}f), i = 0, …, n.

If Q = D or Q = Δ_{h}, that is the differential operator or the finite difference operator respectively, we have that XQ is the set of analytic functions on [a, b] and, respectively, the set of bounded functions in [a, b]. In these cases problem (1.1) admits a unique solution and it has been called Appell interpolation problem [1, 5] or Δ_{h}-Appell interpolation problem [6, 7], respectively.

In this note we want to address the general case and we give the solution of problem (1.1) only if XQ is known. We will call this problem umbral interpolation problem, because its solution can be expressed by a basis of Sheffer polynomials, also said umbral basis. Interpolation and approximation have been studied from an umbral point of view in several papers [10, 15, 17]. In [18, 19] the authors study the sequence of polynomials, which solve the linear interpolation in P_{n}, but not the general case, moreover no connection with real function is given. The paper is organized as follows: in Section 2 we give some preliminary definitions and results; in Section 3 we define the umbral interpolation and provide its solution; in Section 4 we give, as examples, generalizations of Abel–Gontscharoff [11] and central interpolation [22]; in Section 5, some numerical examples, which justify theoretical results, are given; finally, in Section 6 conclusions and further developments are announced.

2010 Mathematics Subject Classification: 11B83; 65F40.

Key words and phrases: Umbral calculus, Sheffer polynomials, interpolation.

Communicated by Gradimir V. Milovanović.

2. Umbral basis for (L, Q)

In order to make the work self-contained we recall some basic notations of the umbral calculus [8, 20, 21]. Let Q be a δ-operator and (p_{n})_{n∈ℕ} be the associated sequence [20], that is, the sequence that satisfies p_{0}(x) = 1, p_{n}(0) = 0, Qp_{n} = np_{n−1} , n = 1, 2, . . .. It is known [20] that (p_{n}(x))_{n∈ℕ} is of binomial type and it is a basis for P_{n}. Let L be a linear functional on X with L(1) ≠ 0 and let us set

(2.1)  β_{n} = L(p_{n}), n = 0, 1, …

and define the sequence of polynomials

\{ \begin{matrix}s_{0}(x) = \frac{1}{β_{0}} \\ s_{n}(x) = \frac{(−1)^{n}}{(β_{0} )^{n+1}}  | \begin{matrix}p_{0}(x) & p_{1}(x) & p_{2}(x) & ⋯ & ⋯ & p_{n−1} (x) & p_{n}(x) \\ β_{0} & β_{1} & β_{2} & ⋯ & ⋯ & β_{n−1} & β_{n} \\ 0 & β_{0} & \binom{2}{1}β_{1} & ⋯ & ⋯ & \binom{n−1}{1}β_{n−2} & \binom{n}{1}β_{n−1} \\ 0 & 0 & β_{0} & ⋯ & ⋯ & \binom{n−1}{2}β_{n−3} & \binom{n}{2}β_{n−2} \\ ⋮ & & & ... & & ⋮ & ⋮ \\ ⋮ & & & & ... & ⋮ & ⋮ \\ 0 & ⋯ & ⋯ & ⋯ & 0 & β_{0} & \binom{n}{n−1}β_{1} \end{matrix} | , \end{matrix}

n = 1, 2, . . .. (2.2)

Remark 2.1. [8] The sequence (s_{n}(x))_{n∈ℕ} is a sequence of polynomials of degree less than or equal to n, for which we have

s_{n}(x) = ∑_{i=0}^{n} ( {n \atop i}) α_{n−i} p_{i}(x), n = 0, 1, …,

with

α_{0} = \frac{1}{β_{0}} , ∑_{i=0}^{n} ( {n \atop i}) α_{i}β_{n−i} = 0, n = 1, 2, . . ..

Remark 2.2 (Recurrence relation, [8]). For the polynomial sequence (s_{n}(x))_{n∈ℕ} the following recurrence relation holds

(2.3)  s_{n}(x) = \frac{1}{β_{0}} ( p_{n}(x) − ∑_{i=0}^{n−1} ( {n \atop i}) β_{n−i} s_{i}(x)) , n = 1, 2, . . ..

UMBRAL INTERPOLATION 167

Remark 2.3. [8] For the polynomial sequence (s_{n}(x))_{n∈ℕ} we have

Qs_{n}(x) = ns_{n−1} (x), n = 1, 2, . . ..

Remark 2.4. We have

(2.4)  L(Q^{i}s_{n}(x)) = i!δ_{i,n} , i = 0, …, n,

where δ_{i,n} is the Kronecker symbol.

Proof. It follows from Remark 2.3, from which we have

L(Q^{i}s_{n}(x)) = n(n − 1) …(n − i + 1)L(s_{n−i} (x)) = i!( {n \atop i}) δ_{i,n} = i!δ_{i,n} .  □

Definition 2.1. The sequence (s_{n}(x))_{n∈ℕ} is a basis for P_{n}, and we call it umbral basis for (L, Q). In the following, often, we will omit for (L, Q).

Remark 2.5. We note explicitly that the polynomial sequence (s_{n}(x))_{n∈ℕ} defined in (2.2) is the Sheffer sequence [21] for (β_{n}, p_{n}(x)) as defined in [8].

3. Umbral interpolation

Let L be a linear functional on X with L(1) ≠ 0, Q a δ-operator on P_{n} and ω_{i} ∈ ℝ, i = 0, 1, …, n; then the problem

L(Q^{i}P_{n}) = i!ω_{i}, i = 0, …, n, P_{n} ∈ P_{n},

is called umbral interpolation problem in P_{n}.

Theorem 3.1. Let Q be a δ-operator on P_{n} and L be a linear functional on X with L(1) ≠ 0. Let (s_{n}(x))_{n∈ℕ} be the umbral basis for (L, Q) and ω_{i} ∈ ℝ, i = 0, 1, …, n. The polynomial P_{n}(x) = ∑ _{i=0}^{n} ω_{i}s_{i}(x), is the unique polynomial of degree less than or equal to n such that L(Q^{i}P_{n}) = i!ω_{i}, i = 0, …, n.

Proof. It is a straightforward consequence of (2.4) and of the linearity of Q. □

Corollary 3.1. For each P_{n}(x) ∈ P_{n} we have

P_{n}(x) = ∑_{i=0}^{n} \frac{L(Q^{i}P_{n})}{i!}s_{i}(x).

Let us consider a function f ∈ XQ. Then we have the following Theorem 3.2 (Main theorem). The polynomial

(3.1)  P_{n}[f](x) = ∑_{i=0}^{n} \frac{L(Q^{i}f)}{i!} s_{i}(x)

is the unique polynomial of degree less than or equal to n such that

L(Q^{i}P_{n}[f]) = L(Q^{i}f), i = 0, …, n.

Proof. It follows from Theorem 3.1. □ Definition 3.1. The polynomial P_{n}[f](x) is called umbral interpolation poly-

nomial of the function f for (L, Q).

Therefore, it is interesting to consider the estimation of the remainder

(3.2)  R_{n}[f](x) = f(x) − P_{n}[f](x), ∀x ∈ [a, b].

Theorem 3.3. For any f(x) ∈ P_{n} and x ∈ [a, b],

R_{n}[f](x) = 0, and R_{n}[p_{n+1} (x)] ≠ 0.

Proof. It follows from (2.1) and (2.3), noting that

L(Q^{i}p_{n}(x)) = n(n − 1) …(n − i + 1)L(p_{n−i} (x)) = i!( {n \atop i}) β_{n−i} .  □

For a fixed x we may consider the remainder R_{n}[f](x) as a linear functional and, therefore, from Peano’s theorem [11, p. 69], we have

Theorem 3.4. Let f ∈ C^{n+1} [a, b]. The following relation holds

(3.3)  ∀x ∈ [a, b], R_{n}[f](x) = \frac{1}{n!} ∫_{a}^{b}K_{n}(x, t)f^{(n+1)} (t) dt,

where K_{n}(x, t) = R_{n}[(x − t)_{+}^{n} ] = (x − t)_{+}^{n} − ∑_{i=0}^{n} \frac{L_{x}[Q^{i}((x − t)_{+}^{n} )]}{i!} s_{i}(x).

Proof. It follows by Theorem 3.3 and Peano’s theorem. □

Remark 3.1. By (3.3), if f^{(n+1)} ∈L^{p}[a, b] and K_{n}(x, t)∈L^{q}[a, b] with \frac{1}{p}+ \frac{1}{q} = 1, applying the Hölder’s inequality, classical error bounds can be obtained.

Now, let us fix z ∈ [a, b] and consider the polynomial

(3.4) \overline{P}_{n}[f, z](x) ≡ f(z)+P_{n}[f](x)−P_{n}[f](z) = f(z)+∑_{i=1}^{n} \frac{L(Q^{i}f)}{i!} (s_{i}(x)−s_{i}(z)).

In the following, to avoid encumbering the notation, we will denote it by \overline{P}_{n}[f](x), omitting the dependence on z. Then we have the following

Theorem 3.5. The polynomial \overline{P}_{n}[f](x) is an approximating polynomial of degree n for f(x), i.e.,

(3.5)  ∀x ∈ [a, b], f(x) = \overline{P}_{n}[f](x) + \overline{R}_{n}[f](x),

with \overline{R}_{n}[p_{i}(x)] = 0, i = 0, …, n and \overline{R}_{n}[p_{n+1} (x)] ≠ 0.

Proof. For each x ∈ [a, b], by (3.2) we get (3.5); the exactness follows from the exactness of the polynomial P_{n}[f](x). □

Theorem 3.6. The polynomial \overline{P}_{n}[f](x) satisfies the interpolatory conditions

\overline{P}_{n}[f](z) = f(z), L(Q^{i} \overline{P}_{n}[f]) = L(Q^{i}f), i = 1, …, n.

Proof. It follows from (2.4). □

We call \overline{P}_{n}[f](x) umbral interpolation polynomial of second kind.

4. Examples

4.1. Abel–Sheffer interpolation. With the previous notation, let be f analytic in [a, b] and Qf = D_{a}f = f′(x + a), a ∈ ℝ, a ≠ 0. Then the associated sequence is the Abel sequence [20]

A_{0}(x, a) = 1, A_{n}(x, a) = x(x − an)^{n−1} , n = 1, 2, . . ..

Now, let L be a linear functional verifying L(1) ≠ 0. Then the umbral interpolation polynomials (3.1) and (3.4) become

(4.1)  P_{n}[f](x) = L(f) + ∑_{i=1}^{n} \frac{L(f^{(i)} (x + ai))}{i!} s_{i}(x),

and, setting z = 0,

\overline{P}_{n}[f](x) = f(0) + ∑_{i=1}^{n} \frac{L(f^{(i)} (x + ai))}{i!} (s_{i}(x) − s_{i}(0)),

where s_{i}(x) is the umbral basis for (L, D_{a}).

Abel–Gontscharoff interpolation. Let L(f) = f(x_{0}), x_{0} ∈ [a, b]. Then the umbral basis for (L, D_{a}) is the sequence

G̃_{0}(x) = 1, G̃_{n}(x, a) = (x − x_{0})(x − x_{0} − an)^{n−1} , n = 1, 2, …,

that is the classical Abel–Gontscharoff sequence [11] on the equidistant points x_{i} = x_{0} + ai, i = 0, …, n. Interpolation polynomial (3.1) becomes

G̃_{n}[f](x) = ∑_{i=0}^{n} \frac{f^{(i)} (x_{0} + ai)}{i!} G̃_{i}(x, a),

i.e., the umbral interpolation is the well-known Abel–Gontscharoff interpolation [11] on the equidistant points x_{i} = x_{0} + ai, i = 0, …, n. Therefore (4.1) can be seen as a generalization of Abel–Gontscharoff interpolation on the equidistant points. For the remainder R_{L,n} [f](x) = G̃_{n}[f](x) − f(x), for any x ∈ [x_{0}, b], by Theorem 3.4, we have

R_{L,n} [f](x) = \frac{1}{n!} ∫_{x_{0}}^{b} K_{n}(x, t)f^{(n+1)} (t) dt

where

K_{n}(x, t) = R_{L,n} [(x − t)_{+}^{n} ] = (x − t)_{+}^{n} − ∑_{i=0}^{n} ( {n \atop i}) (x_{0} + ai − t)_{+}^{n−i} G̃_{i}(x, a).

Remark 4.1. Abel–Gontscharoff interpolation, even in recent years, has been object of study [23]. In the future we will consider a comparison with previous works, especially as regards the error estimation.

170 COSTABILE AND LONGO

Abel-Bernoulli interpolation. . Let L(f) = ∫_{0}^{1} f(x) dx. Then the umbral basis for (L, D_{a}) is the Bernoulli–Abel sequence B̃_{n} (x, a) [8].

Interpolation polynomials (3.1) and (3.4) become

B̃_{n} [f](x) = ∫_{0}^{1} f(x)dx + ∑_{i=1}^{n} \frac{f^{(i−1)} (1 + ai) − f^{(i−1)} (ai)}{i!} B̃_{i}(x, a),
(4.2) \overline{B̃}_{n}[f](x) = f(0) + ∑_{i=1}^{n} \frac{f^{(i−1)} (1 + ai) − f^{(i−1)} (ai)}{i!} (B̃_{i}(x, a) − B̃_{i}(0, a)).

For the remainder R_{L,n} [f](x) = B̃_{n} [f](x)−f(x), for any x ∈ [0, b], by Theorem 3.4, we have

R_{L,n} [f](x) = \frac{1}{n!} ∫_{0}^{b}K_{n}(x, t)f^{(n+1)} (t) dt

where

K_{n}(x, t) = R_{L,n} [(x − t)_{+}^{n} ] = (x − t)_{+}^{n} − ∫_{0}^{1}(x − t)_{+}^{n} dx

− ∑_{i=1}^{n} ( {n \atop i − 1} ) \frac{1}{i}[(1 + ai − t)_{+}^{n−i+1} − (ai − t)_{+}^{n−i+1} ]B̃_{i}(x, a).

For the remainder R_{L,n} [f](x) = \overline{B̃}_{n}[f](x) − f(x), for any x ∈ [0, b], by Theorem 3.4, we have

R_{L,n} [f](x) = \frac{1}{n!} ∫_{0}^{b}K_{n}(x, t)f^{(n+1)} (t) dt

where

K_{n}(x, t) = R_{L,n} [(x − t)_{+}^{n} ] = (x − t)_{+}^{n}

− ∑_{i=1}^{n} ( {n \atop i − 1} ) \frac{1}{i}[(1 + ai − t)_{+}^{n−i+1} − (ai − t)_{+}^{n−i+1} ](B̃_{i}(x, a) − B̃_{i}(0, a)).

Abel-Euler interpolation. We consider L(f) = \frac{f(0)+f(1)}{2} . Then the umbral basis for (L, D_{a}) is the Euler–Abel sequence Ẽ_{n} (x, a) [8].

Interpolation polynomials (3.1) and (3.4) become

(4.3)  Ẽ_{n} [f](x) = \frac{f(0) + f(1)}{2} + ∑_{i=1}^{n} \frac{f^{(i)} (ai) + f^{(i)} (1 + ai)}{2i!} Ẽ_{i}(x, a),
\overline{Ẽ}_{n}[f](x) = f(0) + ∑_{i=1}^{n} \frac{f^{(i)} (ai) + f^{(i)} (1 + ai)}{2i!} (Ẽ_{i}(x, a) − Ẽ_{i}(0, a)).

For the remainder R_{L,n} [f](x) = Ẽ_{n} [f](x) − f(x), for any x ∈ [0, b], we have

R_{L,n} [f](x) = \frac{1}{n!} ∫_{0}^{b}K_{n}(x, t)f^{(n+1)} (t) dt

where

K_{n}(x, t) = R_{L,n} [(x − t)_{+}^{n} ]

= (x − t)_{+}^{n} − ∑_{i=0}^{n} ( {n \atop i}) \frac{1}{2}[(ai − t)_{+}^{n−i} + (1 + ai − t)_{+}^{n−i} ]Ẽ_{i}(x, a).

4.2. δ_{h} -Sheffer interpolation. Let be f bounded in [a, b] and Qf = δ_{h}f(x) = \frac{f(x+h/2)−f(x−h/2)}{h} . Moreover, let δ_{h}^{−1} be the inverse operator of δ_{h}, such that

δ_{h}^{−1} φ(x) = f(x) ⇔ δ_{h}f(x) = φ(x).

Then the associated sequence to δ_{h} is the sequence [20]

x^{[0]} = 1, x^{[n]} ≡ x( x + ( \frac{n}{2} − 1) h)_{n−1}

= x( x + ( \frac{n}{2} − 1) h) ⋯ ( x + ( − \frac{n}{2} + 1) h) , n = 1, 2, . . ..

Now, let L be a linear functional verifying L(1) ≠ 0. Then umbral interpolation polynomials (3.1) and (3.4) become

(4.4)  P_{n}[f](x) = ∑_{i=0}^{n} \frac{L(δ_{h}^{i}f)}{i!} s_{i}(x),

\overline{P}_{n}[f](x) = f(0) + ∑_{i=1}^{n} \frac{L(δ_{h}^{i}f)}{i!} (s_{i}(x) − s_{i}(0)),

where s_{i}(x) is the umbral basis for (L, δ_{h}).

As in the previous example we can consider the following cases:

δ_{h}-central interpolation. Let L(f) = f(0). The umbral basis for (L, δ_{h}) is s_{n}(x) = x^{[n]}.

Interpolation polynomial (3.1) becomes

P_{n}[f](x) = f(0) + ∑_{i=1}^{n} \frac{δ_{h}^{i}f(0)}{i!} x^{[i]}.

It is known as interpolation formula with central differences [22, p. 32], therefore (4.4) can be seen as generalization of central interpolation.

δ_{h}-Bernoulli interpolation. Let L(f) = (Dδ_{h}^{−1} f)_{x=0} . We call the umbral basis for (L, δ_{h}) δ_{h}-Bernoulli polynomial sequence B̂_{n} (x). Interpolation polynomials (3.1) and (3.4) become

B̂_{n} [f](x) = (Dδ_{h}^{−1} f)_{x=0} + ∑_{i=1}^{n} \frac{δ_{h}^{i−1} f′(0)}{i!} B̂_{i}(x),

(4.5)  \overline{B̂}_{n}[f](x) = f(0) + ∑_{i=1}^{n} \frac{δ_{h}^{i−1} f′(0)}{i!} (B̂_{i}(x) − B̂_{i}(0)).

δ_{h}-Euler interpolation. Let L(f) = (M_{h}f)_{x=0} = \frac{f(\frac{1}{2} h)+f(− \frac{1}{2} h)}{2} . We call the umbral basis for (L, δ_{h}) δ_{h}-Euler polynomial sequence Ê_{n} (x). Interpolation polynomials (3.1) and (3.4) become

(4.6)  Ê_{n} [f](x) = \frac{f(h/2) + f(−h/2)}{2} + ∑_{i=1}^{n} \frac{δ_{h}^{i}(f(h/2) + f(−h/2))}{2i!} Ê_{i}(x),

(4.7)  \overline{Ê}_{n}[f](x) = f(0) + ∑_{i=1}^{n} \frac{δ_{h}^{i}(f(h/2) + f( − h/2))}{2i!} (Ê_{i}(x) − Ê_{i}(0)).

5. Numerical examples

Now, we consider some interpolation test problems and report the numerical results obtained by using an ad hoc “Mathematica" code. We compare the error in approximating a given function with Appell, Abel–Sheffer, Δ_{h}-Appell and δ_{h}-Sheffer interpolation polynomials. In particular we compare numerical results obtained by applying:

• Abel-Bernoulli interpolation polynomial defined by (4.2) as follows

\overline{B̃}_{n}[f](x) = f(0) + ∑_{i=1}^{n} \frac{f^{(i−1)} (1 + ai) − f^{(i−1)} (ai)}{i!} (B̃_{i}(x, a) − B̃_{i}(0, a));

• Bernoulli interpolation polynomial defined in [2] as follows

B_{n}[f](x) = f(0) + ∑_{i=1}^{n} \frac{f^{(i−1)} (1) − f^{(i−1)} (0)}{i!} (B_{i}(x) − B_{i}(0));

• δ_{h}-Bernoulli interpolation polynomial defined by (4.5) as follows

\overline{B̂}_{n}[f](x) = f(0) + ∑_{i=1}^{n} \frac{δ_{h}^{i−1} f′(0)}{i!} (B̂_{i}(x) − B̂_{i}(0));

• Bernoulli interpolation polynomial of second kind [6] defined as follows

\overline{B}_{n}^{II} [f](x) = f(0) + ∑_{i=0}^{n−1} f′(ih)(B_{i}^{II} (x) − B_{i}^{II} (0)).

• Abel-Euler interpolation polynomial defined by (4.3) as follows

Ẽ_{n} [f](x) = \frac{f(0) + f(1)}{2} + ∑_{i=1}^{n} \frac{f^{(i)} (ai) + f^{(i)} (1 + ai)}{2i!} Ẽ_{i}(x, a);

• Euler interpolation polynomial defined in [5] as follows

E_{n}[f](x) = ∑_{i=0}^{n} \frac{f^{(i)} (0) + f^{(i)} (1)}{2i!} E_{i}(x);

• δ_{h}-Euler interpolation polynomial defined by (4.6) as follows

Ê_{n} [f](x) = \frac{f(h/2) + f(−h/2)}{2} + ∑_{i=1}^{n} \frac{δ_{h}^{i}(f(h/2) + f(−h/2))}{2i!} Ê_{i}(x);

• Boole interpolation polynomial defined in [6] as follows

E_{n}^{II} [f](x) = ∑_{i=0}^{n} \frac{f(ih) + f((i + 1)h)}{2} E_{i}^{II} (x).

We emphasize that the compared polynomials of the same degree have the same degree of exactness.

Example 5.1. Let us consider the function f(x) = e^{(x+1)/2} , x ∈ [0, 1]. The interpolation error is reported in the following tables (numbers in parentheses indicate decimal exponents):

\overline{B̃}_{n}[f](x) \overline{B}_{n}[f](x) \overline{B̂}_{n}[f](x) \overline{B}_{n}^{II} [f](x)
n = 5 2.774(−6) 1.102(−6) 1.628(−5) 6.949(−7)
n = 6 1.460(−7) 8.619(−8) 8.814(−7) 1.733(−8)
n = 7 1.463(−8) 6.885(−9) 4.160(−8) 4.354(−10)
n = 8 7.589(−10) 5.458(−10) 1.742(−9) 8.609(−12)
Ẽ_{n} [f](x) E_{n}[f](x) Ê_{n} [f](x) E_{n}^{II} [f](x)
n = 5 5.978(−5) 4.460(−5) 2.606(−6) 1.318(−7)
n = 6 9.432(−6) 7.103(−6) 1.171(−7) 2.925(−9)
n = 7 1.483(−6) 1.130(−6) 4.715(−9) 5.817(−11)
n = 8 2.354(−7) 1.799(−7) 1.726(−10) 1.041(−12)

Example 5.2. Let us consider the function f(x) = ln(x^{2} + 10), x ∈ [0, 1]. The interpolation error is reported in the following tables:

\overline{B̃}_{n}[f](x) \overline{B}_{n}[f](x) \overline{B̂}_{n}[f](x) \overline{B}_{n}^{II} [f](x)
n = 5 1.994(−6) 4.526(−6) 1.310(−4) 2.823(−6)
n = 6 2.482(−6) 1.760(−6) 1.737(−6) 3.744(−7)
n = 7 5.442(−7) 3.457(−7) 5.477(−6) 1.579(−8)
n = 8 1.267(−7) 2.559(−7) 6.832(−8) 4.487(−9)
Ẽ_{n} [f](x) E_{n}[f](x) Ê_{n} [f](x) E_{n}^{II} [f](x)
n = 5 1.183(−4) 2.138(−4) 1.974(−4) 4.435(−7)
n = 6 1.257(−4) 1.410(−4) 1.5591(−6) 6.225(−8)
n = 7 6.482(−5) 8.666(−5) 5.843(−7) 1.624(−9)
n = 8 5.736(−5) 7.829(−5) 4.823(−8) 1.041(−10)

Example 5.3. Let us consider the function

f(x) = 10 cos(x) + \frac{sin^{2}(x)}{10}, x ∈ [0, 1].

The interpolation error is reported in the following tables:

\overline{B̃}_{n}[f](x) \overline{B}_{n}[f](x) \overline{B̂}_{n}[f](x) \overline{B}_{n}^{II} [f](x)
n = 5 4.652(−4) 2.319(−4) 4.227(−3) 1.483(−4)
n = 6 1.882(−5) 2.451(−6) 3.614(−6) 5.090(−7)
n = 7 1.338(−5) 2.695(−6) 1.224(−5) 9.767(−8)
n = 8 1.981(−6) 2.058(−6) 4.386(−7) 3.400(−8)

Ẽ_{n} [f](x) E_{n}[f](x) Ê_{n} [f](x) E_{n}^{II} [f](x)
n = 5 9.016(−3) 9.265(−3) 6.613(−4) 2.737(−5)
n = 6 1.881(−3) 3.195(−4) 2.770(−6) 6.299(−8)
n = 7 1.490(−3) 7.061(−4) 1.201(−6) 1.661(−8)
n = 8 6.494(−4) 6.746(−4) 3.200(−7) 4.111(−9)

6. Conclusions

Let Q be a δ-operator on P_{n}, X a linear space of functions such that P_{n} ⊆ X and L a linear functional on X, with L(1) ≠ 0. Let be XQ ⊆ X such that Q^{i}f ∈ X, then for all f ∈ XQ we defined umbral interpolation for the couple (L, Q). In particular generalizations of Abel–Gontscharoff and central interpolation have been considered. Further developments are possible, for example the function series associated to the interpolation polynomial can be considered and its convergence radius can be studied. Furthermore, polynomial (4.7) seems to be of interest for applications on IVP. The multivariate case is interesting too.

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Department of Mathematics (Received 09 03 2015) University of Calabria Rende Italy

    francesco.costabile@unical.it
    longo@mat.unical.it