On Landau-type approximation operators

5 years ago

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Abstract

In this paper, we define and study a general class of convolution operators based on Landau operators. A property of these new operators is that they reproduce the affine functions, a feature less commonly encountered by integral type operators. Approximation properties in different function spaces are obtained, including quantitative Voronovskaya-type results.

Authors

Octavian Agratini
Babeş-Bolyai University, Cluj-Napoca, Romania
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy

Sorin G. Gal
University of Oradea, Romania
Academy of Romanian Scientists

Keywords

Landau operator; modulus of continuity; weighted space; approximation process; upper estimates; quantitative Voronovskaya-type theorems

PDF

requires subscription: https://doi.org/10.1007/s00009-021-01712-w

Cite this paper as:

O. Agratini, S.G. Gal, On Landau-type approximation operators, Mediterranean Journal of Mathematics, 18 (2021) art. no. 64, https://doi.org/10.1007/s00009-021-01712-w

About this paper

Journal

Mediterranean Journal of Mathematics

Publisher Name

Springer

DOI

https://doi.org/10.1007/s00009-021-01712-w

Print ISSN

1660-5446

Online ISSN

1660-5454

Google Scholar Profile

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On Landau type operators

Octavian Agratini Babeş-Bolyai University,
Faculty of Mathematics and Computer Science,
Str. Kogălniceanu, 1, 400084 Cluj-Napoca, Romania
and
Tiberiu Popoviciu Institute of Numerical Analysis,
Romanian Academy,
Str. Fântânele, 57, 400320 Cluj-Napoca, Romania agratini@math.ubbcluj.ro and Sorin G. Gal University of Oradea,
Department of Mathematics and Computer Science,
Str. Universităţii, 1, 410087 Oradea, Romania galso@uoradea.ro

Abstract.

In this paper we define and study a general class of convolution operators based on Landau operators. A property of these is that they reproduce the affine functions, a feature less commonly encountered by integral type operators. Approximation properties in different function spaces are obtained.

Keywords and phrases: Landau operator, Korovkin theorem, modulus of smoothness, weighted space, approximation process.

Mathematics Subject Classification: 41A36, 41A25.

1. Introduction

Edmund Landau [2, Eq. (2)] proved

lim_{n \to \infty} \frac{\int_{0}^{1} f (ξ) {(1 - {(ξ - x)}^{2})}^{n} 𝑑 ξ}{2 \int_{0}^{1} {(1 - u^{2})}^{n} 𝑑 u} = f (x),

(1.1)

the convergence being uniform over a compact interval $[a, b]$ , where $0 < a < b < 1$ . In the first instance, $f$ is supposed given only for $x \in [a, b]$ . As the author states, the function $f$ can be extended by continuity from $[a, b]$ to $[0, 1]$ by usual technique:

f (x) = f (a) for x \in [0, a) and f (x) = f (b) for x \in (b, 1] .

The relation (1.1) was used by Landau to recover the celebrated Weierstrass approximation theorem established in 1885 which says that every continuous function defined on a compact interval can be uniformly approximated as closely as desired by a polynomial function. Examining (1.1) we notice that the author uses a sequence of convolution operators which today bears his name.

A generalization of this class of operators was given by Mamedov and it was described by the following relation [5, Eq. (1)]

(M_{n} f) (x) = \frac{k n^{1 / (2 k)}}{Γ (1 / (2 k))} \int_{0}^{1} f (t) {(1 - {(t - x)}^{2 k})}^{n} 𝑑 t,

(1.2)

where $k$ is a fixed natural number, $x \in [0, 1]$ and $Γ$ indicates Gamma function. In time, more papers appeared concerning these operators, among the most recent one we mention [1].

In this note we propose another generalization of Landau operators which involves a real parameter $α \geq 1$ . Keeping the idea of convolution type operators, this class aim to be associated with functions defined on the whole real axis and the affine functions are fixed points of the operators in question. We obtain evaluations of the approximation error for bounded functions as well as for functions belonging to some weighted spaces.

2. The operators

Let $α \geq 1$ be fixed. For each $n \in ℕ$ , we set

a_{n} = \int_{0}^{1} {(1 - y^{2 α})}^{n} 𝑑 y and τ_{p, n} = \frac{B (n + 1, p λ)}{B (n + 1, λ)}, p \in ℕ,

(2.1)

where $λ = \frac{1}{2 α} \in (0, \frac{1}{2}]$ and $B$ indicates Beta function. Taking in view these quantities, the following statements hold, their proofs are based on elementary calculus.

The sequence ${(a_{n})}_{n \geq 1}$ is strictly decreasing and

a_{n} = λ B (n + 1, λ) .

(2.2)

We have

\int_{0}^{1} y^{i} {(1 - y^{2 α})}^{n} 𝑑 y = λ B (n + 1, (1 + i) λ),

(2.3)

and, for $p \geq 2$ , ${(τ_{p, n})}_{n \geq 1}$ is a positive strictly decreasing sequence verifying the relations

τ_{p, n + 1} = τ_{p, n} \frac{λ + n + 1}{p λ + n + 1} < τ_{p, 1} = \frac{λ + 1}{p (1 + p λ)} < \frac{1}{p} .

(2.4)

Lemma 2.1.

Let the real sequence ${(τ_{p, n})}_{n \geq 1}$ be given by (2.1). For any $λ > 0$ and any $p = 2, 3, \dots$ , we have

lim_{n \to \infty} τ_{p, n} = 0 .

(2.5)

Proof. Relation (2.4) implies

τ_{p, n} = \prod_{k = 0}^{n} \frac{λ + k}{p λ + k} .

On the other hand, the following identity

Γ (λ) = lim_{n \to \infty} \frac{n! n^{λ}}{λ (λ + 1) \dots (λ + n)}

takes place, see, e.g., [6, Exercise 1.7]. Consequently, we get

\frac{Γ (p λ)}{Γ (λ)} = lim_{n \to \infty} n^{(p - 1) λ} τ_{p, n} .

We mention that the existence of the limit of the sequence ${(τ_{p, n})}_{n \geq 1}$ is based on statement (2.4). Since $Γ (p λ) / Γ (λ)$ belongs to $ℝ$ and $lim_{n \to \infty} n^{(p - 1) λ} = \infty$ for any $λ > 0$ and $p = 2, 3, \dots$ , we deduce (2.5). $□$

$B (ℝ)$ denotes the space of all real valued functions defined on $ℝ$ and bounded. The space is endowed with the usual sup-norm $∥ \cdot ∥$ ,

‖ h ‖ = sup_{x \in ℝ} | h (x) | .

We consider the operators defined as follows

(L_{n} f) (x) = \frac{1}{a_{n}} \int_{0}^{1} f (x - y + τ_{2, n}) {(1 - y^{2 α})}^{n} 𝑑 y, x \in ℝ,

(2.6)

where $f$ is Lebesgue measurable on the domain.

The operators can be rewritten as follows

(L_{n} f) (x) = \frac{1}{a_{n}} \int_{x - 1}^{x} f (t + τ_{2, n}) {(1 - {(x - t)}^{2 α})}^{n} 𝑑 t, x \in ℝ .

(2.7)

It is observed that for each $n \in ℕ$ , $L_{n}$ represents a convolution product between the functions $f \circ u_{n}$ and $v_{n}$ , where

u_{n} (t) = t + τ_{2, n} and v_{n} (t) = {(1 - t^{2 α})}^{n}, t \in ℝ .

Also, the operators are linear and positive. If $f \in B (ℝ)$ , then the operators are non-expansive, this means

‖ L_{n} f ‖ \leq ‖ f ‖ .

Replacing the sequence ${(τ_{2, n})}_{n \geq 1}$ with zero and choosing $α = k \in ℕ$ , we obtain the operators introduced by Mamedov, see (1.2).

3. Results

Set $ℕ_{0} = {0} \cup ℕ$ and $e_{j}$ , $j \in ℕ_{0}$ , monomials of degree $j$ , $e_{0} (x) = 1$ , $e_{j} (x) = x^{j}$ , $j \in ℕ$ .

Theorem 3.1.

The operators $L_{n}$ , $n \in ℕ$ , defined by (2.6) reproduce affine functions.

Proof. Taking in view that these operators are linear, it is enough to prove $L_{n} e_{0} = e_{0}$ and $L_{n} e_{1} = e_{1}$ .

Due to the definition of $a_{n}$ , it is clear that $L_{n} e_{0} = e_{0}$ . Further, by using (2.3) and (2.2), we get

(L_{n} e_{1}) (x) = - \frac{1}{a_{n}} \int_{0}^{1} y {(1 - y^{2 α})}^{n} 𝑑 y + x + τ_{2, n} = x,

and the proof is completed. $□$

The purpose of introducing the quantity $τ_{2, n}$ in the definition of $L_{n}$ , $n \in ℕ$ , was precisely to ensure the identity $L_{n} e_{1} = e_{1}$ .

To determine the approximation error, we need to evaluate the second order central moment of our operators.

Lemma 3.2.

For the operators $L_{n}$ , $n \in ℕ$ , defined by (2.6), the following relation

(L_{n} φ_{x}^{2}) (x) = τ_{3, n} - τ_{2, n}^{2}

(3.1)

occurs, where $φ_{x} (t) = | t - x |$ .

Proof. Clearly,

(L_{n} φ_{x}^{2}) (x) = (L_{n} e_{2}) (x) - 2 x (L_{n} e_{1}) (x) + x^{2} (L_{n} e_{0}) (x) .

(3.2)

By using (2.3) and (2.2) we can write

(L_{n} e_{2}) (x) = x^{2} + τ_{3, n} - τ_{2, n}^{2} .

(3.3)

Returning at (3.2) and using Theorem 3.1, we arrive at the desired result. $□$

Theorem 3.3.

Let the operators $L_{n}$ , $n \in ℕ$ , be defined by (2.6). For any compact interval $K \subset ℝ$ , the following relation

lim_{n \to \infty} L_{n} f = f uniformly on K

(3.4)

occurs, provided $f$ is continuous on $ℝ$ .

Proof. We use Korovkin theorem [4]. In accordance with Theorem 3.1, the first two Korovkin test functions are fixed points of the operators. Relation (3.3) and Lemma 2.1 involve

lim_{n \to \infty} (L_{n} e_{2}) (x) = x^{2}, x \in ℝ .

We define the lattice homomorphism $T_{K} : C (ℝ) \to C (K)$ given by

T_{K} (f) = {f |}_{K}

for every $f \in C (ℝ)$ . Based on above statements, we can write

lim_{n \to \infty} T_{K} (L_{n} e_{j}) = T_{K} (e_{j}), j \in {0, 1, 2},

uniformly on $K$ and Korovkin criterion implies (3.4). $□$

We establish the error of approximation with the help of modulus of smoothness defined as follows

	$ω_{f} (δ) \equiv ω (f; δ)$	$= sup {\| f (x^{'}) - f (x^{''}) \| : x^{'}, x^{''} \in ℝ, \| x^{'} - x^{''} \| \leq δ}$
		$= sup_{0 \leq h \leq δ} sup_{x \in ℝ} \| f (x + h) - f (x) \|, δ \geq 0,$

where $f \in B (ℝ)$ .

Theorem 3.4.

Let the operators $L_{n}$ , $n \in ℕ$ , be defined by (2.6). For any Lebesgue integrable function $f$ belonging to $B (ℝ)$ , we get

| (L_{n} f) (x) - f (x) | \leq 2 ω (f; \sqrt{τ_{3, n} - τ_{2, n}^{2}}), x \in ℝ .

(3.5)

Proof. To achieve the statement, we use the following inequality proved by Shisha and Mond [7], that says: if $S$ is a linear positive operator, then one has

	$\| (S f) (x) - f (x) \|$	$\leq \| f (x) \| \| (S e_{0}) (x) - 1 \|$
		$+ ((S e_{0}) (x) + \frac{1}{δ} \sqrt{(S e_{0}) (x) (S φ_{x}^{2}) (x)}) ω (f; δ),$		(3.6)

$δ > 0$ , for every bounded function $f$ . The proof of (3) is mainly based on the following relations:

	$\| f (x) - f (y) \| \leq ω (f; \| x - y \|),$
	$ω (f; μ δ) \leq (1 + μ) ω (f; δ), δ \geq 0, μ \geq 0 .$

Applying the inequality (3) for $L_{n}$ operators, by choosing

δ = \sqrt{τ_{3, n} - τ_{2, n}^{2}}

and taking into account both Theorem 3.1 and the identity (3.1), we obtain the inequality (3.5). $□$

Further, we analyze the behavior of operators in some weighted spaces. For a given $m \in ℕ$ , we consider the weight

ρ_{m} (x) = 1 + x^{2 m}, x \in ℝ,

and the space

B_{ρ_{m}} (ℝ) = {f : ℝ \to ℝ ∣ | f (x) | \leq M_{f} ρ_{m} (x), x \in ℝ},

$M_{f}$ being a positive constant depending only on $f$ . The usual norm of this space is $∥ \cdot ∥_{ρ_{m}}$ defined by

{‖ h ‖}_{ρ_{m}} = sup_{x \in ℝ} \frac{| h (x) |}{ρ_{m} (x)} .

The operators $L_{n}$ , $n \in ℕ$ , are well defined for any Lebesgue integrable function belonging to $B_{ρ_{m}} (ℝ)$ and it is easy to see that

(L_{n} f) (x) \leq {‖ f ‖}_{ρ_{m}} (L_{n} ρ_{m}) (x), x \in ℝ,

(3.7)

takes place.

Lemma 3.5.

Each operator $L_{n}$ defined by (2.6) maps $B_{ρ_{m}} (ℝ)$ into $B_{ρ_{m}} (ℝ)$ .

Proof. Let $n \in ℕ$ be fixed. In view of (3.7), it is enough to show that

(L_{n} ρ_{m}) (x) \leq M ρ_{m} (x), x \in ℝ,

(3.8)

where $M$ is a constant depending on $m$ . We can write successively

	$0 \leq (L_{n} ρ_{m}) (x)$	$= 1 + \frac{1}{a_{n}} \int_{0}^{1} \sum_{j = 0}^{2 m} (\binom{2 m}{j}) {(- y)}^{2 m - j} {(x + τ_{2, n})}^{j} {(1 - y^{2 α})}^{n} d y$
		$\leq 1 + \frac{1}{2 m + 1} \sum_{j = 0}^{2 m} (\binom{2 m + 1}{j}) \sum_{i = 0}^{j} (\binom{j}{i}) \frac{1}{2^{j - i}} {\| x \|}^{i} .$

We used (2.4) both for $p = 2 m + 1 - j$ $(j = \bar{0, 2 m})$ and for $p = 2$ .
The relation we reached allows us to assert that there this a constant $M$ satisfying (3.8). $□$

For weighted functions belonging to the space $B_{ρ_{m}} (ℝ)$ , we give estimates of the error $| L_{n} f (x) - f (x) |$ , $n \in ℕ$ , involving the following weighted modulus of smoothness

Ω_{m} (f; δ) = sup_{\begin{matrix} x \in ℝ \\ 0 \leq h \leq δ \end{matrix}} \frac{| f (x + h) - f (x) |}{1 + {(x + h)}^{2 m}}, δ \geq 0 .

(3.9)

Best of our knowledge, this type of modulus associated to a function defined on $ℝ_{+} = [0, \infty)$ , appeared for the first time in the papers [3], [8]. Its definition formula ensures that it can be also used for functions defined on $ℝ$ which is our case. Based on (3.9), it is obvious that $Ω_{m} (f; \cdot)$ is a monotone increasing function. Following the same line as in paper of Yuksel and Ispir [8, Lemma 2 (iv)] we deduce the following property

Ω_{m} (f; λ δ) \leq (λ + 1) Ω_{m} (f; δ), δ \in ℝ_{+}, λ \in ℝ_{+} .

(3.10)

Theorem 3.6.

Let $L_{n}$ , $n \in ℕ$ , be given by (2.6). For any function $f \in B_{ρ_{m}} (ℝ)$ and Lebesgue integrable, a constant $C$ depending on $m$ exists such that

| (L_{n} f) (x) - f (x) | \leq C (1 + x^{2 m}) Ω_{m} (f; \sqrt{τ_{3, n} - τ_{2, n}^{2}}), x \in ℝ,

(3.11)

where $τ_{p, n}$ is given by (2.1).

Proof. Let $x \in ℝ$ be fixed arbitrarily. Besides the function $φ_{x}$ defined in Lemma 3.2, we introduce the function $ψ_{x, m}$ given as follows

ψ_{x, m} (t) = 1 + {(2 | x | + | t |)}^{2 m}, (x, t) \in ℝ \times ℝ .

By using the definition of $Ω_{m} (f; \cdot)$ and property (3.10), for $δ > 0$ we get

	$\| f (t) - f (x) \|$	$\leq (1 + {(\| x \| + \| t - x \|)}^{2 m}) (1 + \frac{1}{δ} \| t - x \|) Ω_{m} (f; δ)$
		$\leq ψ_{x, m} (t) (1 + \frac{1}{δ} φ_{x} (t)) Ω_{m} (f; δ) .$		(3.12)

Since the operators $L_{n}$ are linear and positive, consequently monotone, relation (3) and Cauchy inequality allow us to write

	$\| (L_{n} f) (x) - f (x) \|$	$\leq L_{n} (\| f - f (x) \|; x)$
		$\leq ((L_{n} ψ_{x, m}) (x) + \frac{1}{δ} \sqrt{L_{n} ψ_{x, m}^{2} (x)} \sqrt{(L_{n} φ_{x}^{2}) (x)}) Ω_{m} (f; δ) .$

Lemma 3.5 implies the existence of some constants depending on $m$ such that

L_{m} ψ_{x, m} \leq M_{1} ρ_{m} and \sqrt{L_{m} ψ_{x, m}^{2}} \leq M_{2} ρ_{m} .

Taking in view (3.1) and choosing $δ = \sqrt{τ_{3, n} - τ_{2, n}^{2}}$ , the relation (3.11) follows. $□$

Let denote by $C_{ρ_{m}}^{*} (ℝ)$ the subspace of all continuous functions belonging to $B_{ρ_{m}} (ℝ)$ with the property that $lim_{| x | \to \infty} | f (x) | / (1 + x^{2 m})$ exists and it is finite. Based on [8, Lemma 2 (ii)], for each $f \in C_{ρ_{m}}^{*} (ℝ)$

lim_{δ \to 0^{+}} Ω_{m} (f; δ) = 0

takes place. Relation (3.11) corroborated with Lemma 2.1 leads us to the following identity

lim_{n \to \infty} {‖ L_{n} f - f ‖}_{ρ_{m}} = 0, f \in C_{ρ_{m}}^{*} (ℝ) .

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