# Uniform Approximation of Periodical Functions by Trigonometric Sums of Special Type

Authors: A. S. Serdyuk; Ie. Yu. Ovsii
Journal: ISRN Mathematical Analysis
 (2014)

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## Abstract

The approximation characteristics of trigonometric sumsUn,pψ of special type on the class Cβ,∞ψ of (ψ,β)-differentiable (in the sense of A. I. Stepanets) periodical functions are studied. Because of agreement between parameters of approximative sums and approximated classes, the solution of Kolmogorov-Nikol’skii problem is obtained in a sufficiently general case. It is shown that in a number of important cases these sums provide higher order of approximation in comparison with Fourier sums, de la Vallée Poussin sums, and others on the class Cβ,∞ψ in the uniform metric. The range of parameters in which the sums Un,pψ give the order of the best uniform approximation on the classes Cβ,∞ψ is indicated.


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## Body

## 1. The Introduction and Problem Definition

LetC be the space of continuous 2π-periodical functions f where the norm is defined by ∥f∥C=maxx|f(x)|.Let us consider the classCβ,∞ψ [1] of functions f∈C such that for given β∈ℝ and fixed sequence ψ(k) (k∈ℕ) of real numbers the series
(1)∑k=1∞‍1ψ(k)(ak(f)cos(kx+βπ2)+bksin(kx+βπ2))
are the Fourier series of some function φ∈S∞, where
(2)S∞={φ:ess supt|φ(t)|⩽1}.
The function φ is called the (ψ,β)-derivative of f and denoted by fβψ. For ψ(k)=k-r, r>0, the class Cβ,∞ψ coincides with Weyl-Nagy class Wβr, and for β=r coincides with Weyl class Wrr. In the case of natural r and β=r the class Cβ,∞ψ is a class Wr of periodical functions whose rth derivatives nearly everywhere do not exceed unity in absolute value. If ψ(k) is a sequence such that
(3)limk→∞krψ(k)=0∀r∈ℝ,
then Cβ,∞ψ consists of infinitely differentiable functions (see [2, Chapter 1, Section 8]). The example of a sequence satisfying condition (3) is ψ(k)=e-αkr, α>0, r>0. In this case the class Cβ,∞ψ will be denoted by Cβ,∞α,r. If ψ(k) satisfies the condition
(4)|ψ(k)|⩽Ke-αk,k∈ℕ,α>0,Cβ,∞ψ consists of analytical functions regularly continuing into the strip |Imz|<α of the complex plane.Following [3, page 147], we set by 𝔐 the set of all continuous convex downwards functions ψ(t), t⩾1, satisfying the condition
(5)limt→∞ψ(t)=0
and associate each ψ∈𝔐 with the characteristic
(6)η(t)=η(ψ;t)=ψ-1(12ψ(t)),t⩾1,
where ψ-1(·) is an inverse function of ψ(·). By using η(t) we define the next subset of 𝔐 as
(7)F={ψ∈𝔐:η′(ψ;t)⩽K,∀t⩾1},iiiiiiiiiiiiiη′(ψ;t):=η′(ψ;t+0).
As is shown in [3, page 153] all functions ψ∈𝔐 for which
(8)0<C1⩽tT(t)⩽C2,∀t⩾1,C1,C2=const,
where T(t)=T(ψ;t)=η(ψ;t)-t, belong to F. The set of such functions is denoted by 𝔐C. The quantity T(t) has a simple geometric interpretation. It is equal to the length of interval where the value of the function ψ(t) decreases by two times. Thereby it is natural to call the function T(t) the half-decay period of ψ. The examples of ψ(·) from 𝔐C are the functions ψ1(t)=t-r, r>0, ψ2(t)=1/trln(t+γ), r>0, γ⩾1, and others. The set F also includes (see [3, page 153]) the subset 𝔐∞+ of all functions ψ∈𝔐 for which the characteristic
(9)μ(t)=μ(ψ;t)=tη(t)-t,t⩾1,
called the modulus of half-decay, tends monotonically to infinity as t→∞. If ψ3(t)=e-αtr, α>0, r>0, then η(ψ3;t)=t((ln2/αtr)+1)1/r and that is why
(10)μ(ψ3;t)=tη(ψ3;t)-t=1((ln2/αtr)+1)1/r-1↑∞,t⟶∞.
Therefore, ψ3∈𝔐∞+⊂F.Thus, inF there exist functions that tend to zero according to the power law as well as the functions that tend to zero faster than any power function. However, the example of the function ψ4(t)=1/ln(t+1)(η(ψ4;t)=(t+1)2-1, so η′(ψ4;t)=2(t+1)) shows that the set F may not contain functions tending to zero lower than any power function.In what follows, we assume that a sequenceψ(k) defining class Cβ,∞ψ is a restriction on the set ℕ of natural numbers of some function ψ(t), t⩾1, from F.We consider for an arbitrary functionf∈C the following sum:
(11)Un,pφ(f;x)=∑k=0n-1‍λn,p(k)Ak(f;x),
where
(12)λn,p(k)=λn,p(k;φ)={1,0⩽k⩽n-p,1-φ(k)φ(n),n-p+1⩽k⩽n-1,p∈ℕ, p∈[1,n], φ(k)(k∈ℕ) is a given monotonically increasing to infinity sequence of real numbers,
(13)Ak(f;x):=ak(f)coskx+bk(f)sinkx,A0(f;x):=a0(f)2,
and a0(f), ak(f), bk(f) are the Fourier coefficients of the function f.When the parametersp and φ(k) are chosen in a certain way, the sum Un,pφ(f;x) coincides with some classical sums such as Zygmund sum [4] (for p=n and φ(k)=ks, s>0)
(14)Zns(f;x)=∑k=0n-1‍(1-ksns)Ak(f;x),s>0,
Fejér sum [5] (for p=n and φ(k)=k)
(15)σn-1(f;x)=∑k=0n-1‍(1-kn)Ak(f;x),
de la Vallée Poussin sum [6] (for p∈ℕ, 1⩽p⩽n, and φ(k)=k-n+p)
(16)Vn,p(f;x)=∑k=0n-1‍λn,p(k)Ak(f;x),
where
(17)λn,p(k)={1,0⩽k⩽n-p,1-k-n+pp,n-p+1⩽k⩽n-1,
and Fourier sum (for p=1)
(18)Sn-1(f;x)=∑k=0n-1‍Ak(f;x).
For p=n, the sum Un,pφ(f;x) coincides with the so-called generalized Zygmund sum [7] (see also [8, 9])
(19)Znφ(f;x)=∑k=0n-1‍(1-φ(k)φ(n))Ak(f;x),
where φ(k) (k∈ℕ) is a given monotonically increasing to infinity sequence of real numbers.The aim of the current paper is an investigation of asymptotic behavior asn→∞ of the quantity
(20)ℰ(Cβ,∞ψ;Un,pψ)=supf∈Cβ,∞ψ∥f(·)-Un,pψ(f;·)∥C,ψ∈F,β∈ℝ,
where Un,pψ(f;·) is the sum Un,pφ(f;·) of type (11) with φ(k)=(k-n+p)/ψ(k), n,p∈ℕ, p⩽n. Note that this research area for Fourier sum, de la Vallée Poussin sum, and Zygmund sum has a long history on different functional classes. It is connected with Kolmogorov, Nikol’skii, Timan, Dzyadyk, Stechkin, Korneichuk, Efimov, Telyakovskii, Stepanets, Motornii, Trigub, Rukasov, and others. For more information on this subject see, for example, [10–13].Following Stepanets [1], we call that the solution of Kolmogorov-Nikol’skii problem is found for sum Un,pψ(f;·) on the class Cβ,∞ψ if the asymptotic equality
(21)ℰ(Cβ,∞ψ;Un,pψ)=ν(n)+o(1)ν(n),n⟶∞,
is obtained, where ν(n)=ν(n,p,ψ,β) is some specific sequence.In the current work, the solution of Kolmogorov-Nikol’skii problem for sumUn,pψ(f;·) is found on Cβ,∞ψ and it is shown that this sum provides higher order of approximation in the uniform metric than Fourier sum, de la Vallée Poussin sum, and other classical sums do.


## 2. The Main Results

Theorem 1.
Letψ∈F, β∈ℝ, n,p∈ℕ, and p⩽n. Then, as n→∞(22)ℰ(Cβ,∞ψ;Un,pψ)=ψ(n)(4π2An,pψ+O(1)),
where
(23)An,pψ={lnp,ifT(n)⩽1,lnpT(n),if1⩽T(n)⩽p,lnT(n)p,ifT(n)⩾p,T(n)=η(ψ;n)-n, η(ψ;n)=ψ-1((1/2)ψ(n)), and O(1) is a quantity uniformly bounded in β, n, and p.As is shown in [3, page 508], if ψ∈F and β∈ℝ, then for the quantity of the best uniform approximation of the class Cβ,∞ψ by trigonometric polynomials of order not more than n-1:
(24)En(Cβ,∞ψ)=supf∈Cβ,∞ψinftn-1∈𝒯n-1∥f(·)-tn-1(·)∥C,
the estimate
(25)En(Cβ,∞ψ)≍ψ(n)
is true (the notation α(n)≍β(n) means that there exist constants K1,K2>0 such that K1β(n)⩽α(n)⩽K2β(n)). Theorem 1 and (25) lead to the following statement.Corollary 2.
Letψ∈F, β∈ℝ, p=p(n), p,n∈ℕ, p⩽n, and T(n)≍p(n). Then the order estimate
(26)ℰ(Cβ,∞ψ;Un,pψ)≍En(Cβ,∞ψ)≍ψ(n)
holds, implying that the sum Un,pψ(f;x) provides the order of the best uniform approximation of the class Cβ,∞ψ.Iflimp→∞n→∞T(n)/p=ξ, where ξ=0 or ξ=∞, then equality (22) gives the solution of Kolmogorov-Nikol’skii problem for Un,pψ(f;x).From (22) and (23) we can write the following.Corollary 3.
Let the conditions of Theorem1 be satisfied. Then, as n→∞(27)ℰ(Cβ,∞ψ;Un,pψ)=ψ(n)(4π2An,pψ+O(1)),
where
(28)An,pψ={lnp,ifT(n)isbounded,ln+pT(n),ifT(n)≪p,ln+T(n)p,ifT(n)≫p,T(n)=η(ψ;n)-n, η(ψ;n)=ψ-1((1/2)ψ(n)), ln+t=max{lnt,0}, O(1) is a quantity uniformly bounded in β, n, and p, and the notation T(n)≪p(T(n)≫p) means that there exists a constant K>0 such that T(n)<Kp(T(n)>Kp).Taking into account that, forψα,r(t)=e-αtr, α>0, r>0, the equality
(29)T(ψα,r;t)=t1-r(ln2αr+O(1))
holds, from Corollary 3, we have the following.Corollary 4.
Letα>0, r>0, β∈ℝ, n,p∈ℕ, and p⩽n. Then, as n→∞(30)ℰ(Cβ,∞α,r;Un,pψα,r)=e-αnr(4π2An,pα,r+O(1)),
where
(31)An,pα,r={lnp,ifr⩾1,ln+pn1-r,ifr<1,n1-r≪p,ln+n1-rp,ifr<1,n1-r≫p,ln+n=max{lnn,0}, and O(1) is a quantity uniformly bounded in β, n, and p.Note that ifp→∞ and n-p→∞, then the sum Un,pψα,1(f;x) provides higher order of approximation on the class Cβ,∞α,1 than de la Vallée Poussin sum Vn,p(f;x) does. Indeed, it is proved in [14, page 981] (see also [15, page 130] and [16, page 10]) that
(32)ℰ(Cβ,∞α,1;Vn,p)=e-α(n-p+1)p×(4π(1-e-2α)+O(1)iiiiii×(e-α(1-e-α)3(n-p+1)+e-αp1-e-α)),
where O(1) is a quantity uniformly bounded in n, p, α, and β. Comparing (30) with (32), we find that if p=p(n) satisfies the condition
(33)p⟶∞,n-p⟶∞,
then
(34)limp→∞n-p→∞ℰ(Cβ,∞α,1;Un,pψα,1)ℰ(Cβ,∞α,1;Vn,p)=0.
As noted earlier, the sum Un,pψ(f;x) for p=1 coincides with Fourier sum Sn-1(f;x) of order n-1. For such a value of p, from Theorem 1 we get the following.Corollary 5.
Letψ∈F, β∈ℝ, and n∈ℕ. Then, as n→∞(35)ℰ(Cβ,∞ψ;Sn-1)=ψ(n)(4π2ln+T(n)+O(1)),
where ln+n=max{lnn,0}, T(n)=η(ψ;n)-n, η(ψ;n)=ψ-1((1/2)ψ(n)), and O(1) is a quantity uniformly bounded in β and n.Equality (35) was obtained by Stepanets (see, for example, [3, page 257]). It gives the solution of Kolmogorov-Nikol’skii problem for Fourier sum when
(36)T(n)⟶∞,n⟶∞.
For ψ(t)=t-r, β=r (in this case Cβ,∞ψ=Wrr and T(n)=(21/r-1)n), equality (35) takes the from
(37)ℰ(Wr;Sn-1)=n-r(4π2lnn+O(1)).
Asymptotic equality (37) was established by Kolmogorov [17] (for r∈ℕ) and Pinkewitch [18] (for r>0).Corollary2 and equality (35) show that in the case when T(n)→∞ and p=p(n) is chosen such that p(n)≍T(n), the sum Un,pψ(f;x) provides higher order of approximation on Cβ,∞ψ in comparison with Fourier sum.Settingp=n in Corollary 3 and taking into account that Un,nψ(f;x) is a generalized Zygmund sum Znφ(f;x)(φ(k)=k/ψ(k), k∈ℕ), we obtain the following.Corollary 6.
Let∈F, φ(k)=k/ψ(k), β∈ℝ, and n∈ℕ. Then, as n→∞(38)ℰ(Cβ,∞ψ;Znφ)=ψ(n)(4π2Anψ+O(1)),
where
(39)Anψ={lnn,ifT(n)isbounded,ln+nT(n),ifT(n)≪n,ln+T(n)n,ifT(n)≫n,T(n)=η(ψ;n)-n, η(ψ;n)=ψ-1((1/2)ψ(n)), ln+n=max{lnn,0}, O(1) is a quantity uniformly bounded in β, n, and the notation T(n)≪n(T(n)≫n) means that there exists a constant K>0 such that T(n)<Kn(T(n)>Kn).Since by definition (8)
(40)T(n)≍n∀ψ∈𝔐C,
from (25), (38), and inclusion 𝔐C⊂F, we have
(41)ℰ(Cβ,∞ψ;Znφ)≍En(Cβ,∞ψ)≍ψ(n),ψ∈𝔐C.
This estimate follows also from Corollary 2 for p=n.Takingψ(t)=t-r, r>0, in (41) (in this case Cβ,∞ψ=Wβ,∞r and Znφ(f;x)=Zns(f;x), s=r+1), we get
(42)ℰ(Wβ,∞r;Zns)≍n-r.
Note that estimate (42) can be obtained from more precise results of Telyakovskii [19] (see Theorem 7).


## 3. Proof of Theorem1

Consider the quantity(43)ρn,p(f;x):=f(x)-Un,pψ(f;x),x∈ℝ,
where f∈Cβ,∞ψ and ψ belongs to the set
(44)𝔐'={ψ∈𝔐:∫1∞‍ψ(t)tdt<∞}
(as shown in [3, page 155], F⊂𝔐'). By Theorem 4.1 [2, page 71], the following equality holds at any point x:
(45)ρn,p(f;x)=∫-∞∞‍fβψ(x+tn)τ^n,p(t)dt,n∈ℕ,
where
(46)τ^n,p(t):=1π∫0∞‍τn,p(u)cos(ut+βπ2)du,(47)τn,p(u)={0,0⩽u⩽n-pn,ψ(n)nu-n+pp,n-pn⩽u⩽1,ψ(nu),u⩾1.
Let us simplify the right-hand side of (45) to obtain the principal term of the quantity ρn,p(f;x). To do this, set
(48)τ^n,p+(t):=1π∫0∞‍τn,p(u)cosutdu,(49)τ^n,p-(t):=1π∫0∞‍τn,p(u)sinutdu.
Taking (48) and (49) into account, equality (45) can be written in the following form:
(50)ρn,p(f;x)=cosβπ2∫-∞∞‍fβψ(x+tn)τ^n,p+(t)dt-sinβπ2∫-∞∞‍fβψ(x+tn)τ^n,p-(t)dt=cosβπ2ρn,p+(f;x)-sinβπ2ρn,p-(f;x).Since the classCβ,∞ψ is invariant under shifts of the argument (if f∈Cβ,∞ψ, then the function f1(·)=f(·+h), h∈ℝ, also belongs to Cβ,∞ψ), for quantity (20) we have
(51)ℰ(Cβ,∞ψ;Un,pψ)=supf∈Cβ,∞ψ|ρn,p(f;0)|.
It follows from (51) that we may restrict our attention to ρn,p(f;x) at x=0.We prove the following result.Lemma 7.
Assume thatψ∈𝔐', β∈ℝ, n,p∈ℕ, p⩽n, and α(n) is an arbitrary sequence of real numbers satisfying the condition
(52)α(n)⩾Kn,n∈ℕ,
where K is some positive constant. Then for any function f∈Cβ,∞ψ, as n→∞, one has
(53)ρn,p(f;0)=(-1)sψ(n)π∫ℐαn‍fβψ(tn)sin(t+(βπ/2))tdt+O(1)rn,
where
(54)ℐαn=(-np,-α(n)n)⋃‍(α(n)n,np),ifα(n)⩽1p,ℐαn=(-α(n)n,-np)⋃‍(np,α(n)n),if1p<α(n)⩽1,ℐαn=(-n,-np)⋃‍(np,n),ifα(n)>1,s=s(α,p)={1,α(n)⩽1p,0,α(n)>1p,rn=ψ(n)+∫1/α(n)∞‍ψ(t+n)tdt+∫α(n)∞‍ψ(n)-ψ(n+1/t)tdt,
and O(1) is a quantity uniformly bounded in β, n, and p.Proof.
By (50) Lemma 7 will be established if we show that
(55)ρn,p+(f;0)=(-1)sψ(n)π∫ℐαn‍fβψ(tn)sinttdt+O(1)rn,(56)ρn,p-(f;0)=(-1)s+1ψ(n)π∫ℐαn‍fβψ(tn)costtdt+O(1)rn.
Since estimate (56) can be obtained in the same way as (55) we only prove estimate (55).
Assume thatα(n)⩽1/p. Represent the quantity ρn,p+(f;0) in the form
(57)ρn,p+(f;0)=∫|t|⩽α(n)n‍fβψ(tn)τ^n,p+(t)dt+∫α(n)n⩽|t|⩽n/p‍fβψ(tn)τ^n,p+(t)dt+∫|t|⩾n/p‍fβψ(tn)τ^n,p+(t)dt.
Using for |t|⩽α(n)n the expression
(58)τ^n,p+(t)=ψ(n)π∫(n-p)/n1‍nu-n+ppcosutdu+1π∫1∞‍ψ(nu)cosutdu,
obtained from (47) and (48), we have
(59)∫|t|⩽α(n)n‍fβψ(tn)τ^n,p+(t)dt=ψ(n)π∫|t|⩽α(n)n‍fβψ(tn)∫(n-p)/n1‍nu-n+ppcosutdudt+1π∫|t|⩽α(n)n‍fβψ(tn)∫1∞‍ψ(nu)cosutdudt.
Taking into account the inclusion fβψ∈S∞, the estimate
(60)|∫(n-p)/n1‍nu-n+ppcosutdu|⩽∫(n-p)/n1‍nu-n+ppdu=p2n,
and condition α(n)⩽1/p, it follows from (59) that
(61)∫|t|⩽α(n)n‍fβψ(tn)τ^n,p+(t)dt=1π∫|t|⩽α(n)n‍fβψ(tn)∫1∞‍ψ(nu)cosutdudt+O(1)ψ(n).
As proved in [3, Chapter 5, Section 11] (see (11.9) and (11.17)), if fβψ∈S∞, then
(62)∫|t|⩽α(n)n‍fβψ(tn)∫1∞‍ψ(nu)cosutdudt=O(1)(ψ(n)+∫1/α(n)∞‍ψ(t+n)tdt).
Therefore, from (61) we obtain
(63)∫|t|⩽α(n)n‍fβψ(tn)τ^n,p+(t)dt=O(1)(ψ(n)+∫1/α(n)∞‍ψ(t+n)tdt).
Consider the second integral on the right-hand side of (57). To find its principal term represent the quantity τ^n,p+(t) in the form
(64)τ^n,p+(t)=ψ(n)π∫(n-p)/n1‍nu-n+ppcosutdu-ψ(n)πtsint-nπt∫1∞‍ψ′(nu)sinutdu,
obtained from (58) by integrating the second integral by parts:
(65)∫1∞‍ψ(nu)cosutdu=-ψ(n)tsint-nt∫1∞‍ψ′(nu)sinutdu.
As is easy to see, estimate (60) implies that
(66)∫α(n)n⩽|t|⩽n/p‍fβψ(tn)∫(n-p)/n1‍nu-n+ppcosutdudt=O(1).
Then by (64)
(67)∫α(n)n⩽|t|⩽n/p‍fβψ(tn)τ^n,p+(t)dt=-ψ(n)π∫α(n)n⩽|t|⩽n/p‍fβψ(tn)sinttdt-nπ∫α(n)n⩽|t|⩽n/p‍fβψ(tn)1t∫1∞‍ψ′(nu)sinutdudt+O(1)ψ(n).
Now we consider the third integral in (57). Integration by parts in the first integral of (64) yields
(68)τ^n,p+(t)=ψ(n)πt2np(cost-cosn-pnt)-nπt∫1∞‍ψ′(nu)sinutdu.
Since
(69)np∫|t|⩾n/p‍fβψ(tn)1t2(cost-cosn-pnt)dt=O(1),
from (68) and (69) we have
(70)∫|t|⩾n/p‍fβψ(tn)τ^n,p+(t)dt=-nπ∫|t|⩾n/p‍fβψ(tn)1t∫1∞‍ψ′(nu)sinutdudt+O(1)ψ(n).
Combining (63), (67), (70), and the estimate
(71)n∫|t|⩾α(n)n‍fβψ(tn)1t∫1∞‍ψ′(nu)sinutdudt=O(1)∫α(n)∞‍ψ(n)-ψ(n+1/t)tdt,
proved in [3] (see (11.31)), from (57) we get
(72)ρn,p+(f;0)=-ψ(n)π∫α(n)n⩽|t|⩽n/p‍fβψ(tn)sinttdt+O(1)(ψ(n)+∫1/α(n)∞‍ψ(t+n)tdtdddddhd+∫α(n)∞‍ψ(n)-ψ(n+1/t)tdt).
Assume thatα(n)>1/p. The following equality holds:
(73)ρn,p+(f;0)=∫|t|⩽n/p‍fβψ(tn)τ^n,p+(t)dt+∫n/p⩽|t|⩽α(n)n‍fβψ(tn)τ^n,p+(t)dt+∫|t|⩾α(n)n‍fβψ(tn)τ^n,p+(t)dt.
By using representation (58) and estimate (60) it is not hard to establish that
(74)∫|t|⩽n/p‍fβψ(tn)τ^n,p+(t)dt=1π∫|t|⩽n/p‍fβψ(tn)∫1∞‍ψ(nu)cosutdudt+O(1)ψ(n).
In view of (65) and (68), we have
(75)τ^n,p+(t)=ψ(n)πsintt+1π∫1∞‍ψ(nu)cosutdu+ψ(n)πt2np(cost-cosn-pnt).
This and
(76)np∫n/p⩽|t|⩽α(n)n‍fβψ(tn)1t2(cost-cosn-pnt)dt=O(1)np∫n/pα(n)n‍dtt2=O(1)
give us the estimate for the second integral in (73):
(77)∫n/p⩽|t|⩽α(n)n‍fβψ(tn)τ^n,p+(t)dt=ψ(n)π∫n/p⩽|t|⩽α(n)n‍fβψ(tn)sinttdt+1π∫n/p⩽|t|⩽α(n)n‍fβψ(tn)∫1∞‍ψ(nu)cosutdudt+O(1)ψ(n).
To find the estimate of the third integral on the right-hand side of (73) we use representation (68). We have
(78)∫|t|⩾α(n)n‍fβψ(tn)τ^n,p+(t)dt=ψ(n)πnp∫|t|⩾α(n)n‍fβψ(tn)1t2(cost-cosn-pnt)dt-nπ∫|t|⩾α(n)n‍fβψ(tn)1t∫1∞‍ψ′(nu)sinutdudt.
Since
(79)np∫|t|⩾α(n)n‍fβψ(tn)1t2(cost-cosn-pnt)dt=O(1)np∫α(n)n∞‍dtt2=O(1),
we hence obtain from (78) by using (71) that
(80)∫|t|⩾α(n)n‍fβψ(tn)τ^n,p+(t)dt=O(1)(ψ(n)+∫α(n)∞‍ψ(n)-ψ(n+1/t)tdt).
Combining (73), (74), (77), and (80) and taking into account (62), we get
(81)ρn,p+(f;0)=ψ(n)π∫n/p⩽|t|⩽α(n)n‍fβψ(tn)sinttdt+O(1)(ψ(n)+∫1/α(n)∞‍ψ(t+n)tdtiiiiiiiiiiiiiii+∫α(n)∞‍ψ(n)-ψ(n+1/t)tdt).
For the proof of lemma in the case whenα(n)>1, it is enough to apply the estimate [1, page 119]  (see Lemma 5)
(82)|∫|t|⩾α*(n)‍φ(tn)sin(t+(γπ/2))tdt|=O(1),φ∈S∞,n⟶∞,γ∈ℝ,
where {α*(n)}={α(n):α(n)⩾nπ}, to the first integral of (81). Indeed, using (82) we can write
(83)∫n/p⩽|t|⩽α(n)n‍fβψ(tn)sinttdt=∫n/p⩽|t|⩽n‍fβψ(tn)sinttdt+O(1).
Thus, for α(n)>1 estimate (55) follows from (81) and (83). Lemma 7 is proved.Setting in Lemma7(84)α(n)=α(ψ;n)=1η(ψ;n)-n,
where η(ψ;n) is defined by (6) and ψ∈F (F⊂𝔐'), and taking into account the following estimates from [3, pages 155, 156]:
(85)∫η(ψ;n)∞‍ψ(t)t-ndt=O(1)ψ(n),∀ψ∈F,∫nη(ψ;n)‍ψ(n)-ψ(t)t-ndt=O(1)ψ(n),∀ψ∈F,(86)μ(n)=μ(ψ;n)=nη(ψ;n)-n⩾K>0,∀ψ∈F,
we obtain the following statement.Corollary 8.
Assume thatψ∈F, β∈ℝ, n,p∈ℕ, and p⩽n. Then for any f∈Cβ,∞ψ, as n→∞, the following equality holds:
(87)ρn,p(f;0)=(-1)sψ(n)π×∫ℐn‍fβψ(tn)sin(t+(βπ/2))tdt+O(1)ψ(n),
where
(88)ℐn=(-np,-μ(n))⋃‍(μ(n),np),ifμ(n)⩽np,ℐn=(-μ(n),-np)⋃‍(np,μ(n)),ifnp<μ(n)⩽n,ℐn=(-n,-np)⋃‍(np,n),ifμ(n)>n,μ(n)=μ(ψ;n)=nη(ψ;n)-n,s=s(ψ,n,p)={1,μ(n)⩽np,0,μ(n)>np,
and O(1) is a quantity uniformly bounded in β, n, and p.We now set(89)tk=(2k+1-β)π2,k∈ℤ,xk=tk+π2,k∈ℤ,(90)ln(t)={1xk,t∈[tk,tk+1],k=k0,k0+1,…,k1-1,iiiiiiiiiiiiiiiiiiiiiiiiiiik2,k2+1,…,k3-1,0,t∈(-∞,tk0)∪(tk3,∞),
where k0, k1, k2, and k3 are selected in such a way that
(91)tk0-1<-np⩽tk0,tk1<-μ(n)⩽tk1+1,tk2-1<μ(n)⩽tk2,tk3<np⩽tk3+1,
if μ(n)⩽n/p,
(92)tk0-1<-μ(n)⩽tk0,tk1<-np⩽tk1+1,tk2-1<np⩽tk2,tk3<μ(n)⩽tk3+1,
if n/p<μ(n)⩽n, and
(93)tk0-1<-n⩽tk0,tk1<-np⩽tk1+1,tk2-1<np⩽tk2,tk3<n⩽tk3+1,
if μ(n)>n.In the above notation the integral on the right-hand side of (87) can be represented in the form
(94)∫ℐn‍fβψ(tn)sin(t+(βπ/2))tdt=∫(tk0,tk1)∪(tk2,tk3)‍fβψ(tn)ln(t)sin(t+βπ2)dt+Rn,p(1)+Rn,p(2),
where
(95)Rn,p(1)=∫(tk0,tk1)∪(tk2,tk3)‍fβψ(tn)(1t-ln(t))sin(t+βπ2)dt,(96)Rn,p(2)=(∫-atk0‍+∫tk1-b‍+∫btk2‍+∫tk3a‍)fβψ(tn)×sin(t+(βπ/2))tdt,(97)a={np,μ(n)⩽np,μ(n),np<μ(n)⩽n,n,μ(n)>n,(98)b={μ(n),μ(n)⩽np,np,μ(n)>np.We claim that(99)Rn,p(1)=O(1),(100)Rn,p(2)=O(1).
Making elementary transformations in (95) and taking into account that |fβψ(·)|⩽1, we have
(101)Rn,p(1)=O(1)×(∑k=k0k1-1‍∫tktk+1‍|1t-1xk|dt+∑k=k2k3-1‍∫tktk+1‍|1t-1xk|dt).
Since
(102)∫tkxk‍|1t-1xk|dt>∫xktk+1‍|1t-1xk|dt,k∈ℤ,
we can write the estimate
(103)∫tktk+1‍|1t-1xk|dt<2∫tkxk‍|1t-1xk|dt=2∫tkxk‍xk-t|txk|dt⩽π∫tkxk‍dtt2<π∫tktk+1‍dtt2.
Combining (101) and (103), we obtain
(104)Rn,p(1)=O(1)(∫tk0tk1‍dtt2+∫tk2tk3‍dtt2)=O(1)(1|tk1|+1tk2)=O(1)max{1μ(n),1}.
By (86) μ(n) is bounded below; therefore, (99) follows from (104).Since(105)tk0+a<tk0-tk0-1=π,-b-tk1⩽tk1+1-tk1=π,tk2-b<tk2-tk2-1=π,a-tk3⩽tk3+1-tk3=π,
it follows from (96) that
(106)Rn,p(2)=O(1)(tk0+a|tk0|+-b-tk1b+tk2-bb+a-tk3tk3)=O(1)(1|tk0|+1b+1tk3)=O(1)max{1μ(n),1}=O(1).Combining (87), (94), (99), and (100), we obtain
(107)ρn,p(f;0)=(-1)sψ(n)π∫(tk0,tk1)∪(tk2,tk3)‍fβψ(tn)ln(t)sin(t+βπ2)dt+O(1)ψ(n).We proceed to the finding of the estimate ofℰ(Cβ,∞ψ;Un,pψ). Substituting (107) in (51), we have
(108)ℰ(Cβ,∞ψ;Un,pψ)⩽ψ(n)π∫(tk0,tk1)∪(tk2,tk3)‍|ln(t)sin(t+βπ2)|dt+O(1)ψ(n).
We shall show that
(109)∫(tk0,tk1)∪(tk2,tk3)‍|ln(t)sin(t+βπ2)|dt=4πAn,pψ+O(1),
where
(110)An,pψ={lnnpμ(n),ifμ(n)⩽np,lnpμ(n)n,ifnp<μ(n)⩽n,lnp,ifμ(n)>n.Bearing in mind the symmetry of the functionsin(t+βπ/2) relative to xk of the segment [tk,tk+1], k∈ℤ, we obtain
(111)∫tktk+1‍|sin(t+βπ2)|dt=2∫tkxk‍|sin(t+βπ2)|dt=2∫(k+(1/2))π(k+1)π‍|sint|dt=2∫0π/2‍costdt=2.
From this we get that
(112)∫(tk0,tk1)∪(tk2,tk3)‍|ln(t)sin(t+βπ2)|dt=∑k=k0k1-1‍1|xk|∫tktk+1‍|sin(t+βπ2)|dt+∑k=k2k3-1‍1xk∫tktk+1‍|sin(t+βπ2)|dt=2(∑k=k0k1-1‍1|xk|+∑k=k2k3-1‍1xk).
Since
(113)tk+1-tk=π,k∈ℤ,
this gives us the equality
(114)∑k=k0k1-1‍1|xk|=1π∑k=k0k1-1‍∫tktk+1‍1|xk|dt=-1π∫tk0tk1‍dtt+1π∑k=k0k1-1‍∫tktk+1‍(-1xk+1t)dt.
Using (103), we have
(115)|∑k=k0k1-1‍∫tktk+1‍(-1xk+1t)dt|⩽∑k=k0k1-1‍∫tktk+1‍|1t-1xk|dt<π∑k=k0k1-1‍∫tktk+1‍dtt2=π∫tk0tk1‍dtt2<π|tk1|⩽Kmax{1μ(n),1}=O(1).
Comparing (114) and (115), we find that
(116)∑k=k0k1-1‍1|xk|=-1π∫tk0tk1‍dtt+O(1).
Using the notations of (97) and (98), we represent (116) in the form
(117)∑k=k0k1-1‍1|xk|=-1π∫-a-b‍dtt+1π∫-a-b‍dtt-1π∫tk0tk1‍dtt+O(1)=-1π∫-a-b‍dtt+∫tk1-b‍dtt+∫-atk0‍dtt+O(1).Since(118)-b-tk1⩽π,tk0+a<π,
we have
(119)∫tk1-b‍dtt+∫-atk0‍dtt=O(1)(1b+1|tk0|)=O(1)1|tk0|=O(1)max{1μ(n),1}=O(1).
Comparing (117) and (119), we obtain
(120)∑k=k0k1-1‍1|xk|=-1π∫-a-b‍dtt+O(1)=1πlnab+O(1).
Thus,
(121)∑k=k0k1-1‍1|xk|=1πAn,pψ+O(1),
where An,pψ is defined by (110).In an analogous way one can prove that(122)∑k=k2k3-1‍1xk=1πAn,pψ+O(1).
Relation (109) follows from (112), (121), and (122).Comparing (108) and (109), we find
(123)ℰ(Cβ,∞ψ;Un,pψ)⩽ψ(n)(4π2An,pψ+O(1)).We shall show that the equality sign can be put in (123). Denote by φ0(t) the 2π-periodic function such that
(124)∫-ππ‍φ0(t)dt=0,|φ0(t)|⩽1,φ0(t)=sign(ln(nt)sin(nt+βπ2)),t∈[-1,1],
where ln(t) is defined by (90). Obviously, such a function exists. According to Subsection 7.2 of [2, pages 136, 137] in Cβ,∞ψ, ψ∈F, there exists a function f0(t) such that φ0(t) is its (ψ,β)-derivative. From (107) and (109), we have
(125)|ρn,p(f0;0)|=ψ(n)π|∫(tk0,tk1)∪(tk2,tk3)‍φ0(tn)ln(t)sin(t+βπ2)dt|+O(1)ψ(n)=ψ(n)π∫(tk0,tk1)∪(tk2,tk3)‍|ln(t)sin(t+βπ2)|dt+O(1)ψ(n)=ψ(n)(4π2An,pψ+O(1)).
Since ℰ(Cβ,∞ψ;Un,pψ)⩾|ρn,p(f0,0)|, it follows from (123) and (125) that if ψ∈F, β∈ℝ, n,p∈ℕ, p⩽n, and n→∞, then
(126)ℰ(Cβ,∞ψ;Un,pψ)=ψ(n)(4π2An,pψ+O(1)),
where
(127)An,pψ={lnp,ifT(n)⩽1,lnpT(n),if1⩽T(n)⩽p,lnT(n)p,ifT(n)⩾p,T(n)=η(ψ;n)-n, η(ψ;n)=ψ-1((1/2)ψ(n)), and O(1) is a quantity uniformly bounded in β, n, and p.Theorem1 is proved.



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*Source: 165389-2014-01-05.xml*