PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 96 (110) (2014), 169–180 DOI: 10.2298/PIM1410169K

NEW MODULI OF SMOOTHNESS

K. A. Kopotun, D. Leviatan, and I. A. Shevchuk

Abstract. We discuss various properties of the new modulus of smoothness

ω_{k,r}^{φ} (f^{(r)} , t)_{p} := sup_{0<h⩽t} ∥W_{kh}^{r} (·)Δ_{hφ(·)}^{k} (f^{(r)} , ·)∥_{𝕃_{p} [−1,1]} ,

where φ(x) := \sqrt{1 − x^{2}} and W_{δ}(x) = ((1 − x − δφ(x)/2)(1 + x − δφ(x)/2))^{1/2} . Related moduli with more general weights are also considered.

1. Introduction

1.1. Trigonometric approximation. Let 𝕃̃_{p}, 1 ⩽ p ⩽ ∞, denote the space of 2π-periodic measurable functions for which the norm ∥f∥_{𝕃_{p}} := ( ∫_{−π}^{π} |f(x)|^{p}dx)^{1/p}

is finite. Here, by 𝕃̃_{∞} we mean the space of continuous 2π-periodic functions ℂ̃ equipped with the uniform norm, i.e., ∥f∥_{ℂ} := max_{x∈[−π,π]} |f(x)|.

Let T_{n}, n ∈ ℕ, be the space of (n − 1)st degree trigonometric polynomials

T_{n}(x) = ∑_{j=0}^{n−1} (a_{j} cosjx + b_{j} sinjx).

For f ∈ 𝕃̃_{p}, denote by

(1.1)  Δ_{h}^{k}(f, x) = ∑_{i=0}^{k} ( {k \atop i}) (−1)^{k−i} f(x + (i − k/2)h)

the kth symmetric difference of the function f, and by

ω_{k}(f, t)_{p} := inf_{h∈[0,t]} ∥Δ_{h}^{k}(f, ·)∥_{𝕃_{p}}

its kth modulus of smoothness. Finally, let Ẽ_{n} (f)_{p} := inf_{T_{n} ∈T_{n}} ∥f − T_{n}∥_{𝕃_{p}} denote the degree of approximation of f by trigonometric polynomials from T_{n}.

In 1908, de la Vallée Poussin (see [20, Section 7], for example) posed a problem on a connection between the rate of polynomial approximation of functions and their differential properties. To quote de la Vallée Poussin [20, p. 119], “It is the memoir by D. Jackson [9] which answers most completely the direct question, and that of S. Bernstein [3] which answers most completely the inverse problem”. These results were generalized by de la Vallée Poussin in [21], though as he writes in [20, p. 119], “I combined the results obtained by the two authors above named, and filled them out in many points; I changed or simplified the proofs; but I contributed little in the way of new materials to the construction”.

2010 Mathematics Subject Classification: Primary 41A17; Secondary 41A10, 42A10, 41A25, 41A27.

The first author acknowledges support of NSERC of Canada.

In 1911, Jackson [9, Theorem VIII] (see also [8, p. 428]) proved the following inequality (which is now commonly known as one of “Jackson’s inequalities”):

Ẽ_{n} (f)_{∞} ⩽ cω_{1}(f, n^{−1} )_{∞} , n ⩾ 1.

This result was later extended by Zygmund [22, Theorems 8 and 8′], Bernstein [2], Akhiezer [1, Section 89], and Stechkin [15, Theorem 1] as follows.

Theorem D̃_{0} (Direct theorem, r = 0). Let k ∈ ℕ. If f ∈ 𝕃̃_{p}, 1 ⩽ p ⩽ ∞,

then Ẽ_{n} (f)_{p} ⩽ c(k)ω_{k}(f, n^{−1} )_{p}, n ⩾ 1.

We note that “r = 0” and the subscript “0” in “D̃_{0}” will become clear once one compares this result with Theorem D̃_{r} below.

Matching inverse theorems are due to Bernstein [3], de la Vallée Poussin [21, Section 39], Quade [19, Theorem 1], Salem [14, Chapter V], Zygmund [22, Theorems 8, 8′, 9, and 9′], the Timan brothers [18], and Stechkin [15, Theorem 8].

Theorem 1.1. Let k ∈ ℕ and f ∈ 𝕃̃_{p}, 1 ⩽ p ⩽ ∞. Then

ω_{k}(f, n^{−1} )_{p} ⩽ \frac{c(k)}{n^{k}} ∑_{ν=1}^{n} ν^{k−1} Ẽ_{ν} (f)_{p}, n ⩾ 1.

This theorem can be restated in the following form.

Theorem Ĩ_{0} (Inverse theorem, r = 0). Let k ∈ ℕ and let ϕ : [0, 1] → [0, ∞) be a nondecreasing function such that ϕ(0+) = 0. If a function f ∈ 𝕃̃_{p}, 1 ⩽ p ⩽ ∞, is such that Ẽ_{n} (f)_{p} ⩽ ϕ (n^{−1} ), n ⩾ 1, then

ω_{k}(f, t)_{p} ⩽ c(k)t^{k} ∫_{t}^{1} \frac{ϕ(u)}{u^{k+1}} du, 0 < t ⩽ 1/2.

These direct and inverse theorems yield a constructive characterization of the class L̃ip(α, p) = {f ∈ 𝕃̃_{p} | ω_{⌊α⌋+1} (f, t)_{p} ⩽ ct^{α} }.

Theorem ℂ̃_{0} (Constructive characterization, r = 0). Let f ∈ 𝕃̃_{p}, 1 ⩽ p ⩽ ∞, and α > 0. If ω_{k}(f, t)_{p} ⩽ t^{α}, then Ẽ_{n} (f)_{p} ⩽ c(k)n^{−α} , n ⩾ 1.

Conversely, if 0 < α < k and Ẽ_{n} (f)_{p} ⩽ n^{−α} , n ⩾ 1, then ω_{k}(f, t)_{p} ⩽ c(k, α)t^{α}.

Jackson’s inequalities of the second type involve differentiable functions. Let 𝕎̃_{p}^{r}, r ∈ ℕ, be the space of 2π-periodic functions f such that f^{(r−1)} is

absolutely continuous and f^{(r)} ∈ 𝕃̃_{p}, where by 𝕎̃_{∞}^{r} we mean ℂ̃^{r}.

The following result is an immediate consequence of Theorem D̃_{0} and the well known property ω_{k+r} (f, t)_{p} ⩽ t^{r}ω_{k}(f^{(r)} , t)_{p}, t > 0.

Theorem D̃_{r} (Direct theorem, r ∈ ℕ). Let k ∈ ℕ and r ∈ ℕ. If f ∈ 𝕎̃_{p}^{r}, then

Ẽ_{n} (f)_{p} ⩽ c(k, r)n^{−r} ω_{k}(f^{(r)} , n^{−1} )_{p}, n ⩾ 1.

The following inverse theorems are due to Bernstein [3], de la Vallée Poussin [21, Section 39], Quade [19, Theorem 1], Zygmund [22, Theorems 8, 8′, 9 and 9′], Stechkin [15, Theorem 11], and A. Timan [17], [16, Theorem 6.1.3].

Theorem 1.2. Let r ∈ ℕ and f ∈ 𝕃̃_{p}, 1 ⩽ p ⩽ ∞. If ∑ _{ν=1}^{∞} ν^{r−1} Ẽ_{ν} (f)_{p} < ∞,

then f is a.e. identical with a function from 𝕎̃_{p}^{r}. In addition, for any k ∈ ℕ,

ω_{k}(f^{(r)} , n^{−1} )_{p} ⩽ \frac{c(k, r)}{n^{k}} ∑_{ν=1}^{n} ν^{k+r−1} Ẽ_{ν} (f)_{p} + c(k, r) ∑_{ν=n+1}^{∞} ν^{r−1} Ẽ_{ν} (f)_{p}, n ⩾ 1.

This theorem can be restated as follows.

Theorem Ĩ_{r} 1 (Inverse theorem, r ∈ ℕ). Let k ∈ ℕ, r ∈ ℕ and ϕ : [0, 1] → [0, ∞) be a nondecreasing function such that ϕ(0+) = 0 and

∫_{0}^{1} \frac{ϕ(t)}{t^{r+1}} dt < ∞.

If f ∈ 𝕃̃_{p} be such that Ẽ_{n} (f)_{p} ⩽ ϕ (n^{−1} ), n ⩾ 1, then f is a.e. identical with a function from 𝕎̃_{p}^{r}, and

ω_{k}(f^{(r)} , t)_{p} ⩽ c(k, r)( ∫_{0}^{t} \frac{ϕ(u)}{u^{r+1}} du + t^{k} ∫_{t}^{1} \frac{ϕ(u)}{u^{k+r+1}} du) , 0 < t ⩽ 1/2.

Finally, we have a constructive characterization of functions f ∈ 𝕎̃^{r} such that f^{(r)} ∈ L̃ip(α − r, p).

Theorem ℂ̃_{r} (Constructive characterization, r ∈ ℕ). Let r ∈ ℕ, α > r, f ∈ 𝕎̃_{p}^{r}, 1 ⩽ p ⩽ ∞, and ω_{k}(f^{(r)} , t)_{p} ⩽ t^{α−r} . Then Ẽ_{n} (f)_{p} ⩽ c(k, r)n^{−α} , n ⩾ 1.

Conversely, if f ∈ 𝕃̃_{p}, 1 ⩽ p ⩽ ∞, r < α < k + r, and Ẽ_{n} (f)_{p} ⩽ n^{−α} , n ⩾ 1, then f is a.e. identical with a function from 𝕎̃_{p}^{r} and ω_{k}(f^{(r)} , t)_{p} ⩽ c(k, r, α)t^{α−r} .

1.2. Algebraic approximation. Let 𝕃_{p}[−1, 1], 1 ⩽ p ⩽ ∞ denote the usual 𝕃_{p} space equipped with the norm ∥f∥_{p} := ( ∫_{−1}^{1} |f(x)|^{p}dx)^{1/p}, where by L_{∞} [−1, 1] we mean C[−1, 1] equipped with the uniform norm.

Let P_{n} denote the space of algebraic polynomials of degree < n and set

E_{n}(f)_{p} := inf_{P_{n} ∈P_{n}} ∥f − P_{n}∥_{p}

the degree of best approximation of f by algebraic polynomials in 𝕃_{p}.

Define

Δ_{h}^{k}(f, x; [−1, 1]) := \{ \begin{matrix}Δ_{h}^{k}(f, x), & x ± kh/2 ∈ [−1, 1], \\ 0, & otherwise \end{matrix} ,

where Δ_{h}^{k}(f, x) was defined in (1.1).

Finally, define the Ditzian–Totik (DT) moduli of smoothness [6], by

(1.2)  ω_{k}^{φ} (f, t)_{p} := sup_{0<h⩽t} ∥Δ_{hφ(·)}^{k} (f, ·; [−1, 1])∥_{p}

where φ(x) := (1 − x^{2})^{1/2} .

It is well known that the DT moduli of smoothness yield results which are completely analogous to Theorems D̃_{0}, Ĩ_{0} and ℂ̃_{0}. Namely, we have the following results (see [6]).

Theorem D_{0} . Let k ∈ ℕ. If f ∈ 𝕃_{p}[−1, 1], 1 ⩽ p ⩽ ∞, then

E_{n}(f)_{p} ⩽ c(k)ω_{k}^{φ} (f, n^{−1} )_{p}, n ⩾ k.

Theorem I_{0} . Let k ∈ ℕ and ϕ : [0, 1] → [0, ∞) be a nondecreasing function such that ϕ(0+) = 0. If a function f ∈ 𝕃_{p}[−1, 1], 1 ⩽ p ⩽ ∞, is such that E_{n}(f)_{p} ⩽ ϕ(n^{−1} , n ⩾ k, then

ω_{k}^{φ} (f, t)_{p} ⩽ c(k)t^{k} ∫_{t}^{1} \frac{ϕ(u)}{u^{k+1}} du, t ∈ [0, 1/2].

Theorem C_{0} . Let α > 0 and f ∈ 𝕃_{p}[−1, 1], 1 ⩽ p ⩽ ∞. If ω_{k}^{φ} (f, t)_{p} ⩽ t^{α}, then E_{n}(f)_{p} ⩽ c(k)n^{−α} , n ⩾ k.

Conversely, if 0 < α < k and E_{n}(f)_{p} ⩽ n^{−α} , n ⩾ k, then ω_{k}^{φ} (f, t)_{p} ⩽ c(k, α)t^{α}.

The purpose of this paper is to discuss our new moduli of smoothness (introduced in [10]) that allow to obtain the analogs of Theorems D̃_{r}, Ĩ_{r} and ℂ̃_{r}.

2. New moduli of smoothness

2.1. Definitions. For 1 ⩽ p < ∞ and r ∈ ℕ, denote

𝔹_{p}^{r} := {f : f^{(r−1)} ∈ AC_{loc}(−1, 1) and ∥f^{(r)} φ^{r}∥_{p} < +∞}.

If p = ∞, then

𝔹_{∞}^{r} := {f : f ∈ C^{r}(−1, 1) and lim_{x→±1} f^{(r)} (x)φ^{r}(x) = 0}.

Finally, if r = 0, then 𝔹_{p}^{0} := 𝕃_{p}[−1, 1], 1 ⩽ p < ∞ and 𝔹_{∞}^{0} := ℂ[−1, 1].

For f ∈ 𝔹_{p}^{r}, define

ω_{k,r}^{φ} (f^{(r)} , t)_{p} := sup_{0<h⩽t} ∥W_{kh}^{r}(·)Δ_{hφ(·)}^{k} (f^{(r)} , ·)∥_{p},

where

W_{δ}(x) := \{ \begin{matrix}((1 − x − δφ(x)/2)(1 + x − δφ(x)/2))^{1/2} , \\ 1 ± x − δφ(x)/2 ∈ [−1, 1], \\ 0, otherwise. \end{matrix}

Note that, if r = 0, then ω_{k,0}^{φ}(f, t)_{p} = ω_{k}^{φ} (f, t)_{p} are the usual DT moduli defined in (1.2).

It turns out (see [10, Lemma 3.2]) that if f ∈ 𝔹_{p}^{r}, then lim_{t→0+} ω_{k,r}^{φ} (f^{(r)}, t)_{p} = 0 .

2.2. Weighted DT moduli of smoothness. Let

\overrightarrow{Δ} _{h}^{k}f(x) := \{ \begin{matrix}∑ _{i=0}^{k} \binom{k}{i}(−1)^{k−i} f(x + ih), & if x, x + kh ∈ [−1, 1], \\ 0, & otherwise, \end{matrix}
\overleftarrow{Δ} _{h}^{k}f(x) := \{ \begin{matrix}∑ _{i=0}^{k} \binom{k}{i}(−1)^{i}f(x − ih), & if x − kh, x ∈ [−1, 1], \\ 0, & otherwise, \end{matrix}

be the forward and backward kth differences, respectively. Note that

\overrightarrow{Δ} _{h}^{k}f(x) := Δ_{h}^{k}(f, x + kh/2) and \overleftarrow{Δ} _{h}^{k}f(x) := Δ_{h}^{k}(f, x − kh/2).

Let

(2.1)  w(x) := w_{α,β} (x) := (1 − x)^{α} (1 + x)^{β}, α, β ⩾ 0,

and denote 𝕃_{p}(w) := 𝕃_{p}(w_{α,β} ) := {f : [−1, 1] → ℝ | ∥w_{α,β} f∥_{p} < ∞} . For f ∈ 𝕃_{p}(w), the weighted DT moduli of smoothness were defined (see [6, (8.2.10) and Appendix B]) by

(2.2)  ω_{k}^{φ} (f, t)_{w,p} := sup_{0<h⩽t} ∥wΔ_{hφ}^{k} f∥_{𝕃_{p} [−1+2k^{2} h^{2} ,1−2k^{2} h^{2} ]}

+ sup_{0<h⩽2k^{2} t^{2}} ∥w\overrightarrow{Δ} _{h}^{k}f∥_{𝕃_{p} [−1,−1+2k^{2} t^{2} ]}

+ sup_{0<h⩽2k^{2} t^{2}} ∥w\overleftarrow{Δ} _{h}^{k}f∥_{𝕃_{p} [1−2k^{2} t^{2} ,1]} .

The first term on the right-hand side of (2.2) is the main part modulus which is denoted by Ω_{k}^{φ} (f, t)_{w,p} (see [6, (8.1.2)]) and is further discussed in Section 5.

It was shown in [6, Theorem 6.1.1] that ω_{k}^{φ} (f, t)_{w,p} is equivalent to the following weighted K-functional K_{k,φ} (f, t^{k})_{w,p} (with 0 < t ⩽ t_{0}):

K_{k,φ} (f, t^{k})_{w,p} := inf_{g^{(k−1)} ∈AC_{loc}} (∥(f − g)w∥_{p} + t^{k}∥wφ^{k}g^{(k)}∥_{p}).

2.3. Properties of the new moduli. For r ⩾ 0 and f ∈ 𝔹_{p}^{r}, we denote

K_{k,r}^{φ} (f^{(r)} , t^{k})_{p} := inf_{g∈𝔹_{p}^{k+r}} (∥(f^{(r)} − g^{(r)} )φ^{r}∥_{p} + t^{k}∥g^{(k+r)} φ^{k+r} ∥_{p}).

Then, we have the following equivalence results (see [10, Theorem 2.7]).

Theorem 2.1. If k ∈ ℕ, r ∈ ℕ_{0}, 1 ⩽ p ⩽ ∞ and f ∈ 𝔹_{p}^{r}, then, for all

0 < t ⩽ 2/k,

cK_{k,r}^{φ} (f^{(r)} , t^{k})_{p} ⩽ ω_{k,r}^{φ} (f^{(r)} , t)_{p} ⩽ cK_{k,r}^{φ} (f^{(r)} , t^{k})_{p},

where constants c > 0 and may depend only on k, r and p.

Corollary 2.1. If k ∈ ℕ, r ∈ ℕ_{0}, 1 ⩽ p ⩽ ∞ and f ∈ 𝔹_{p}^{r}, then, for all

0 < t ⩽ 2/k,

cK_{k,φ} (f^{(r)} , t^{k})_{φ^{r} ,p} ⩽ ω_{k,r}^{φ} (f^{(r)} , t)_{p} ⩽ cK_{k,φ} (f^{(r)} , t^{k})_{φ^{r} ,p} .

Also, the following was proved in [10, Theorem 7.1].

174 KOPOTUN, LEVIATAN, AND SHEVCHUK

Theorem 2.2. If f ∈ 𝔹_{p}^{r+1} , 1 ⩽ p ⩽ ∞, r ∈ ℕ_{0} and k ⩾ 2, then

ω_{k,r}^{φ} (f^{(r)} , t)_{p} ⩽ ctω_{k−1,r+1}^{φ} (f^{(r+1)} , t)_{p}.

The following sharp Marchaud inequality was proved in [4]. Theorem 2.3. [4, Theorem 7.5] For α > −1/p, β > −1/p, 1 < p < ∞, m ∈ ℕ

and a weight w defined in (2.1), we have

K_{m,φ} (f, t^{m})_{w,p} ⩽ Ct^{m}( ∫_{t}^{1} \frac{K_{m+1,φ} (f, u^{m+1} )_{w,p}^{q}}{u^{mq+1}} du + E_{m}(f)_{w,p}^{q}) ^{1/q}
K_{m,φ} (f, t^{m})_{w,p} ⩽ Ct^{m}( ∑_{n<1/t} n^{qm−1} E_{n}(f)_{w,p}^{q}) ^{1/q},

where q = min(2, p) and E_{n}(f)_{w,p} is the degree of best weighted approximation of f by polynomials from P_{n}, i.e., E_{n}(f)_{w,p} := inf {∥(f − P_{n})w∥_{p} | P_{n} ∈ P_{n}} .

Corollary 2.2. For 1 < p < ∞, r ∈ ℕ_{0}, m ∈ ℕ and f ∈ 𝔹_{p}^{r}, we have

ω_{m,r}^{φ} (f^{(r)} , t)_{p} ⩽ Ct^{m}( ∫_{t}^{1} \frac{ω_{m+1,r}^{φ} (f^{(r)} , u)_{p}^{q}}{u^{mq+1}} du + E_{m}(f^{(r)} )_{φ^{r} ,p}^{q} ) ^{1/q},
ω_{m,r}^{φ} (f^{(r)} , t)_{p} ⩽ Ct^{m}( ∑_{n<1/t} n^{qm−1} E_{n}(f^{(r)} )_{φ^{r} ,p}^{q} ) ^{1/q},

where q = min(2, p).

The following sharp Jackson inequality was proved in [5]. Theorem 2.4. [5, Theorem 6.2] For α > −1/p, β > −1/p, 1 < p < ∞, m ∈ ℕ

and a weight w defined in (2.1), we have

2^{−nm} ( ∑_{j=j_{0}}^{n} 2^{mjs} E_{2^{j}} (f)_{w,p}^{s}) ^{1/s} ⩽ CK_{m,φ} (f, 2^{−nm} )_{w,p},

2^{−nm} ( ∑_{j=j_{0}}^{n} 2^{mjs} K_{m+1,φ} (f, 2^{−j(m+1)} )_{w,p}^{s}) ^{1/s} ⩽ CK_{m,φ} (f, 2^{−nm} )_{w,p},

where 2^{j_{0}} ⩾ m and s = max(p, 2).

Corollary 2.3. For 1 < p < ∞, r ∈ ℕ_{0}, m ∈ ℕ and f ∈ 𝔹_{p}^{r}, we have

2^{−nm} ( ∑_{j=j_{0}}^{n} 2^{mjs} E_{2^{j}} (f^{(r)} )_{φ^{r} ,p}^{s} ) ^{1/s} ⩽ Cω_{m,r}^{φ} (f^{(r)} , 2^{−n} )_{p}
, 2^{−nm} ( ∑_{j=j_{0}}^{n} 2^{mjs} ω_{m+1,r}^{φ} (f^{(r)} , 2^{−j} )_{p}^{s}) ^{1/s} ⩽ Cω_{m,r}^{φ} (f^{(r)} , 2^{−n} )_{p},

where 2^{j_{0}} ⩾ m and s = max(p, 2).

Corollary 2.4. For 1 < p < ∞, r ∈ ℕ_{0}, m ∈ ℕ and f ∈ 𝔹_{p}^{r}, we have

t^{m}( ∫_{t}^{1/m} \frac{ω_{m+1,r}^{φ} (f^{(r)} , u)_{p}^{s}}{u^{ms+1}} du) ^{1/s} ⩽ Cω_{m,r}^{φ} (f^{(r)} , t)_{p}, 0 < t ⩽ 1/m,

where s = max(p, 2).

3. Algebraic polynomial approximation in 𝕃_{p}

In [10], we proved the following results analogous to Theorems D̃_{r}, Ĩ_{r} and ℂ̃_{r} (see also [11, Theorem 3.2] for the inverse result for p = ∞).

Theorem D_{r} . If f ∈ 𝔹_{p}^{r}, 1 ⩽ p ⩽ ∞, then

(3.1)  E_{n}(f)_{p} ⩽ c(k, r)n^{−r} ω_{k,r}^{φ} (f^{(r)} , n^{−1} )_{p}, n ⩾ k + r.

Note that it follows from the DT estimates that if f ∈ 𝔹_{p}^{r}, then

E_{n}(f)_{p} ⩽ c(r)n^{−r} ∥f^{(r)} φ^{r}∥_{p}, n ⩾ r,

which is asymptotically weaker than (3.1).

It is also known that if, for some r ⩾ 1, f^{(r)} ∈ 𝕃_{p}[−1, 1], 1 ⩽ p ⩽ ∞, then

E_{n}(f)_{p} ⩽ c(k, r)n^{−r} ω_{k}^{φ} (f^{(r)} , n^{−1} )_{p}, n ⩾ k + r.

But we should emphasize that here we have to assume that f^{(r)} ∈ 𝕃_{p}[−1, 1], as the DT-moduli are not well defined if the function is not in 𝕃_{p}[−1, 1] and, clearly, ω_{k,r}^{φ} (f^{(r)} , n^{−1} )_{p} is smaller than ω_{k}^{φ} (f^{(r)} , n^{−1} )_{p}.

Theorem I_{r} . Let r ∈ ℕ_{0}, k ⩾ 1, and N ∈ ℕ, and let ϕ : [0, 1] → [0, ∞) be a

nondecreasing function such that ϕ(0+) = 0 and

∫_{0}^{1} r \frac{ϕ(u)}{u^{r+1}} du < ∞.

If f ∈ 𝕃_{p}[−1, 1], 1 ⩽ p ⩽ ∞, and E_{n}(f)_{p} ⩽ ϕ(n^{−1} ), for all n ⩾ N, then f is a.e. identical with a function from 𝔹_{p}^{r}, and

ω_{k,r}^{φ} (f^{(r)} , t)_{p} ⩽ c(k, r) ∫_{0}^{t} r \frac{ϕ(u)}{u^{r+1}} du + c(k, r)t^{k} ∫_{t}^{1} \frac{ϕ(u)}{u^{k+r+1}} du
+ c(N, k, r)t^{k}E_{k+r} (f)_{p}, t ∈ [0, 1/2].

If, in addition, N ⩽ k + r, then

ω_{k,r}^{φ} (f^{(r)} , t)_{p} ⩽ c(k, r) ∫_{0}^{t} r \frac{ϕ(u)}{u^{r+1}} du + c(k, r)t^{k} ∫_{t}^{1} \frac{ϕ(u)}{u^{k+r+1}} du, t ∈ [0, 1/2].

Taking N = 1 and appropriately choosing the function ϕ, we get the following corollary of Theorem I_{r} in terms of the degrees of approximation.

Corollary 3.1. Given 1⩽ p <∞, k ∈ℕ, r ∈ℕ_{0}. If ∑ _{n=1}^{∞} rn^{r−1} E_{n}(f)_{p}<+∞ ,

then f is a.e. identical with a function from 𝔹_{p}^{r}, and

ω_{k,r}^{φ} (f^{(r)} , t)_{p} ⩽ c ∑_{n>1/t} rn^{r−1} E_{n}(f)_{p} + ct^{k} ∑_{1⩽n⩽1/t} n^{k+r−1} E_{n}(f)_{p}, t ∈ [0, 1/2].

Theorem C_{r} . Let r ∈ ℕ_{0}, α > r, k ⩾ 1 and f ∈ 𝔹_{p}^{r}, 1 ⩽ p ⩽ ∞. If

ω_{k,r}^{φ} (f^{(r)} , t)_{p} ⩽ t^{α−r} , then E_{n}(f)_{p} ⩽ cn^{−α} , n ⩾ k + r.

Conversely, if r < α < r + k and f ∈ 𝕃_{p}[−1, 1] and E_{n}(f)_{p} ⩽ n^{−α} , n ⩾ N, then f is a.e. identical with a function from 𝔹_{p}^{r}, and

ω_{k,r}^{φ} (f^{(r)} , t)_{p} ⩽ c(α, k, r)t^{α−r} + c(N, k, r)t^{k}E_{k+r} (f)_{p}, t ∈ [0, 1/2].

If, in addition, N ⩽ k + r, then ω_{k,r}^{φ} (f^{(r)} , t)_{p} ⩽ c(α, k, r)t^{α−r} .

4. Further characterizations

In addition to characterizations in the previous section, we can also characterize certain smoothness classes of functions via the growth of certain weighted norms of their polynomials of best approximation.

Theorem 4.1. Let f ∈ 𝕃_{p}[−1, 1], 1 ⩽ p ⩽ ∞, k ∈ ℕ, r ∈ ℕ_{0}, r < α < r + k,

and suppose that P_{n} denotes the (n − 1)st degree polynomial of best approximation of f in 𝕃_{p}[−1, 1]. Then

(4.1)  ∥φ^{r+k} P_{n}^{(r+k)} ∥_{p} ⩽ cn^{r+k−α} , n ⩾ r + k,

if and only if f is a.e. identical with a function from 𝔹_{p}^{r}, and

(4.2)  ω_{k,r}^{φ} (f^{(r)} , t)_{p} ⩽ ct^{α−r} , t > 0.

Proof. By virtue of [6, Theorem 7.3.1] we conclude that, for every k ∈ ℕ and r ∈ ℕ_{0},

∥φ^{r+k} P_{n}^{(r+k)} ∥_{p} ⩽ cn^{k+r} ω_{k+r}^{φ} (f, n^{−1} )_{p}.

Hence, if f ∈ 𝔹_{p}^{r} and (4.2) is valid, then (4.1) follows immediately from the inequality

(4.3)  ω_{k+r}^{φ} (f, t)_{p} ⩽ ct^{r}ω_{k,r}^{φ} (f^{(r)} , t)_{p}

which is an immediate consequence of Theorem 2.2.

Conversely, if (4.1) holds, then it follows by [6, Theorem 7.3.2] that E_{n}(f)_{p} ⩽ cn^{−α} , n ⩾ r + k. Hence (4.2) follows from Theorem C_{r}. □

We note that while inequality (4.3) cannot be reversed for a general function f, the following is an immediate consequence of Theorem C_{r}.

Corollary 4.1. Let r ∈ ℕ_{0}, k ⩾ 1, f ∈ 𝕃_{p}[−1, 1], 1 ⩽ p ⩽ ∞, r < α < r + k.

If

ω_{r+k}^{φ} (f, t)_{p} ⩽ ct^{α} , t > 0,

then f is a.e. identical with a function from 𝔹_{p}^{r}, and

ω_{k,r}^{φ} (f^{(r)} , t)_{p} ⩽ ct^{α−r} , t > 0.

5. Further results for Weighted DT moduli

The proofs (and therefore the results) of [10] may be extended to the weighted DT moduli with weight w which satisfies the conditions of [6, Section 6.1]. So, in particular, we have the hierarchy relations between the weighted moduli of the function (of course, provided its derivative exists), extending Theorem 2.2.

Theorem 5.1. Let 0 < r < k, and assume that f is such that f^{(r−1)} is locally absolutely continuous in (−1, 1) and wφ^{r}f^{(r)} ∈ 𝕃_{p}[−1, 1], 1 ⩽ p ⩽ ∞. Then

(5.1)  ω_{k}^{φ} (f, t)_{w,p} ⩽ ct^{r}ω_{k−r}^{φ} (f^{(r)} , t)_{wφ^{r} ,p} , t > 0.

Remark 5.1. The inequality (5.1) extends [6, Corollary 6.3.3(b)], as we do not require the condition of β(c) ⩾ 1, for c = ±1, that appears there.

Proof. Recall that the main part modulus Ω_{k}^{φ} is defined in [6, (8.1.2)] by Ω_{k}^{φ} (f, t)_{w,p} := sup_{0<h⩽t} ∥wΔ_{hφ}^{k} f∥_{𝕃_{p} [−1+2k^{2} h^{2} ,−1+2k^{2} h^{2} ]} . Then, [6, (6.2.9)] implies that

ω_{k}^{φ} (f, t)_{w,p} ⩽ c ∫_{0}^{t}(Ω_{k}^{φ} (f, τ)_{w,p} /τ) dτ.

Also, by [6, (6.3.2)], we have Ω_{k}^{φ} (f, t)_{w,p} ⩽ ctΩ_{k−1}^{φ} (f′, t)_{wφ,p} . Hence,

ω_{k}^{φ} (f, t)_{w,p} ⩽ c ∫_{0}^{t} Ω_{k−1}^{φ} (f′, τ)_{wφ,p} dτ
⩽ ctΩ_{k−1}^{φ} (f′, t)_{wφ,p} ⩽ ctω_{k−1}^{φ} (f′, t)_{wφ,p} ,

where for the second inequality we used the monotonicity of Ω_{k−1}^{φ} (f′, t)_{wφ,p} , and for the third one we applied [6, (6.2.9)]. Applying this inequality r times we get the desired estimate. □

For the Jacobi weights w = w_{α,β} defined in (2.1), it was proved by Ky [12, Theorem 4] (see also Luther and Russo [13, Corollary 2.2]) that there is an n_{0} ∈ ℕ such that

(5.2)  E_{n}(f)_{w,p} ⩽ cω_{k}^{φ} (f, n^{−1} )_{w,p}, n ⩾ n_{0}.

Thus, by (5.1), we have the following Jackson-type result.

Theorem 5.2. Let 0 < r < k and assume that f^{(r−1)} is locally absolutely continuous in (−1, 1) and wφ^{r}f^{(r)} ∈ 𝕃_{p}[−1, 1], 1 ⩽ p ⩽ ∞. Then

E_{n}(f)_{w,p} ⩽ cn^{−r} ω_{k−r}^{φ} (f^{(r)} , n^{−1} )_{wφ^{r} ,p} , n ⩾ n_{0}.

It was proved in [6, Theorem 8.2.4] that

ω_{k}^{φ} (f, t)_{w,p} ⩽ ct^{k} ∑_{0<n⩽1/t} n^{k−1} E_{n}(f)_{w,p}, t ⩽ t_{0}.

This readily implies that, if 0 < α < k and E_{n}(f)_{w,p} ⩽ n^{−α} , for n ⩾ 1, then

ω_{k}^{φ} (f, t)_{w,p} ⩽ ct^{α} , t ⩽ t_{0}.

In fact, it is possible to prove the following more general result.

Theorem 5.3. Let 0 ⩽ r < α < k, and let f be such that wf ∈ 𝕃_{p}[−1, 1],

1 ⩽ p ⩽ ∞. If, for an N ∈ ℕ,

(5.3)  E_{n}(f)_{w,p} ⩽ n^{−α} , n ⩾ N,

then f is a.e. identical with a function that has a locally absolutely continuous derivative f^{(r−1)} in (−1, 1), and

ω_{k−r}^{φ} (f^{(r)} , t)_{wφ^{r} ,p} ⩽ c(w, α, k, r)t^{α−r} + c(w, N, k, r)t^{k−r} E_{k}(f)_{w,p}, t > 0.

In particular, if N ⩽ k, then ω_{k−r}^{φ} (f^{(r)} , t)_{wφ^{r} ,p} ⩽ c(w, α, k, r)t^{α−r} , t > 0.

Proof. Let P_{k} ∈ P_{k} be a polynomial of best approximation to f in the weighted norm ∥w · ∥_{p}, and set F := f − P_{k}. Then E_{n}(F)_{w,p} = ∥wF ∥_{p} = E_{k}(f)_{w,p}, n < k, and E_{n}(F)_{w,p} = E_{n}(f)_{w,p}, n ⩾ k. Hence, in particular, E_{n}(F)_{w,p} ⩽ E_{k}(f)_{w,p}, for all n ∈ ℕ.

Combining [6, Theorem 8.2.1] and (5.3), we obtain

Ω_{k}^{φ} (F, t)_{w,p} ⩽ ct^{k} ∑_{0<n⩽1/t} n^{k−1} E_{n}(F)_{w,p}

⩽ ct^{k}N^{k}E_{k}(f)_{w,p} + ct^{k} ∑_{N⩽n⩽1/t} n^{k−1} E_{n}(f)_{w,p}

⩽ c(N)t^{k}E_{k}(f)_{w,p} + ct^{α} , t > 0.

Hence,

∫_{0}^{1}(Ω_{k}^{φ} (F, τ)_{w,p} /τ^{r+1} ) dτ ⩽ ∫_{0}^{1} (cτ^{α−r−1} + c(N)t^{k−r−1} E_{k}(f)_{w,p}) dτ < ∞,

which, by [6, Theorem 6.3.1(a)], implies that F^{(r−1)} is locally absolutely continuous in (−1, 1) and

Ω_{k−r}^{φ} (F^{(r)} , t)_{wφ^{r} ,p} ⩽ c ∫_{0}^{t}(Ω_{k}^{φ} (F, τ)_{w,p} /τ^{r+1} ) dτ
⩽ c ∫_{0}^{t} (cτ^{α−r−1} + c(N)τ^{k−r−1} E_{k}(f)_{w,p})dτ
⩽ ct^{α−r} + c(N)t^{k−r} E_{k}(f)_{w,p}, t > 0.

Finally, taking into account that

ω_{k−r}^{φ} (F^{(r)} , t)_{wφ^{r} ,p} = ω_{k−r}^{φ} (f^{(r)} , t)_{wφ^{r} ,p} , t > 0,

we apply [6, (6.2.9)] to get

ω_{k−r}^{φ} (f^{(r)} , t)_{wφ^{r} ,p} = ω_{k−r}^{φ} (F^{(r)} , t)_{wφ^{r} ,p}
⩽ c ∫_{0}^{t}(Ω_{k−r}^{φ} (F^{(r)} , τ)_{wφ^{r} ,p} /τ) dτ
⩽ ct^{α−r} + c(N)t^{k−r} E_{k}(f)_{w,p}.

This completes the proof. □

Finally, we have the following result analogous to Corollary 4.1 which immediately follows from (5.2) and Theorem 5.3.

Theorem 5.4. Let wf ∈ 𝕃_{p}[−1, 1], 1 ⩽ p ⩽ ∞ and 0 ⩽ r < α < k. If

ω_{k}^{φ} (f, t)_{w,p} ⩽ ct^{α} , t > 0,

then f is a.e. identical with a function that has a locally absolutely continuous derivative f^{(r−1)} in (−1, 1), and ω_{k−r}^{φ} (f^{(r)} , t)_{wφ^{r} ,p} ⩽ ct^{α−r} , t > 0.

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Department of Mathematics University of Manitoba Winnipeg, Manitoba R3T 2N2 Canada

kopotunk@cc.umanitoba.ca

Raymond and Beverly Sackler School of Mathematical Sciences Tel Aviv University Tel Aviv 69978 Israel

leviatan@post.tau.ac.il

Faculty of Mechanics and Mathematics Taras Shevchenko National University of Kyiv Kyiv 01601 Ukraine

shevchuk@univ.kiev.ua