Pointwise Analog of the Stečkin Approximation Theorem

Abstract

We show the pointwise version of the Stečkin theorem on approximation by de la Vallée-Poussin means. The result on norm approximation is also derived.

1. Introduction

Let 𝐿^𝑝(1≤𝑝<∞)[𝐶] be the class of all 2𝜋-periodic real-valued functions integrable in the Lebesgue sense with 𝑝th power (continuous) over 𝑄=[−𝜋,𝜋], and let 𝑋^𝑝=𝐿^𝑝 when 1≤𝑝<∞ or 𝑋^𝑝=𝐶 when 𝑝=∞.

Let us define the norms of 𝑓∈𝑋^𝑝 as 𝑓=𝑓_𝑋𝑝=𝑓(⋅)_𝑋𝑝∫∶=^𝑄𝑓(𝑥)^𝑝𝑑𝑥^1/𝑝when1≤𝑝<∞,sup_𝑥∈𝑄𝑓(𝑥𝑓)when𝑝=∞,_𝑥,𝛿=𝑓_{𝑋𝑝,𝑥,𝛿}=𝑓(⋅)_{𝑋𝑝,𝑥,𝛿}∶=sup_0<ℎ≤𝛿𝑓(⋅)^∘_{𝑋𝑝,𝑥,ℎ}∶=sup_0<ℎ≤𝛿1∫2ℎ^{𝑥+ℎ𝑥−ℎ}𝑓(𝑡)^𝑝𝑑𝑡^1/𝑝when1≤𝑝<∞,sup_0<ℎ≤𝛿sup_{|𝑡|0<≤ℎ}𝑓(𝑥+𝑡)when𝑝=∞, (1) where 𝑓^∘_𝑥,𝛿=𝑓^∘_{𝑋𝑝,𝑥,𝛿}=𝑓(⋅)^∘_{𝑋𝑝,𝑥,𝛿}1∶=∫2𝛿^{𝑥+𝛿𝑥−𝛿}𝑓(𝑡)^𝑝𝑑𝑡^1/𝑝when1≤𝑝<∞sup_{|ℎ|0<≤𝛿}𝑓(().𝑥+ℎ)when𝑝=∞𝛿>0 (2) We note additionally that 𝑓_𝐶,𝑥,0=𝑓^∘_𝐶,𝑥,0=𝑓(𝑥). (3) Consider the trigonometric Fourier series of 𝑓(𝑥)=𝑎𝑆𝑓^𝑜𝑓2+^∞_𝑘=0𝑎_𝑘𝑓cos𝑘𝑥+𝑏_𝑘𝑓sin𝑘𝑥 (4) with the partial sums 𝑆_𝑘𝑓.

Let 𝜎_𝑛,𝑚𝑓(𝑥)1∶=𝑚+1^𝑛_{𝑘=𝑛−𝑚}𝑆_𝑘𝑓(𝑥)(𝑚≤𝑛=0,1,2,…). (5)

As a measure of approximation by the previous quantities, we use the pointwise characteristics 𝑤_𝑥𝑓(𝛿)=𝑤_𝑥𝑓(𝛿)_𝑋^𝑝∶=Δ_𝑥𝑓(⋅)_{𝑋𝑝,𝑥,𝛿}=sup_0<ℎ≤𝛿1∫2ℎ^ℎ−ℎΔ_𝑥𝑓(𝑡)^𝑝𝑑𝑡^1/𝑝when1≤𝑝<∞,sup_0<ℎ≤𝛿sup_{|𝑡|0<≤ℎ}Δ_𝑥𝑓(𝑡)when𝑝=∞, (6) compared to [1], and 𝑤^∘_𝑥𝑓(𝛿)=𝑤^∘_𝑥𝑓(𝛿)_𝑋^𝑝∶=Δ_𝑥𝑓(⋅)^∘_{𝑋𝑝,𝑥,𝛿}=1∫2𝛿^𝛿−𝛿Δ_𝑥𝑓(𝑡)^𝑝𝑑𝑡^1/𝑝when1≤𝑝<∞,sup_{|𝑡|0<≤𝛿}Δ_𝑥𝑓(𝑡)when𝑝=∞, (7) and also Ω_𝑥𝑓𝜋𝑛+1=Ω_𝑥𝑓𝜋𝑛+1_𝑋^𝑝1∶=𝑛+1^𝑛_𝑘=0𝑤_𝑥𝑓𝜋𝑘+1_𝑋^𝑝,Ω^∘_𝑥𝑓𝜋𝑛+1=Ω^∘_𝑥𝑓𝜋𝑛+1_𝑋^𝑝=1𝑛+1^𝑛_𝑘=0𝑤^∘_𝑥𝑓𝜋𝑘+1_𝑋^𝑝, (8) where Δ_𝑥𝑓(𝑡)∶=𝑓(𝑥+𝑡)−𝑓(𝑥), (9) constructed on the base of definition of 𝑋^𝑝-points  (Lebesgue points (𝐿^𝑝-points) or points of continuity (𝐶-points)). We also use the modulus of continuity of 𝑓 in the space 𝑋^𝑝 defined by the formula 𝜔𝑓(𝛿)=𝜔𝑓(𝛿)_𝑋^𝑝∶=sup_{|ℎ|0<≤𝛿}Δ_⋅𝑓(ℎ)_𝑋𝑝 (10) and its arithmetic mean 𝜋Ω𝑓𝜋𝑛+1=Ω𝑓𝑛+1_𝑋^𝑝=1𝑛+1^𝑛_𝑘=0𝜋𝜔𝑓𝑘+1_𝑋^𝑝. (11)

We can observe that, for 𝑓∈𝑋𝑝 and 𝑝≥𝑝, 𝑤_⋅.𝑓(𝛿)_𝑋^𝑝_𝐶≤𝜔𝑓(𝛿)_𝐶, (12) whence Ω_⋅.𝑓(𝛿)_𝑋^𝑝_𝐶≤Ω𝑓(𝛿)_𝐶,𝑤^∘_⋅.𝑓(𝛿)_𝑋^𝑝_𝑋𝑝≤𝜔𝑓(𝛿)_𝑋^𝑝, (13) whence Ω^∘_⋅.𝑓(𝛿)_𝑋^𝑝_𝑋𝑝≤Ω𝑓(𝛿)_𝑋^𝑝. (14)

Let us introduce one more measure of pointwise approximation analogical to the best approximation of function 𝑓 by trigonometric polynomials 𝑇 of the degree at most 𝑛(𝑇∈𝐻_𝑛)𝐸_𝑛𝑓_𝑋𝑝∶=inf_{𝑇∈𝐻𝑛}𝑓(⋅)−𝑇(⋅)_𝑋𝑝; (15) Namely, 𝐸_𝑛𝑓,𝑥;𝛿=𝐸_𝑛𝑓,𝑥;𝛿_𝑋𝑝∶=inf_{𝑇∈𝐻𝑛}𝑓(⋅)−𝑇(⋅)_{𝑋𝑝,𝑥,𝛿}=inf_{𝑇∈𝐻𝑛}sup_0<ℎ≤𝛿1∫2ℎ^ℎ−ℎ𝑓()(𝑥+𝑡−𝑇𝑥+𝑡)|^𝑝𝑑𝑡^1/𝑝when1≤𝑝<∞,inf_{𝑇∈𝐻𝑛}sup_{|ℎ|0<≤𝛿}𝑓()(𝐸𝑥+ℎ−𝑇𝑥+ℎ)when𝑝=∞,_∘𝑛𝑓,𝑥;𝛿=𝐸^∘_𝑛𝑓,𝑥;𝛿_𝑋𝑝∶=inf_{𝑇∈𝐻𝑛}𝑓(⋅)−𝑇(⋅)^∘_{𝑋𝑝,𝑥,𝛿}. (16) We will also use its arithmetic mean 𝐹_𝑛,𝑚𝑓,𝑥=𝐹_𝑛,𝑚𝑓,𝑥_𝑋𝑝1∶=𝑚+1^𝑚_𝑘=0𝐸_𝑛𝜋𝑓,𝑥;𝑘+1_𝑋^𝑝,𝐹^∘_𝑛,𝑚𝑓,𝑥=𝐹^∘_𝑛,𝑚𝑓,𝑥_𝑋𝑝1∶=𝑚+1^𝑚_𝑘=0𝐸^∘_𝑛𝜋𝑓,𝑥;𝑘+1_𝑋^𝑝. (17)

Using these characteristics, we will show the pointwise version of the Stečkin [2] generalization of the Fejér-Lebesgue theorem. As a corollary, we will obtain the mentioned original result of Stečkin on norm approximation as well as the result of Tanović-Miller [3].

By 𝐾 we will designate either an absolute constant or a constant depending on some parameters, not necessarily the same of each occurrence.

2. Statement of the Results

At the beginning, we formulate the partial solution of the considered problem.

Theorem 1 (see [4]). If 𝑓∈𝑋^𝑝, then, for any positive integer 𝑚≤𝑛 and all real 𝑥, 𝜎_𝑛,𝑚𝑓(𝑥)−𝑓(𝑥)≤𝜋²𝐸^∘_𝑛−𝑚𝜋𝑓,𝑥,2𝑛−𝑚+1_𝑋^𝑝+6𝐹^∘_{𝑛−𝑚,𝑚}𝑓,𝑥_𝑋𝑝+₍₎^{𝜋/(𝑚+1)}_{𝜋/2𝑛−𝑚+1}𝐸^∘_𝑛−𝑚𝑓,𝑥,𝑡_𝑋𝑝𝑡𝑑𝑡+𝐸^∘_𝑛−𝑚𝑓,𝑥;0_𝑋𝑝,𝜎_𝑛,𝑚𝑓(𝑥)−𝑓(𝑥)≤6+𝜋²𝐹_{𝑛−𝑚,𝑚}𝑓,𝑥_𝑋𝑝×1+ln𝑛+1𝑚+1+𝐸_𝑛−𝑚𝑓,𝑥;0_𝑋𝑝. (18)

Now, we can present the main result on pointwise approximation.

Theorem 2. If 𝑓∈𝑋^𝑝, then, for any positive integer 𝑚≤𝑛 and all real 𝑥, 𝜎_𝑛,𝑚𝑓(𝑥)−𝑓(𝑥)≤𝐾^𝑛_𝜈=0𝐹_{𝑛−𝑚+𝜈,𝑚}𝑓,𝑥_𝑋𝑝+𝐹_{𝑛−𝑚+𝜈,𝜈}𝑓,𝑥_𝑋𝑝𝑚+𝜈+1+𝐸_2𝑛𝑓,𝑥;0_𝑋𝑝. (19)

Remark 3. Theorem 2, in case 𝑓∈𝐶, immediately yields the result of Stečkin [2]. The result of Stečkin type also holds if instead of 𝐶 we consider the spaces 𝑋^𝑝 with 1<𝑝<∞. Then, in the proof, we need the Hardy-Littlewood estimate of the maximal function.

At every 𝑋^𝑝-point 𝑥 of 𝑓Ω_𝑥𝑓𝛾_𝑋𝑝=𝑜_𝑥(1)as𝛾→0+ (20) and thus from Theorem 1, we obtain the corollary which states the result of the Tanović-Miller type [3].

Corollary 4. If 𝑓∈𝑋^𝑝, then, for any positive integer 𝑚≤𝑛 at every 𝑋^𝑝-point  𝑥 of 𝑓, 𝜎_𝑛,𝑚𝑓(𝑥)−𝑓(𝑥)=𝑜_𝑥(1)1+ln𝑛+1𝑚+1as𝑛→∞. (21)

3. Auxiliary Results

In order to prove our theorems, we require some lemmas.

Lemma 5. If 𝑇_𝑛 is the trigonometric polynomial of the degree at most 𝑛 of the best approximation of 𝑓∈𝑋^𝑝 with respect to the norm ‖⋅‖_𝑋^𝑝, then it is also the trigonometric polynomial of the degree at most 𝑛 of the best approximation of 𝑓∈𝑋^𝑝 with respect to the norm ‖⋅‖_𝑋^𝑝_,𝑥,𝛿 for any 𝛿∈[0,𝜋].

Proof. From the inequalities 𝐸_𝑛𝑓,⋅,𝛿_{𝑋𝑝𝑋𝑝}≥𝐸^∘_𝑛𝑓,⋅,𝛿_{𝑋𝑝𝑋𝑝}=𝑓−𝑇_𝑛,𝛿^∘_{𝑋𝑝,⋅,𝛿𝑋𝑝}=𝑓−𝑇_{𝑛,𝛿𝑋𝑝}≥𝑓−𝑇_𝑛𝑋𝑝=𝐸_𝑛𝑓_𝑋𝑝,𝐸^∘_𝑛𝑓,⋅,𝛿_{𝑋𝑝𝑋𝑝}≤𝑓−𝑇_𝑛^∘_{𝑋𝑝,⋅,𝛿𝑋𝑝}=𝑓−𝑇_𝑛𝑋𝑝=𝐸_𝑛𝑓_𝑋𝑝, (22) where 𝑇_𝑛,𝛿 and 𝑇_𝑛 are the trigonometric polynomials of the degree at most 𝑛 of the best approximation of 𝑓∈𝑋^𝑝 with respect to the norms ‖⋅‖^∘_𝑋^𝑝_,𝑥,𝛿 and ‖⋅‖_𝑋^𝑝, respectively, we obtain relation 𝑓−𝑇_{𝑛,𝛿𝑋𝑝}=𝑓−𝑇_𝑛𝑋𝑝=𝐸_𝑛𝑓_𝑋𝑝, (23) whence 𝑇_𝑛,𝛿=𝑇_𝑛 for any 𝛿∈[0,𝜋] by uniqueness of the trigonometric polynomial of the degree at most 𝑛 of the best approximation of 𝑓∈𝑋^𝑝 with respect to the norm ‖⋅‖_𝑋^𝑝(e.g. see [5] p. 96). We can also observe that for such 𝑇_𝑛 and any ℎ∈[0,𝛿], 𝑓−𝑇_𝑛^∘_{𝑋𝑝,𝑥,ℎ}=𝐸^∘_𝑛𝑓,𝑥,ℎ_𝑋𝑝≤𝐸_𝑛𝑓,𝑥,𝛿_𝑋𝑝≤𝑓−𝑇_{𝑛𝑋𝑝,𝑥,𝛿}. (24) Hence, 𝐸_𝑛𝑓,𝑥,𝛿_𝑋𝑝=𝑓−𝑇_{𝑛𝑋𝑝,𝑥,𝛿} (25) and our proof is complete.

Lemma 6. If 𝑛∈ℕ₀ and 𝛿>0, then 𝐸_𝑛(𝑓,𝑥;𝛿)_𝑋𝑝 is nonincreasing function of 𝑛 and nondecreasing function of 𝛿. These imply that for 𝑚,𝑛∈ℕ the function 𝐹_𝑛,𝑚(𝑓,𝑥)_𝑋𝑝 is nonincreasing function of 𝑛 and 𝑚 simultaneously.

Proof. The first part of our statement follows from the property of the norm ‖⋅‖_𝑥,𝛿 and supremum. The second part is a consequence of the calculation 𝐹_𝑛,𝑚+1𝑓,𝑥_𝑋𝑝𝐹_𝑛,𝑚𝑓,𝑥_𝑋𝑝=𝑚+1𝐸𝑚+21+^𝑛𝑓,𝑥;𝜋/(𝑚+2)_𝑋𝑝∑^𝑚_𝑘=0𝐸_𝑛𝑓,𝑥;𝜋/(𝑘+1)_𝑋𝑝≤𝑚+1𝐸𝑚+21+^𝑛𝑓,𝑥;𝜋/(𝑚+1)_𝑋𝑝∑^𝑚_𝑘=0𝐸_𝑛𝑓,𝑥;𝜋/(𝑚+1)_𝑋𝑝=𝑚+11𝑚+21+𝑚+1=1. (26)

Lemma 7. Let 𝑚,𝑛,𝑞∈ℕ₀ such that 𝑚≤𝑛 and 𝑞≥𝑚+1. If 𝑓∈𝑋^𝑝, then 𝜎_{𝑛+𝑞,𝑚}𝑓(𝑥)−𝜎_𝑛,𝑚𝑓(𝑥)≤𝐾𝐹_{𝑛−𝑚,𝑚}𝑓,𝑥_𝑋𝑝^𝑞−1_𝜈=01.𝑚+𝜈+1 (27)

Proof. It is clear that 𝜎_𝑛,𝑚𝑓(𝑥)=1𝑚+1^𝑛_{𝑘=𝑛−𝑚}1𝜋^𝜋_−𝜋𝑓(𝑥+𝑡)𝐷_𝑘(𝑡)=1𝑑𝑡𝜋^𝜋_−𝜋𝑓(𝑥+𝑡)𝑉_𝑛,𝑚(𝑡)𝑑𝑡, (28) where 𝑉_𝑛,𝑚(𝑡)=1𝑚+1^𝑛_{𝑘=𝑛−𝑚}𝐷_𝑘(𝑡),𝐷_𝑘(𝑡)=sin((2𝑘+1)𝑡/2)().2sin𝑡/2 (29) Hence, by orthogonality of the trigonometric system, 𝜎_{𝑛+𝑞,𝑚}𝑓(𝑥)−𝜎_𝑛,𝑚𝑓(𝑥)=1𝜋^𝜋_−𝜋𝑓(𝑥+𝑡)−𝑇_𝑛−𝑚(×𝑉𝑥+𝑡)_{𝑛+𝑞,𝑚}(𝑡)−𝑉_𝑛,𝑚(𝑡)=1𝑑𝑡𝜋(𝑚+1)^𝑛_{𝑘=𝑛−𝑚}^𝜋_−𝜋𝑓(𝑥+𝑡)−𝑇_𝑛−𝑚(×𝐷𝑥+𝑡)_𝑘+𝑞(𝑡)−𝐷_𝑘(𝑡)=1𝑑𝑡𝜋()×𝑚+1_{𝑛𝑘=𝑛−𝑚𝜋−𝜋}𝑓(𝑥+𝑡)−𝑇_𝑛−𝑚(×))𝑥+𝑡)sin2𝑘+2𝑞+1𝑡/2−sin((2𝑘+1𝑡/2()=12sin𝑡/2𝑑𝑡𝜋(𝑚+1)^𝑛_{𝑘=𝑛−𝑚}^𝜋_−𝜋𝑓(𝑥+𝑡)−𝑇_𝑛−𝑚(×𝑥+𝑡)sin𝑞𝑡/2cos2𝑘+𝑞1𝑡/2sin(𝑡/2)𝑑𝑡, (30)with trigonometric polynomial 𝑇_𝑛−𝑚 of the degree at most 𝑛−𝑚 of the best approximation of 𝑓.
Using the notations 𝐼₁=−𝜋𝑞,𝜋𝑞,𝐼₂=−𝜋𝜋𝑚+1,−𝑞∪𝜋𝑞,𝜋,𝐼𝑚+1₃=𝜋−𝜋,−∪𝜋𝑚+1,𝑚+1,𝜋 (31) we get 1=𝜋()×𝑚+1_{𝑛𝑘=𝑛−𝑚𝐼1}+_𝐼2+_𝐼3×𝑓(𝑥+𝑡)−𝑇_𝑛−𝑚(×𝑥+𝑡)sin𝑞𝑡/2cos2𝑘+𝑞+1𝑡/2sin(𝑡/2)𝑑𝑡=₁+₂+₃,₁≤1𝜋(𝑚+1)^𝑛_{𝑘=𝑛−𝑚𝐼1}𝑓(𝑥+𝑡)−𝑇_𝑛−𝑚(=𝑞𝑥+𝑡)𝑞𝑑𝑡𝜋_𝐼1𝑓(𝑥+𝑡)−𝑇_𝑛−𝑚(𝑥+𝑡)𝑑𝑡≤2𝐸_𝑛−𝑚𝜋𝑓,𝑥;𝑞_𝑋𝑝. (32) We next evaluate the sums ∑₂ and ∑₃ using the partial integrating and Lemma 5. Thus, ₂≤_𝐼2𝑓(𝑥+𝑡)−𝑇_𝑛−𝑚(𝑥+𝑡)𝑡1𝑑𝑡=22𝑡^𝑡_−𝑡𝑓(𝑥+𝑢)−𝑇_𝑛−𝑚(𝑥+𝑢)𝑑𝑢^{𝑡=𝜋/(𝑚+1)}_{𝑡=𝜋/𝑞}+2^{𝜋/(𝑚+1)}_𝜋/𝑞1𝑡12𝑡^𝑡_−𝑡𝑓(𝑥+𝑢)−𝑇_𝑛−𝑚(𝑥+𝑢)𝑑𝑢𝑑𝑡≤2𝐸_𝑛−𝑚𝜋𝑓,𝑥;𝑚+1_𝑋^𝑝+2^𝑡_−𝑡1𝑡𝐸_𝑛−𝑚𝑓,𝑥;𝑡_𝑋𝑝𝑑𝑡≤4𝐸_𝑛−𝑚𝜋𝑓,𝑥;𝑚+1_𝑋^𝑝𝑞1+ln𝑚+1≤4𝐸_𝑛−𝑚𝜋𝑓,𝑥;𝑚+1_𝑋^𝑝1+^𝑞−1_𝜈=01,𝑚+𝜈+1₃≤1𝑚+1×_𝐼3𝑓(𝑥+𝑡)−𝑇_𝑛−𝑚(𝑥+𝑡)𝑡×^𝑛_{𝑘=𝑛−𝑚}cos𝑘𝑡+𝑞+12𝑡≤1𝑑𝑡×𝑚+1_𝐼3𝑓(𝑥+𝑡)−𝑇_𝑛−𝑚(𝑥+𝑡)𝑡×2sin((𝑛+1)𝑡/2)cos2𝑛−𝑚+𝑞+1𝑡/2()≤𝜋2sin𝑡/2𝑑𝑡𝑚+1_𝐼3𝑓(𝑥+𝑡)−𝑇_𝑛−𝑚(𝑥+𝑡)𝑡²=𝜋𝑑𝑡21𝑚+12𝑡^𝑡_−𝑡𝑓(𝑥+𝑢)−𝑇_𝑛−𝑚(𝑥+𝑢)𝑑𝑢₍₎^𝑡=𝜋_{𝑡=𝜋/𝑚+1}+4^𝜋_{()𝜋/𝑚+1}1𝑡²12𝑡^𝑡_−𝑡𝑓(𝑥+𝑢)−𝑇_𝑛−𝑚(≤𝜋𝑥+𝑢)𝑑𝑢𝑑𝑡𝑚+12𝐸_𝑛−𝑚𝑓,𝑥;𝜋_𝑋𝑝+4^𝜋_{𝜋/(𝑚+1)}1𝑡²𝐸_𝑛−𝑚𝑓,𝑥;𝑡_𝑋𝑝=𝑑𝑡2𝜋𝐸𝑚+1^𝑛−𝑚𝑓,𝑥;𝜋_𝑋𝑝+2₁^𝑚+1𝐸_𝑛−𝑚𝑓,𝑥;𝜋/𝑢_𝑋𝑝𝜋²/𝑢²𝜋𝑑𝑢𝑢²=2𝜋𝐸𝑚+1^𝑛−𝑚𝑓,𝑥;𝜋_𝑋𝑝+2𝜋^𝑚−1_𝑘=0^𝑘+2_𝑘+1𝐸_𝑛−𝑚𝜋𝑓,𝑥;𝑢_𝑋^𝑝=𝑑𝑢2𝜋𝐸𝑚+1^𝑛−𝑚𝑓,𝑥;𝜋_𝑋𝑝+2𝜋^𝑚−1_𝑘=0𝐸_𝑛−𝑚𝜋𝑓,𝑥;𝑘+1_𝑋^𝑝≤2𝜋+2/𝜋𝑚+1^𝑚−1_𝑘=0𝐸_𝑛−𝑚𝜋𝑓,𝑥;𝑘+1_𝑋^𝑝, (33) which proves Lemma 7.

Before formulating the next lemmas, we define a new difference. Let 𝑚,𝑛∈ℕ₀ and 𝑚≤𝑛. Denote that 𝜏_𝑛,𝑚𝑓(𝑥)∶=(𝑚+1)𝜎_{𝑛+𝑚+1,𝑚}𝑓(𝑥)−𝜎_𝑛,𝑚𝑓(𝑥). (34)

Lemma 8. Let 𝑚,𝑛,𝜇∈ℕ₀ such that 2𝜇≤𝑚≤𝑛. If 𝑓∈𝑋^𝑝, then 𝜏_𝑛,𝑚𝑓(𝑥)−𝜏_{𝑛−𝜇,𝑚−𝜇}𝑓(𝑥)≤𝐾𝜇𝐹_{𝑛−𝜇+1,𝜇−1}𝑓,𝑥_𝑋𝑝𝑚ln𝜇. (35)

Proof. The proof follows by the method of Leindler [6]. Namely, 𝜏_𝑛,𝑚𝑓(𝑥)−𝜏_{𝑛−𝜇,𝑚−𝜇}𝑓(𝑥)=^𝑛+𝑚+1_{𝑘=𝑛+𝑚−2𝜇+2}−2^𝑛_{𝑘=𝑛−𝜇+1}𝑆_𝑘𝑓(𝑥)(𝑥,𝜏−𝑓)_𝑛,𝑚𝑓(𝑥)−𝜏_{𝑛−𝜇,𝑚−𝜇}𝑓(𝑥≤_{𝑛+𝑚−𝜇+1𝑘=𝑛+𝑚−2𝜇+2})−^𝑛_{𝑘=𝑛−𝜇+1}𝑆_𝑘𝑓(𝑥)(𝑥+−𝑓)_{𝑛+𝑚+1𝑘=𝑛+𝑚−𝜇+2}−^𝑛_{𝑘=𝑛−𝜇+1}𝑆_𝑘𝑓(𝑥)(𝑥𝜎−𝑓)=𝜇_{𝑛+𝑚−𝜇+1,𝜇−1}𝑓(𝑥)−𝜎_{𝑛,𝜇−1}𝑓(𝑥𝜎)+𝜇_{𝑛+𝑚+1,𝜇−1}𝑓(𝑥)−𝜎_{𝑛,𝜇−1}𝑓(𝑥). (36) By Lemma 6, for 2𝜇≤𝑚, 𝜏_𝑛,𝑚𝑓(𝑥)−𝜏_{𝑛−𝜇,𝑚−𝜇}𝑓(𝑥)≤𝐾𝜇𝐹_{𝑛−𝜇+1,𝜇−1}𝑓,𝑥_𝑋𝑝1+ln𝑛−𝜇+1+𝜇−1𝜇+𝐾𝜇𝐹_{𝑛−𝜇+1,𝜇−1}𝑓,𝑥_𝑋𝑝1+ln𝑚+𝜇−1𝜇≤𝐾𝜇𝐹_{𝑛−𝜇+1,𝜇−1}𝑓,𝑥_𝑋𝑝𝑚1+ln𝜇, (37) and our proof is complete.

Lemma 9. Let 𝑚,𝑛∈ℕ₀ and 𝑚≤𝑛. If 𝑓∈𝑋^𝑝, then 𝜏_𝑛,𝑚𝑓(𝑥)≤𝐾^𝑛_{𝑘=𝑛−𝑚}𝐹_{𝑘,𝑘−𝑛+𝑚}𝑓,𝑥_𝑋𝑝. (38)

Proof. Our proof runs parallel with the proof of Theorem 1 in [2].
If 𝑚=0, then 𝜏_𝑛,0𝑓(𝑥)=𝜎_𝑛+1,0𝑓(𝑥)−𝜎_𝑛,0𝑓(𝑥)≤𝐾𝐹_𝑛,0𝑓,𝑥_𝑋𝑝, (39) and if 𝑚=1, then 𝜏_𝑛,1𝑓(𝑥)≤2𝜎_𝑛+2,1𝑓(𝑥)−𝜎_𝑛,1𝑓(𝑥)≤𝐾𝐹_𝑛−1,1𝑓,𝑥_𝑋𝑝≤𝐾𝐹_𝑛−1,1𝑓,𝑥_𝑋𝑝+𝐹_𝑛−1,1𝑓,𝑥_𝑋𝑝≤𝐾𝐹_𝑛−1,1𝑓,𝑥_𝑋𝑝+𝐹_𝑛,1𝑓,𝑥_𝑋𝑝 (40) by Lemmas 6 and 7.
Next, we construct the same decreasing sequence (𝑚_𝑠) of integers that was given by Stečkin. Let 𝑚₀=𝑚,𝑚_𝑠=𝑚_𝑠−1−𝑚_𝑠−12(𝑠=1,2,…), (41) where [𝑦] denotes the integral part of 𝑦. It is clear that there exists smallest index 𝑡≥1 such that 𝑚_𝑡=1 and 𝑚=𝑚₀>𝑚₁>⋯>𝑚_𝑡=1. (42) By the definition of the numbers 𝑚_𝑠, we have 𝑚_𝑠≥𝑚_𝑠−12,𝑚_𝑠−1−𝑚_𝑠=𝑚_𝑠−12≥𝑚_𝑠−13(𝑠=1,2,…,𝑡) (43) whence 𝑚_𝑡−1=2,𝑚_𝑡−1−𝑚_𝑡𝑚=1,_𝑠−1−𝑚_𝑠≤𝑚_𝑠≤3𝑚_𝑠−𝑚_𝑠+1(𝑠=1,2,…,𝑡−1) (44) follow.
Under these notations, we get the following equality: 𝜏_𝑛,𝑚𝑓(𝑥)=^𝑡_𝑠=1𝜏_{𝑛−𝑚+𝑚𝑠−1,𝑚𝑠−1}𝑓(𝑥)−𝜏_{𝑛−𝑚+𝑚𝑠,𝑚𝑠}𝑓(𝑥)+𝜏_{𝑛−𝑚+𝑚𝑡,𝑚𝑡}𝑓(𝑥), (45) whence, by 𝑚_𝑡=1, 𝜏_𝑛,𝑚𝑓(𝑥)≤^𝑡_𝑠=1𝜏_{𝑛−𝑚+𝑚𝑠−1,𝑚𝑠−1}𝑓(𝑥)−𝜏_{𝑛−𝑚+𝑚𝑠,𝑚𝑠}𝑓(𝑥+𝜏_{𝑛−𝑚+𝑚𝑡,𝑚𝑡})𝑓(𝑥) (46) follows.
It is easy to see that the terms in the sum ∑^𝑡_𝑠=1, by Lemma 8, with 𝜇=𝑚_𝑠−1−𝑚_𝑠 and 𝑚=𝑚_𝑠−1 do not exceed 𝐾𝑚_𝑠−1−𝑚_𝑠𝐹_{𝑛−𝑚+𝑚𝑠+1,𝑚𝑠−1−𝑚𝑠}𝑓,𝑥_𝑋𝑝𝑚ln^𝑠−1𝑚_𝑠−1−𝑚_𝑠,where(𝑠=1,2,…,𝑡−1), (47) and by Lemma 7, we get 𝜏_{𝑛−𝑚+1,1}𝑓(𝑥)≤2𝜎_{𝑛−𝑚+2,1}𝑓(𝑥)−𝜎_{𝑛−𝑚+1,1}𝑓(𝑥)≤𝐾𝐹_{𝑛−𝑚,1}𝑓,𝑥_𝑋𝑝. (48) Thus, 𝜏_𝑛,𝑚𝑓(𝑥)≤𝐾^𝑡−1_𝑠=13𝑚_𝑠−𝑚_𝑠+1𝐹_{𝑛−𝑚+𝑚𝑠+1,𝑚𝑠}𝑓,𝑥_𝑋𝑝ln3+𝐾𝐹_{𝑛−𝑚+2,𝑚−2}𝑓,𝑥_𝑋𝑝+𝐾𝐹_{𝑛−𝑚,1}𝑓,𝑥_𝑋𝑝, (49) whence, by the monotonicity of 𝐹_𝜈,𝜇(𝑓,𝑥)_𝑋𝑝, 𝜏_𝑛,𝑚𝑓(𝑥)≤𝐾^𝑡−1𝑚_𝑠=1^𝑠_{𝜈=𝑚𝑠+1+1}𝐹_{𝑛−𝑚+𝜈+1,𝜈}𝑓,𝑥_𝑋𝑝+²_𝜈=0𝐹_{𝑛−𝑚+𝜈,𝑚−𝜈−1}𝑓,𝑥_𝑋𝑝+𝐾𝐹_{𝑛−𝑚,1}𝑓,𝑥_𝑋𝑝≤𝐾^𝑚1+1_𝜈=0𝐹_{𝑛−𝑚+𝜈,𝜈}𝑓,𝑥_𝑋𝑝+𝐾𝐹_{𝑛−𝑚,1}𝑓,𝑥_𝑋𝑝≤𝐾^𝑚_𝜈=0𝐹_{𝑛−𝑚+𝜈,𝜈}𝑓,𝑥_𝑋𝑝+𝐾𝐹_{𝑛−𝑚,1}𝑓,𝑥_𝑋𝑝≤𝐾^𝑛_{𝑘=𝑛−𝑚}𝐹_{𝑘,𝑘−𝑛+𝑚}𝑓,𝑥_𝑋𝑝+𝐾𝐹_{𝑛−𝑚,1}𝑓,𝑥_𝑋𝑝. (50)

4. Proofs of the Results

Proof of Theorem 2. The proof follows the lines of the proofs of Theorem 4 in [2] and Theorem in [6]. Therefore, let 𝑛>0 and 𝑚≤𝑛 be fixed. Let us define an increasing sequence (𝑛_𝑠∶𝑠=0,1,…,𝑡) of indices introduced by Stečkin in the following way. Set 𝑛₀=𝑛. Assuming that the numbers 𝑛₀,…,𝑛_𝑠 are already defined and 𝑛_𝑠<2𝑛, we define 𝑛_𝑠+1 as follows. Let 𝜈_𝑠 denote the smallest natural number such that 𝐹_{𝑛𝑠−𝑚+𝜈𝑠,𝜈}𝑓,𝑥_𝑋𝑝≤12𝐹_{𝑛𝑠−𝑚,𝜈}𝑓,𝑥_𝑋𝑝(𝜈=0,1,…,𝑛). (51) According to the magnitude of 𝜈_𝑠, we define 𝑛_𝑠+1=𝑛_𝑠−𝑚+1for𝜈_𝑠𝑛≤𝑚,_𝑠+𝜈_𝑠for𝑚+1≤𝜈_𝑠<2𝑛+𝑚−𝑛_𝑠,2𝑛+𝑚for𝜈_𝑠≥2𝑛+𝑚−𝑛_𝑠. (52) If 𝑛_𝑠+1<2𝑛, we continue the procedure, and if once 𝑛_𝑠+1≥2𝑛, then we stop the construction and define 𝑡∶=𝑠+1.
By the previous definition of (𝑛_𝑠), we have the following obvious properties: 𝑡≥1,𝑛=𝑛₀<𝑛₁<⋯<𝑛_𝑡,2𝑛≤𝑛_𝑡𝑛≤2𝑛+𝑚,_𝑠+1−𝑛_𝑠≥𝑚+1(𝑠=0,1,…,𝑡−1), (53) and relations 𝐹_{𝑛𝑠+1−𝑚,𝜈}𝑓,𝑥_𝑋𝑝≤12𝐹_{𝑛𝑠−𝑚,𝜈}𝑓,𝑥_𝑋𝑝1for𝑠=0,1,…,𝑡−2,2𝐹_{𝑛𝑠−𝑚,𝜈}𝑓,𝑥_𝑋𝑝≤𝐹_{𝑛𝑠+1−𝑚−1,𝜈}𝑓,𝑥_𝑋𝑝for𝑠=0,1,…𝑡−1, (54) whenever 𝑛_𝑠+1−𝑛_𝑠>𝑚+1.
Let us start with 𝜎_𝑛,𝑚𝑓(𝑥)(𝑥=−𝑓)_{𝑡−1𝑠=0}𝜎_{𝑛𝑠,𝑚}𝑓(𝑥)−𝑓(𝑥)−𝜎_{𝑛𝑠+1,𝑚}𝑓(𝑥)(𝑥+𝜎−𝑓)_{𝑛𝑡,𝑚}𝑓(𝑥)(𝑥≤−𝑓)_{𝑡−1𝑠=0}𝜎_{𝑛𝑠+1,𝑚}𝑓(𝑥)−𝜎_{𝑛𝑠,𝑚}𝑓(𝑥)+𝜎_{𝑛𝑡,𝑚}𝑓(𝑥)(𝑥=−𝑓)_{𝑡−1𝑠=0}1𝜏𝑚+1^{𝑛𝑠,𝑚}𝑓(𝑥)+𝜎_{𝑛𝑡,𝑚}𝑓(𝑥)−𝑓(𝑥). (55) Using Theorem 1 and that 2𝑛≤𝑛_𝑡≤2𝑛+𝑚, we get 𝜎_{𝑛𝑡,𝑚}𝑓(𝑥)−𝑓(𝑥)≤𝐾𝐹_{𝑛𝑡−𝑚,𝑚}𝑓,𝑥_𝑋𝑝𝑛1+ln^𝑡+1+𝑓(𝑥)𝑚+1−𝑇_{𝑛𝑡−𝑚}(𝑥)≤𝐾^𝑛_𝜈=0𝐹_{𝑛−𝑚+𝜈,𝑚}𝑓,𝑥_𝑋𝑝+𝑓(𝑥)𝑚+𝜈+1−𝑇_{𝑛𝑡−𝑚}(𝑥)≤𝐾^𝑛_𝜈=0𝐹_{𝑛−𝑚+𝜈,𝑚}𝑓,𝑥_𝑋𝑝+𝐹_{𝑛−𝑚+𝜈,𝜈}𝑓,𝑥_𝑋𝑝+𝑓(𝑥)𝑚+𝜈+1−𝑇_{𝑛𝑡−𝑚}(𝑥)≤𝐾^𝑛_𝜈=0𝐹_{𝑛−𝑚+𝜈,𝑚}𝑓,𝑥_𝑋𝑝+𝐹_{𝑛−𝑚+𝜈,𝜈}𝑓,𝑥_𝑋𝑝𝑚+𝜈+1+𝐸_{𝑛𝑡−𝑚}𝑓,𝑥;0_𝑋𝑝≤𝐾^𝑛_𝜈=0𝐹_{𝑛−𝑚+𝜈,𝑚}𝑓,𝑥_𝑋𝑝+𝐹_{𝑛−𝑚+𝜈,𝜈}𝑓,𝑥_𝑋𝑝𝑚+𝜈+1+𝐸_2𝑛𝑓,𝑥;0_𝑋𝑝. (56)
The estimate of the sum in the right hand side of (55) we derive from the following one 1𝜏𝑚+1^{𝑛𝑠,𝑚}𝑓(𝑥)≤𝐾^{𝑛𝑠+1−𝑛𝑠−1}_𝜈=0𝐹_{𝑛𝑠−𝑚+𝜈,𝑚}𝑓,𝑥_𝑋𝑝+𝐹_{𝑛𝑠−𝑚+𝜈,𝑚}𝑓,𝑥_𝑋𝑝.𝑚+𝜈+1 (57) We split the proof of this inequality in two parts. If  𝑛_𝑠+1−𝑛_𝑠=𝑚+1, then by Lemma 9, 1𝜏𝑚+1^{𝑛𝑠,𝑚}𝑓(𝑥1)≤𝐾𝑚+1^𝑛𝑠_{𝑘=𝑛𝑠−𝑚}𝐹_{𝑘,𝑘−𝑛𝑠+𝑚}𝑓,𝑥_𝑋𝑝≤𝐾^{𝑛𝑠+1−𝑛𝑠−1}_𝜈=0𝐹_{𝑛𝑠−𝑚+𝜈,𝜈}𝑓,𝑥_𝑋𝑝.𝑚+𝜈+1 (58) If 𝑛_𝑠+1−𝑛_𝑠>𝑚+1, then, by Lemma 7, 1𝜏𝑚+1^{𝑛𝑠,𝑚}𝑓(𝑥)≤𝐾𝐹_{𝑛𝑠−𝑚,𝑚}𝑓,𝑥_𝑋𝑝^{𝑛𝑠+1−𝑛𝑠−1}_𝜈=01,𝑚+𝜈+1 (59) and since (1/2)𝐹_{𝑛𝑠−𝑚,𝑚}(𝑓,𝑥)_𝑋𝑝≤𝐹_{𝑛𝑠+1−𝑚−1,𝑚}(𝑓,𝑥)_𝑋𝑝, we have 1𝜏𝑚+1^{𝑛𝑠,𝑚}𝑓(𝑥)≤2𝐾𝐹_{𝑛𝑠+1−𝑚−1,𝑚}𝑓,𝑥_𝑋𝑝^{𝑛𝑠+1−𝑛𝑠−1}_𝜈=01𝑚+𝜈+1≤2𝐾^{𝑛𝑠+1−𝑛𝑠−1}_𝜈=0𝐹_{𝑛𝑠−𝑚+𝜈,𝑚}𝑓,𝑥_𝑋𝑝.𝑚+𝜈+1 (60)
Consequently, ^𝑡−1_𝑠=01𝜏𝑚+1^{𝑛𝑠,𝑚}𝑓(𝑥)≤2𝐾^𝑡−1𝑛_𝑠=0^{𝑠+1−𝑛𝑠−1}_𝜈=0𝐹_{𝑛𝑠−𝑚+𝜈,𝑚}𝑓,𝑥_𝑋𝑝+𝐹_{𝑛𝑠−𝑚+𝜈,𝜈}𝑓,𝑥_𝑋𝑝.𝑚+𝜈+1 (61) Since 𝑛_𝑠+1−𝑛_𝑠≤2𝑛+𝑚−𝑛−1=𝑛+𝑚−1 for all 𝑠≤𝑡−1, changing the order of summation, we get ^𝑡−1_𝑠=01𝜏𝑚+1^{𝑛𝑠,𝑚}𝑓(𝑥)≤2𝐾^{𝑛+𝑚−1}_𝜈=01𝑚+𝜈+1×_{𝑠∶𝑛𝑠+1−𝑛𝑠>𝜈}𝐹_{𝑛𝑠−𝑚+𝜈,𝑚}𝑓,𝑥_𝑋𝑝+𝐹_{𝑛𝑠−𝑚+𝜈,𝜈}𝑓,𝑥_𝑋𝑝. (62)
Using the inequality 𝐹_{𝑛𝑠+1−𝑚,𝜈}𝑓,𝑥_𝑋𝑝≤12𝐹_{𝑛𝑠−𝑚,𝜈}𝑓,𝑥_𝑋𝑝for𝜈=0,1,2,…,𝑛_𝑠+1−𝑛_𝑠−1,𝑠=0,1,2,…,𝑡−2, (63) we obtain _{𝑠∶𝑛𝑠+1−𝑛𝑠>𝜈}𝐹_{𝑛𝑠−𝑚+𝜈,𝑚}𝑓,𝑥_𝑋𝑝+𝐹_{𝑛𝑠−𝑚+𝜈,𝜈}𝑓,𝑥_𝑋𝑝=𝐹_{𝑛𝑝−𝑚+𝜈,𝑚}𝑓,𝑥_𝑋𝑝+𝐹_{𝑛𝑝−𝑚+𝜈,𝜈}𝑓,𝑥_𝑋𝑝+_{𝑠≥𝑝+1∶𝑛𝑠+1−𝑛𝑠>𝜈}𝐹_{𝑛𝑠+1−𝑚+𝜈,𝑚}𝑓,𝑥_𝑋𝑝+𝐹_{𝑛𝑠+1−𝑚+𝜈,𝜈}𝑓,𝑥_𝑋𝑝≤𝐹_{𝑛𝑝−𝑚+𝜈,𝑚}𝑓,𝑥_𝑋𝑝+𝐹_{𝑛𝑝−𝑚+𝜈,𝜈}𝑓,𝑥_𝑋𝑝+_{𝑠∶𝑠≥𝑝+1}𝐹_{𝑛𝑠−𝑚,𝑚}𝑓,𝑥_𝑋𝑝𝐹_{𝑛𝑠−𝑚,𝑚}𝑓,𝑥_𝑋𝑝+𝐹_{𝑛𝑠−𝑚,𝜈}𝑓,𝑥_𝑋𝑝≤𝐹_{𝑛𝑝−𝑚+𝜈,𝑚}𝑓,𝑥_𝑋𝑝+𝐹_{𝑛𝑝−𝑚+𝜈,𝜈}𝑓,𝑥_𝑋𝑝+2𝐹_{𝑛𝑝+1−𝑚,𝑚}𝑓,𝑥_𝑋𝑝+𝐹_{𝑛𝑝+1−𝑚,𝜈}𝑓,𝑥_𝑋𝑝≤3𝐹_{𝑛𝑝−𝑚+𝜈,𝑚}𝑓,𝑥_𝑋𝑝+𝐹_{𝑛𝑝−𝑚+𝜈,𝜈}𝑓,𝑥_𝑋𝑝, (64) where 𝑝 denotes the smallest index 𝑠 having the property 𝑛_𝑠+1−𝑛_𝑠>𝜈. Hence, ^𝑡−1_𝑠=01𝜏𝑚+1^{𝑛𝑠,𝑚}𝑓(𝑥)≤𝐾^{𝑛+𝑚−1}_𝜈=0𝐹_{𝑛−𝑚+𝜈,𝑚}𝑓,𝑥_𝑋𝑝+𝐹_{𝑛−𝑚+𝜈,𝜈}𝑓,𝑥_𝑋𝑝𝑚+𝜈+1≤𝐾^𝑛_𝜈=0𝐹_{𝑛−𝑚+𝜈,𝑚}𝑓,𝑥_𝑋𝑝+𝐹_{𝑛−𝑚+𝜈,𝜈}𝑓,𝑥_𝑋𝑝,𝑚+𝜈+1 (65) and our proof follows.