Return to Article Details New estimates related with the best polynomial approximation

New estimates related with
the best polynomial approximation

Jorge Bustamante $^{*}$

March 21, 2023; accepted: May 2, 2023; published online: July 5, 2023.

In some old results, we find estimates the best approximation $E_{n, p} (f)$ of a periodic function satisfying $f^{(r)} \in L_{2 π}^{p}$ in terms of the norm of $f^{(r)}$ (Favard inequality). In this work, we look for a similar result under the weaker assumption $f^{(r)} \in L_{2 π}^{q}$ , with $1 < q < p < \infty$ . We will present inequalities of the form $E_{n, p} (f) \leq C (n) ‖ D^{(r)} f ‖_{q}$ , where $D^{(r)}$ is a differential operator. We also study the same problem in spaces of non-periodic functions with a Jacobi weight.

MSC. 42A10, 41A10.

Keywords. Favard inequality; best approximation.

$^{*}$ Benemerita Universidad Autonoma de Puebla, Faculty of Physics and Mathematics, Puebla, Mexico, e-mail: jbusta@fcfm.buap.mx.

1 Introduction

Let $C_{2 π}$ denote the Banach space of all real $2 π$ -periodic continuous functions $f$ defined on the real line $R$ with the sup norm

‖ f ‖_{\infty} = max_{x \in [- π, π]} ∣ f (x) ∣ .

For $1 \leq p < \infty$ , the Banach space $L_{2 π}^{p}$ consists of all $2 π$ -periodic, $p$ -th power Lebesgue integrable (class of) functions $f$ on $R$ with the norm

‖ f ‖_{p} = (\frac{1}{2 π} \int_{- π}^{π} ∣ f (x) ∣^{p} d x)^{1 / p} .

In order to simplify we write $X_{2 π}^{p} = L_{2 π}^{p}$ for $1 \leq p < \infty$ and $X_{2 π}^{\infty} = C_{2 π}$ .

Let $T_{n}$ denote the family of all trigonometric polynomials of degree non greater that $n$ . For $n \in N_{0}$ and $f \in X_{2 π}^{p}$ , the best approximation is defined by

E_{n, p} (f) = inf_{T \in T_{n}} ‖ f - T ‖_{p} .

By $W_{p, 2 π}^{r}$ we mean the family of all functions $f \in X_{2 π}^{p}$ such that $f, \dots, f^{(r - 1)}$ are absolutely continuous and $f^{(r)} \in X_{2 π}^{p}$ .

The following result is known (for instance, see [ 4 , p. 166 ] where other notations were used).

Theorem 1

If $1 \leq p \leq \infty$ , $r, n \in N$ and $g \in W_{p, 2 π}^{r}$ , then

E_{n, p} (g) \leq \frac{F_{r}}{(n + 1)^{r}} E_{n, p} (g^{(r)}),

where $F_{r}$ is the Favard constant.

Here we want to consider the following problem. Is there an estimate similar to 1, if $1 \leq q < p \leq \infty$ , $r, n \in N$ , $g \in X_{2 π}^{p}$ and $g \in W_{q, 2 π}^{r}$ (with the necessary adjustment)?

In [ 1 ] , Ganzburg considered a similar problem, when the derivative of a continuous functions is not continuous.

Theorem 2 Ganzburg [ 1 ]

If $f \in C_{2 π}$ is absolutely continuous and $f^{'} \in L_{2 π}^{1}$ , then

E_{n, \infty} (f) \leq \frac{1}{2} E_{n, 1} (f^{'}) .

In section 2 we extend the result of Ganzburh to the case when $1 < q < p < \infty$ , $f \in X_{2 π}^{p}$ and $f \in W_{q, 2 π}^{r}$ .

For $1 \leq p < \infty$ , we denote by $L_{α, β}^{p}$ the space of all measurable functions $f$ satisfying

‖ f ‖_{p, α, β} = {\int_{- 1}^{1} ∣ f (x) ∣^{p} ϱ_{α, β} (x) d x}^{1 / p} < \infty,

where $ϱ_{α, β} (x) = (1 - x)^{α} (1 + x)^{β}$ and $α, β > - 1$ .

For $n \in N_{0}$ and $f \in L_{(α, β)}^{p}$ , define the best approximation of $f$ of order $n$ by

E_{n} (f)_{p, α, β} = inf {‖ f - P_{n} ‖_{p, α, β}; P_{n} \in P_{n}},

here $P_{n}$ is the family of all algebraic polynomials of degree not bigger than $n$ .

In section 3, for $1 < q < p$ and functions $f \in L_{(α, β)}^{p}$ such that $D_{α, β}^{r} (f) \in L_{(α, β)}^{q}$ (see 8), we estimate $E_{n} (f)_{p, α, β}$ in terms of $E_{n} (D_{α, β}^{r} (f))_{q, α, β}$ . We will assume that $α \geq β \geq - 1 / 2$ .

2 The periodical case

Let

B_{r} (t) = \sum_{k = 1}^{\infty} \frac{\cos (k t - r π / 2)}{k^{r}}

be the Bernoulli kernel.

Theorem 3

Assume $1 < q < p < \infty$ and let $s$ satisfy

\frac{1}{q} - \frac{1}{p} = 1 - \frac{1}{s} .

If $r \in N$ , $f \in X_{2 π}^{p}$ and $f \in W_{q, 2 π}^{r}$ , then

E_{n, p} (f) \leq 2 E_{n, s} (B_{r}) E_{n, q} (f^{(r)}) .

for each $n \in N$ .

Proof. Fix polynomials $T_{n}, B_{n} \in T_{n}$ such that

E_{n, q} (f^{(r)}) = ‖ f^{(r)} - T_{n} ‖_{q} a n d ‖ B_{r} - B_{n} ‖_{s} = E_{n, s} (B_{r}) .

Define

\begin{aligned} M_{n} (f, x) & = \frac{1}{π} \int_{- π}^{π} f^{(r)} (t) B_{n} (x - t) d t, \\ N_{n} (f, x) & = \frac{1}{π} \int_{- π}^{π} T_{n} (t) (B_{r} (x - t) - B_{n} (x - t)) d t, \end{aligned}

and

L_{n} (f, x) = \frac{1}{2 π} \int_{0}^{2 π} f (t) d t + M_{n} (f, x) + N_{n} (f, x) .

Since $M_{n} (f, x)$ and $N_{n} (f, x)$ are convolution with trigonometric polynomials of degree not bigger than $n$ , one has $M_{n} (f), N_{n} (f), L_{n} (f) \in T_{n}$ .

Recall that (see [ 4 , p. 30 ] )

f (x) = \frac{1}{2 π} \int_{- π}^{π} f (t) d t + \frac{1}{π} \int_{- π}^{π} f^{(r)} (t) B_{r} (x - t) d t .

Therefore

\begin{aligned} L_{n} (f, x) - f (x) = \\ = \frac{1}{π} \int_{- π}^{π} f^{(r)} (t) B_{n} (x - t) + T_{n} (t) (B_{r} (x - t) - B_{n} (x - t)) - (f^{(r)} (t) B_{r} (x - t)) d t \\ = \frac{1}{π} \int_{- π}^{π} (T_{n} (t) - f^{(r)} (t)) (B_{r} (x - t) - B_{n} (x - t)) d t . \end{aligned}

That is,

\frac{1}{2} (L_{n} (f, x) - f (x)) = (T_{n} - f^{(r)}) * (B_{r} - B_{n}) (x) .

Now an application of the Young inequality yields

\frac{1}{2} ‖ L_{n} (f) - f ‖_{p} \leq ‖ T_{n} - f^{(r)} ‖_{q} ‖ B_{r} - B_{n} ‖_{s} = E_{n, s} (B_{r}) E_{n, q} (f^{(r)}) . ◻

In theorem 4 we extend the result of Ganzburg presented in the introduction. We consider $r > 1$ , because $B_{1}$ is not a continuous function. On the other hand the series 2 converges uniformly for $r \geq 2$ .

Theorem 4

If $r \in N$ , $r > 1$ , $1 < p \leq \infty$ , $f \in X_{2 π}^{p}$ and $f \in W_{1}^{r}$ , then

E_{n, \infty} (f) \leq 2 E_{n, p} (B_{r}) E_{n, 1} (f^{(r)}) .

for each $n \in N$ .

Proof. It is known (see [ 4 , p. 36 ] ) that if $f \in X_{2 π}^{p}$ and $h \in L^{1}$ , then $f * h \in X_{2 π}^{p}$ and

‖ f * h ‖_{p} \leq ‖ h ‖_{1} ‖ f ‖_{p} .

If $f \in X_{2 π}^{p}$ and $f \in W_{1}^{r}$ , we fix polynomials $T_{n}, B_{n} \in T_{n}$ such that

E_{n, 1} (f^{(r)}) = ‖ f^{(r)} - T_{n} ‖_{1} a n d ‖ B_{r} - B_{n} ‖_{p} = E_{n, p} (B_{r}) .

It follows from 4 that

E_{n, p} (f) \leq 2 ‖ T_{n} - f^{(r)} ‖_{1} ‖ B_{r} - B_{n} ‖_{p} = 2 E_{n, p} (B_{r}) E_{n, 1} (f^{(r)}) . ◻

Remark 5

We have not found a study of the quantities $E_{n, s} (B_{r})$ for $s > 1$ . But, if $r, s > 1$ and $S_{n} (x)$ is the partial sum of the series 2, then

\begin{aligned} E_{n, s} (B_{r}) & \leq ‖ B_{r} - S_{n} ‖_{s} = (\frac{1}{2 π} \int_{- π}^{π} | \sum_{k = n + 1}^{\infty} \frac{\cos (k x)}{k^{r}} |^{s} d x)^{1 / s} \\ \leq (\frac{1}{2 π} \int_{- π}^{π} (\sum_{k = n + 1}^{\infty} \frac{1}{k^{r}})^{s} d x)^{1 / s} = \sum_{k = n + 1}^{\infty} \frac{1}{k^{r}} \leq \frac{1}{(r - 1) (n + 1)^{r - 1}} . \end{aligned}

â–¡

3 The case of non-periodic functions

Let ${J_{n}^{(α, β)}}$ be the orthogonal system of Jacobi polynomials on $[- 1, 1]$ with respect to the weight $ϱ_{α, β} (x) = (1 - x)^{α} (1 + x)^{β}$ , normalized by the conditions $J_{n}^{(α, β)} (1) = 1$ .

If $α \geq β \geq - 1 / 2$ and $1 \leq p \leq \infty$ , it is known that the exists a generalized translation $τ_{t} : L_{α, β}^{p} \to L_{α, β}^{p}$ with the following properties

‖ τ_{t} (f) ‖_{p, α, β} \leq ‖ f ‖_{p, α, β}, t \in [- 1, 1], f \in L_{α, β}^{p},

and

c_{k} (τ_{t} (f)) = c_{k} (f) J_{k}^{(α, β)} (t), k \in N_{0},

where

c_{k} (g) = c_{k}^{(α, β)} (g) = \int_{- 1}^{1} g (x) J_{k}^{(α, β)} (x) ϱ_{α, β} (x) d x, k \in N_{0} .

for all $g \in L_{α, β}^{1}$ .

The generalized translation allows us to define a convolution in $L_{α, β}^{p}$ by setting

(f * g) (x) = \int_{- 1}^{1} τ_{s} (f, x) g (s) ϱ_{α, β} (s) d s .

If $f \in L_{(α, β)}^{p}$ and $g \in L_{(α, β)}^{1}$ , for almost every $x \in [- 1, 1]$ the integral

(f * g) (x) = \int_{- 1}^{1} τ_{y} (f, x) g (y) ϱ^{(α, β)} (y) d y

exists, $f * g \in L_{(α, β)}^{1}$ , $f * g = g * f$ and

‖ f * g ‖_{p} \leq ‖ g ‖_{1} ‖ f ‖_{p} .

Moreover

c_{k} (f * g) = c_{k} (f) c_{k} (g), k \in N_{0} .

For the properties of the translation quoted above see [ 2 ] and [ 3 ] .

We also need the following property (see [ 5 ] ). Assume $1 \leq r, q < \infty$ , $r^{- 1} + q^{- 1} > 1$ , and $p^{- 1} = r^{- 1} + q^{- 1} - 1$ . If $φ \in L_{(α, β)}^{r}$ and $g \in L_{(α, β)}^{q}$ , then $φ * g \in L_{(α, β)}^{p}$ and

‖ φ * g ‖_{p, α, β} \leq ‖ φ ‖_{r, α, β} ‖ g ‖_{q, α, β} .

As in Rafalson [ 5 ] , let $Ω_{α, β}^{r}$ be the family of all functions $f \in C^{2 r - 1} (- 1, 1)$ such that, for $0 \leq k \leq r - 1$ , the function

ψ_{r}^{k} (f, x) := ((1 - x)^{α + r} (1 + x)^{β + r} f^{(r)} (x))^{(k)}

is absolutely continuous in $(- 1, 1)$ and $ψ_{r}^{k} (f, \pm 1) = 0$ .

For $f \in Ω_{α, β}^{r}$ , define

D_{α, β}^{r} (f, x) = \frac{1}{(1 - x)^{α}} \frac{1}{(1 + x)^{β}} ((1 - x)^{α + r} (1 + x)^{β + r} f^{(r)} (x))^{(k)} .

For $r \in N$ and $t \in (- 1, 1)$ , consider the function

Φ_{α, β}^{r} (t) = C_{r} \int_{- 1}^{t} \frac{(t - z)^{r - 1}}{(1 - z)^{1 + α} (1 + z)^{r + β}} \int_{- 1}^{z} (1 - u)^{α} (1 + z)^{r + β - 1} d u d z,

where

C_{r} = C (r, α, β) := \frac{(- 1)^{r}}{2^{α + β + r}} \frac{Γ (r + α + β + 1)}{Γ^{2} (r) Γ (α + 1) Γ (r + β)} .

It is known [ 5 , Lemma 8 ] that, if $f \in Ω_{α, β}^{r}$ , then

f (x) - S_{r - 1}^{(α, β)} (f, x) = D_{α, β}^{r} (f) * Φ_{α, β}^{r} (x)

almost everywhere with respect to the Lebesgue measure. Here $S_{r}^{(α, β)} (f)$ is the partial sum of order $r$ of the Jacobi expansion of $f$ .

Let us recall a known result.

Theorem 6

(Rafalson, [ 5 , Th. 1 ] ) Assume $α \geq β \geq - 1$ , $r, n \in N$ , $n \geq r - 1$ , and one of the following conditions hold:

(i) r > α + 1, a n d 1 \leq p \leq \infty,

(i i) r = α + 1, a n d 1 \leq p < \infty,

(i i i) r > α + 1, a n d 1 \leq r < \frac{1 + α}{1 + α - r} .

If $f \in Ω_{α, β}^{r}$ , then

E_{n} (f)_{p, α, β} \leq E_{n} (Φ_{α, β}^{r})_{p, α, β} E_{n} (D_{α, β}^{r} (f))_{p, α, β} .

We present an analogous of theorem 3 in the frame of Jacobi spaces.

Theorem 7

Assume $α \geq β \geq - 1 / 2$ , $1 < q < p < \infty$ and chose $s$ from the condition $(???)$ . If $r \in N$ , $f \in Ω_{α, β}^{r}$ and $D_{α, β}^{r} (f) \in L_{(α, β)}^{q}$ , then

E_{n} (f)_{p, α, β} \leq E_{n} (Φ_{α, β}^{r})_{s, α, β} E_{n} (D_{α, β}^{r} (f))_{q, α, β} .

for each $n \in N$ , $n \geq r - 1$ .

Proof. Let $P_{n} \in P_{n}$ ( $B_{n} \in P_{n}$ ) be the polynomial of the best approximation of order $n$ of $D_{α, β}^{r} (f)$ ( $Φ_{α, β}^{r}$ ) in the norm of $L_{(α, β)}^{q}$ ( $L_{(α, β)}^{s}$ ). That is,

E_{n} (D_{α, β}^{r} (f))_{q, α, β} = ‖ D_{α, β}^{r} (f) - P_{n} ‖_{q, α, β}

and

E_{n} (Φ_{α, β}^{r})_{s, α, β} = ‖ Φ_{α, β}^{r} - B_{n} ‖_{s, α, β}

Define

\begin{aligned} M_{n} (g, x) & = \int_{- 1}^{1} (D_{α, β}^{r} (f) (t) τ_{t} (B_{n}, x) ϱ^{(α, β)} (t) d t, \\ N_{n} (g, x) & = \int_{- 1}^{1} P_{n} (t) τ_{t} (Φ_{α, β}^{r} - B_{n}) (x) ϱ^{(α, β)} (t) d t, \end{aligned}

and

L_{n} (g, x) = S_{r - 1}^{(α, β)} (f, x) + M_{n} (g, x) + N_{n} (g, x) .

Since $M_{n} (g)$ and $N_{n} (g)$ are convolutions with algebraic polynomials of degree non greater than $n$ and $n \geq r - 1$ , it follows from 4 that $M_{n} (g), N_{n} (g)$ , $L_{n} (g) \in P_{n}$ .