About Lebesgue inequalities on the classes of generalized Poisson integrals
aa r X i v : . [ m a t h . C A ] M a y A. S. Serdyuk (Institute of Mathematics of NAS of Ukraine, Ukraine)
T. A. Stepanyuk (University of L¨ubeck, Germany; Institute of Mathematicsof NAS of Ukraine, Ukraine)
About Lebesgue inequalities on the classes of generalized Poissonintegrals
For the functions f , which can be represented in the form of the convolution f ( x ) = a + π π R − π ∞ P k =1 e − αk r cos( kt − βπ ) ϕ ( x − t ) dt , ϕ ⊥ α > , r ∈ (0 , β ∈ R , we establish theLebesgue-type inequalities of the form k f − S n − ( f ) k C ≤ e − αn r (cid:18) π ln n − r αr + γ n (cid:19) E n ( ϕ ) C . These inequalities take place for all numbers n that are larger than some number n = n ( α, r ), which con-structively defined via parameters α and r . We prove that there exists a function, such that the sign ” ≤ ” ingiven estimate can be changed for ”=”. Keywords:
Lebesgue inequalities, Fourier sums, classes of convolutions ofperiodic functions, best approximation.
1. Introduction
Let L p , 1 ≤ p < ∞ , be the space of 2 π –periodic functions f summable tothe power p on [0 , π ), in which the norm is given by the formula k f k p = (cid:16) π R | f ( t ) | p dt (cid:17) p ; L ∞ be the space of measurable and essentially bounded 2 π –periodic functions f with the norm k f k ∞ = ess sup t | f ( t ) | ; C be the space ofcontinuous 2 π –periodic functions f , in which the norm is specified by the equal-ity k f k C = max t | f ( t ) | .By ρ n ( f ; x ) we denote the deviation of the function f from its partial Fouriersum of order n − ρ n ( f ; x ) := f ( x ) − S n − ( f ; x ) , where S n − ( f ; x ) = a n − X k =1 ( a k cos kx + b k sin kx ) ,a k = a k ( f ) = 1 π π Z − π f ( t ) cos ktdt, b k = b k ( f ) = 1 π π Z − π f ( t ) sin ktdt, and by E n ( f ) C we denote the best uniform approximation of the function f byelements of the subspace τ n − of trigonometric polynomials t n − ( · ) of the order n − E n ( f ) C := inf t n − ∈ τ n − k f − S n − ( f ) k C . The norms k ρ n ( f ; · ) k C can be estimated via E n ( f ) C , using the Lebesgueinequality k ρ n ( f ; · ) k C ≤ (1 + L n − ) E n ( f ) C , n ∈ N . (1)Here the sequence of numbers L n − = 1 π π Z − π | D n − ( t ) | dt = 2 π π Z | sin(2 n − t | sin t dt, where D n − ( t ) := 12 + ∞ X k =1 cos kt = sin( n − ) t t , are called the Lebesgue constants of the Fourier sums.The asymptotic equality for Lebesgue constants L n was obtained in [1]: L n = 4 π ln n + O (1) , n → ∞ . For more exact estimates for the differences L n − π ln( n + a ), a >
0, as n ∈ N the reader can be referred to the works [2]–[8]. In particular, it follows from [5](see also [7, p.97]) that (cid:12)(cid:12)(cid:12)(cid:12) L n − − π ln n (cid:12)(cid:12)(cid:12)(cid:12) < , , n ∈ N . Then, the inequality (1) can be written in the form k ρ n ( f ; · ) k C ≤ (cid:18) π ln n + R n (cid:19) E n ( f ) C , (2)where | R n | < , C the inequality (2) is asymptotically exact. At thesame there exist subsets of functions from C and for elements of these subsetsthe inequality (2) is not exact even by order (see, e.g., [9, p. 434]).In the paper [11] the following estimate was proved k ρ n ( f ; · ) k C ≤ K n − X ν = n E ν ( f ) C ν − n + 1 , f ∈ C, n → ∞ , (here K is some absolute constant) and it was proved that this constant isexact by the order on the classes C ( ε ) with a given majorant of the best ap-proximations C ( ε ) := { f ∈ C : E ν ( f ) C ≤ ε ν , ν ∈ N } , { ε ν } ∞ ν =0 is a sequenceof nonnegative numbers, such that ε ν ↓ ν → ∞ . This estimate sharpensLebesgue classical inequality for ”fast” decreasing E ν .In [9]–[14] (see also [5]) for the classes C ψβ C of the functions f ∈ C , whichare defined with a help of convolutions f ( x ) = a π π Z − π Ψ β ( x − t ) ϕ ( t ) dt, ϕ ⊥ , ϕ ∈ C, a ∈ R , (3)with summable kernels Ψ β ( t ) whose Fourier series has the formΨ β ( t ) ∼ ∞ X k =1 ψ ( k ) cos (cid:0) kt − βπ (cid:1) , ψ ( k ) ≥ , β ∈ R , asymptotically best possible analogs of Lebesgue-type inequalities were found.In these inequalities the norms of deviations of Fourier sums k ρ n ( f ; · ) k C areexpressed via the best approximations E n ( ϕ ) C of the function ϕ (the function ϕ ,which is connected with f with a help of equality (3) is called ( ψ, β )–derivativeof the function f and is denoted by f ψβ ).Denote by C α,rβ C, α > , r > , the set of all 2 π –periodic functions, suchthat for all x ∈ R can be represented in the form of convolution f ( x ) = a π π Z − π P α,r,β ( x − t ) ϕ ( t ) dt, a ∈ R , ϕ ⊥ , (4)where ϕ ∈ C , and P α,r,β ( t ) is a generalized Poisson kernel of the form P α,r,β ( t ) = ∞ X k =1 e − αk r cos (cid:0) kt − βπ (cid:1) , α > , r > , β ∈ R . If f and ϕ are connected with a help of equality (4), then the function f inthis equality is called the generalized Poisson integral of the function ϕ and isdenoted by J α,rβ ( ϕ ). The function ϕ in the equality (4) is called the generalizedderivative of the function f and is denoted by f α,rβ .It is clear that the sets of generalized Poisson integrals C α,rβ C are subsets ofthe sets C ψβ C , if to put ψ ( k ) = e − αk r , α > r >
0. In this case for all t ∈ R the equality holds f ψβ ( t ) = f α,rβ ( t ).It should be noticed that for any r > C α,rβ C belong to set ofinfinitely differentiable 2 π –periodic functions D ∞ , i.e., C α,rβ C ⊂ D ∞ (see, e.g.,[5, p. 128]) For r ≥ C α,rβ C consist of functions f , admitting aregular extension into the strip | Im z | ≤ c, c > r > C α,rβ C consist of functions regular on the whole complex plane, i.e., of entirefunctions (see, e.g., [5, p. 131]). Besides, it follows from the Theorem 1 in [10]that for any r > C α,rβ C ⊂ J /r , where J a , a > , areknown Gevrey classes J a = (cid:26) f ∈ D ∞ : sup k ∈ N (cid:16) k f ( k ) k C ( k !) a (cid:17) /k < ∞ (cid:27) . In the paper of Stepanets [9] the general results were obtained. From them,in particular, it follows that for any f ∈ C α,rβ C , r ∈ (0 , α > β ∈ R , forany n ∈ N the following asymptotically best possible inequality holds k ρ n ( f ; x ) k C ≤ e − αn r (cid:18) π ln n − r + O (1) (cid:19) E n ( f α,rβ ) C , (5)where O (1) is a quantity uniformly bounded with respect to f ∈ C α,rβ C , n ∈ N and β ∈ R .Herewith the behavior (speed of increasing) of the quantity O (1) in theinequality (5) with respect to values of parameters α and r in the work [9] wasnot considered.In present paper we establish the asymptotically best possible Lebesgue-typeinequalities for the functions f ∈ C α,rβ C , in which for all n , starting from somenumber n = n ( α, r ), an additional term is estimated by absolute constant.Herewith the number n is defined constructively via parameters of the problem(the inequality (6)), and an absolute constant is written in an explicit form 20 π .Obtained results complement the results of the papers [15]–[16], and also clarifythe estimate (5), which was obtained in [9].
2. Main results
Let us formulate now the main results of the paper.For arbitrary α > r ∈ (0 ,
1) we denote by n = n ( α, r ) the smallestinteger n ∈ N , such that1 αr n r (cid:16) πn − r αr (cid:17) + αrn − r ≤ π ) . (6) Theorem 1.
Let α > , r ∈ (0 , , β ∈ R and n ∈ N . Then, for any function f ∈ C α,rβ C and all n ≥ n ( α, r ) the following inequality holds k ρ n ( f ; · ) k C ≤ e − αn r (cid:18) π ln n − r αr + γ n (cid:19) E n ( f α,rβ ) C . (7) Moreover, for arbitrary function f ∈ C α,rβ C one can find a function F ( x ) = F ( f, n, x ) from the set C α,rβ C , such that E n ( F α,rβ ) C = E n ( f α,rβ ) C , such that for n ≥ n ( α, r ) the equality holds k ρ n ( F ; · ) k C = e − αn r (cid:18) π ln n − r αr + γ n (cid:19) E n ( f α,rβ ) C . (8) In (7) and (8) for the quantity γ n = γ n ( α, r, β ) the estimate holds | γ n | ≤ π .Proof. Let f ∈ C α,rβ C . Then, for arbitrary x ∈ R the following integral repre-sentation takes place ρ n ( f ; x ) = f ( x ) − S n − ( f ; x ) = 1 π π Z − π f α,rβ ( t ) P ( n ) α,r,β ( x − t ) dt, (9)where P ( n ) α,r,β ( t ) := ∞ X k = n e − αk r cos (cid:16) kt − βπ (cid:17) , < r < , α > , β ∈ R . (10)Whereas the function P ( n ) α,r,β ( t ) is orthogonal to any trigonometric polynomial t n − ∈ τ n − , then because of (9) ρ n ( f ; x ) = f ( x ) − S n − ( f ; x ) = 1 π π Z − π δ n ( t ) P ( n ) α,r,β ( x − t ) dt, (11)where δ n ( x ) = δ n ( α, r, β ; x ) := f α,rβ ( x ) − t n − ( x ) . (12)By t ∗ n − ∈ τ n − we denote the polynomial of the best uniform approximationof the function f α,rβ , namely, such that k f α,rβ − t ∗ n − k C = E n ( f α,rβ ) C . Then, in view of (11), we have k f ( · ) − S n − ( f ; · ) k C ≤ π k P ( n ) α,r,β k E n ( f α,rβ ) C . (13)As it follows from the formula (20) of the paper [17] (see also [18]) for arbi-trary r ∈ (0 , α > β ∈ R , 1 ≤ s < ∞ , s + s ′ = 1, n ∈ N and n ≥ n ( α, r, s ′ )the relation holds1 π k P ( n ) α,r,β k s = e − αn r n − rs ′ (cid:18) k cos t k s π s ( αr ) s ′ I s (cid:16) πn − r αr (cid:17) + δ (1) n,s (cid:16) αr ) s ′ I s (cid:16) πn − r αr (cid:17) n r + 1 n − rs ′ (cid:17)(cid:19) , (14)where n = n ( α, r, p ) is a smallest number n , such that1 αr n r + αrχ ( p ) n − r ≤ , p = 1 , π ) · p − p , < p < ∞ , π ) , p = ∞ , (15)where χ ( p ) = p for 1 ≤ p < ∞ and χ ( p ) = 1 for p = ∞ and I s ( υ ) := (cid:18) υ R √ t +1) s dt (cid:19) s , ≤ s < ∞ , ess sup t ∈ [0 ,υ ] | √ t +1 | = 1 , s = ∞ , (16)and for the quantity δ (1) n,s = δ (1) n,s ( α, r, β ) the following estimate holds | δ (1) n,s | ≤ (14 π ) .Putting in the formula (14) s = 1, we get that for r ∈ (0 , α > β ∈ R , n ∈ N and n ≥ n ( α, r, ∞ ) the relation takes place1 π k P ( n ) α,r,β k = e − αn r (cid:18) π I (cid:16) πn − r αr (cid:17) + δ (1) n, (cid:16) αr I (cid:16) πn − r αr (cid:17) n r + 1 (cid:17)(cid:19) . (17)According to the formula (112) of the work [18] I (cid:16) πn − r αr (cid:17) = πn − rαr Z dt √ t + 1 = ln πn − r αr + Θ α,r,n , (18)where 0 < Θ α,r,n <
1. It is easy to show that for n ≥ n ( α, r ) the followinginequality holds4 π (ln π + Θ α,r,n ) + | δ (1) n, | (cid:18) αrn r ln πn − r αr + Θ α,r,n αrn r + 1 (cid:19) < π (ln π + 1) + (14 π ) (cid:18) π ) + 1 (cid:19) < , then formulas (17) and (18) imply that for n ≥ n ( α, r ) k P ( n ) α,r,β k = e − αn r (cid:18) π ln n − r αr + γ ∗ n (cid:19) , (19)where for the quantity γ ∗ n = γ ∗ n ( α, r, β ) the estimate is true | γ ∗ n | < ϕ ∈ C one can construct a function Φ( · ) = Φ( ϕ, · ) ∈ C , such that E n (Φ) C = E n ( ϕ ) C and for any n ≥ n ( α, r ) the equality holds1 π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) π Z − π Φ( t ) P ( n ) α,r,β (0 − t ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = e − αn r (cid:18) π ln n − r αr + γ n (cid:19) E n ( ϕ ) C , (20)where | γ n | < π .In this case for arbitrary function f ∈ C α,rβ C there exists a function Φ( · ) =Φ( f α,rβ , · ), such that E n (Φ) C = E n ( f α,rβ ) C and for n ≥ n ( α, r ) formula (20)is true, where as function ϕ we take the function f α,rβ . Let us assume F ( · ) = J α,rβ (Φ( · ) − a ), where a = a (Φ) = π π R − π Φ( t ) dt . The function F is the function,which we have looked for, because F ∈ C α,rβ C , and E n ( F α,rβ ) C = E n (Φ − a ) C = E n (Φ) C = E n ( f α,rβ ) C and moreover, formulas (7), (9), (11) and (20) yield (8).To prove (20) we need more detailed information about the character ofoscillation of the kernel P ( n ) α,r,β ( t ). We denote by n ∗ = n ∗ ( α, r ) the smallestnumber, for which the inequality holds1 αrn r + αrn r − < π . (21) Lemma 1.
Let α > , r ∈ (0 , , β ∈ R and n ∈ N . For n ≥ n ∗ the function P ( n ) α,r,β ( t ) has exactly n simple zeros z k on the period [0 , π ) , where the function P ( n ) α,r,β ( t ) takes values with alternating signs.Proof. According to formulas (44) and (47) of the work [18] we can write P ( n ) α,r,β ( t ) = g α,r,n ( t ) cos (cid:18) nt − βπ (cid:19) + h α,r,n ( t ) sin (cid:18) nt − βπ (cid:19) = q g α,r,n ( t ) + h α,r,n ( t ) cos (cid:18) nt − βπ − arctg h α,r,n ( t ) g α,r,n ( t ) (cid:19) = q g α,r,n ( t ) + h α,r,n ( t ) cos( n · y ( t )) , (22)where g α,r,n ( t ) = ∞ X k =0 e − α ( k + n ) r cos kt, (23) h α,r,n ( t ) = ∞ X k =0 e − α ( k + n ) r sin kt, (24) y ( t ) = y ( α, r, n ; t ) = t − βπ n − n arctg h α,r,n ( t ) g α,r,n ( t ) . (25)Lemma 1 will be proved, if one can show that for n ≥ n ∗ the function y ( t ) ofthe form (25) increasing on [0 , π ] from a value y (0) = − βπ to a value y (2 π ) =2 π − βπ . In this case the function cos( n · y ( t )), and also the function P ( n ) α,r,β ( t )(taking into account (22) and also the strict inequality q g α,r,n ( t ) + h α,r,n ( t ) > , π ) exactly 2 n simple zeros z k of the form z k = y − (cid:18) π + kπn (cid:19) , k = 0 , ..., n − , (26)where y − ( · ) is inverse function to y ( · ). In points z k the function cos( ny ( t ))(and also the function P ( n ) α,r,β ( t )) takes values with alternating signs.Let us consider the derivative of the function y ( t ): y ′ ( t ) = 1 − n (cid:16) h α,r,n ( t ) g α,r,n ( t ) (cid:17) ′ h α,r,n ( t ) g α,r,n ( t ) = 1 + 1 n − h ′ α,r,n ( t ) g α,r,n ( t ) + h α,r,n ( t ) g ′ α,r,n ( t ) g α,r,n ( t ) + h α,r,n ( t ) . (27)Let us estimate the absolute value of the last term in formula (27)1 n (cid:12)(cid:12)(cid:12)(cid:12) − h ′ α,r,n ( t ) g α,r,n ( t ) + h α,r,n ( t ) g ′ α,r,n ( t ) g α,r,n ( t ) + h α,r,n ( t ) (cid:12)(cid:12)(cid:12)(cid:12) = 1 n q g ′ α,r,n ( t ) + h ′ α,r,n ( t ) p g α,r,n ( t ) + h α,r,n ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − h ′ α,r,n ( t ) q g ′ α,r,n ( t ) + h ′ α,r,n ( t ) g α,r,n ( t ) p g α,r,n ( t ) + h α,r,n ( t ) + h α,r,n ( t ) p g α,r,n ( t ) + h α,r,n ( t ) g ′ α,r,n ( t ) q g ′ α,r,n ( t ) + h ′ α,r,n ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ n q g ′ α,r,n ( t ) + h ′ α,r,n ( t ) p g α,r,n ( t ) + h α,r,n ( t ) ≤ M n ( α, r ) n , (28)where M n ( α, r ) := sup t ∈ R q g ′ α,r,n ( t ) + h ′ α,r,n ( t ) p g α,r,n ( t ) + h α,r,n ( t ) . (29)From formula (99) from [18] we have M n ≤ π (cid:18) n − r αr + αrn r (cid:19) . This and inequality (21) yield that y ′ ( t ) > n ≥ n ∗ ( α, r ), so the function y ( t ) strictly increasing. Lemma 1 is proved.Let us prove now the estimate (20). Let ϕ ∈ C . Denote by Φ δ ( t ) the 2 π –periodic function, which coincides with the functionΦ ( t ) = E n ( ϕ ) C sign P ( n ) α,r,β ( − t )everywhere, except δ –neighborhoods ( δ < min k ∈ Z { z k +1 − z k } ) of points z k , whereit is linear function and its graph connects the points with coordinates ( z k − δ, Φ ( z k − δ )) and ( z k + δ, Φ ( z k + δ )).The function Φ δ ( · ) is continuous. As the condition (6) is more strong thanthe condition (21), then n ( α, r ) ≥ n ∗ ( α, r ). On the basis of Lemma 1 for n ≥ n ( α, r ) the function Φ δ has on [0 , π ) exactly 2 n zeros z k of the form(26), where it takes values with alternating signs, and in the middle of eachinterval ( z k , z k +1 ) it takes the maximum absolute values with alternating signs0 ± E n ( ϕ ) C . Then, by Chebyshev theorem about alternance, the polynomial t ∗ n − of the best approximation of the function Φ δ in the uniform metric will beidentically equal to zero and E n (Φ δ ) C = E n ( ϕ ) C . Therefore1 π π Z − π Φ δ ( t ) P ( n ) α,r,β (0 − t ) dt = 1 π π Z − π Φ ( t ) P ( n ) α,r,β ( − t ) dt + R n ( δ ) , (30)where R n ( δ ) = R n ( α, r, β, δ ) = 1 π π Z − π (Φ δ ( t ) − Φ ( t )) P ( n ) α,r,β ( − t ) dt. (31)As, 1 π π Z − π Φ ( t ) P ( n ) α,r,β ( − t ) dt = E n ( ϕ ) C π Z − π sign P ( n ) α,r,β ( − t ) P ( n ) α,r,β ( − t ) dt = E n ( ϕ ) C π Z − π | P ( n ) α,r,β ( − t ) | dt = E n ( ϕ ) C π Z − π | P ( n ) α,r,β ( t ) | dt = E n ( ϕ ) C k P ( n ) α,r,β k , then on the basis of (30) and (19) we have that1 π π Z − π Φ δ ( t ) P ( n ) α,r,β (0 − t ) dt = e − αn r (cid:18) π ln n − r αr + γ ∗ n (cid:19) E n ( ϕ ) C + R n ( δ ) , (32)where | γ ∗ n | < δ small enough, that the following inequality holds δ < π (10 π − αrn r n . (33)For values δ , which satisfy the condition (33), for n ≥ n ( α, r ) the followingestimate holds | R n ( δ ) | < (20 π − e − αn r E n ( ϕ ) C . (34)Indeed, according to (31) | R n ( δ ) | ≤ π k P ( n ) α,r,β k C π Z − π | Φ δ ( t ) − Φ ( t ) | dt ≤ π ∞ X k =0 e − α ( k + n ) r π Z − π | Φ δ ( t ) − Φ ( t ) | dt (35)1and, as follows from the formula (91) of the work [18], for n ≥ n ( α, r ), wederive the inequality ∞ X k =0 e − α ( k + n ) r < e − αn r n − r αr . (36)Then, whereas according to definitions of the functions Φ δ and Φ , the followingequality holds π Z − π | Φ δ ( t ) − Φ ( t ) | dt = E n ( ϕ ) C n − X k =0 z k + δ Z z k − δ | − t + z k − δ | δ dt = 2 nδE n ( ϕ ) C , then from (33), (35) and (36) we obtain the inequalities | R n ( δ ) | < π e − αn r n − r αr δE n ( ϕ ) C < π − e − αn r E n ( ϕ ) C . This proves (34).Hence, let us put Φ( t ) = Φ δ ( t ), choosing some δ , such that δ < min k ∈ Z { z k +1 − z k } . It should be noticed that for such choice of δ , the condition (33) is satisfied.Then, because of (32) and (34) for n ≥ n ( α, r ) for the function Φ( t ) theestimate (20) is true. Theorem 1 is proved.Notice that the statement of Lemma 1 takes place not only for n ≥ n ∗ , butfor all n ∈ N . To be sure in it, we use the following statement of the work [19]. Proposition 1.
Let the coefficients a k of the trigonometric series ∞ X k = n a k sin( kx + γ ) , γ ∈ (cid:16) , π i , n ∈ N , (37) satisfy the conditions ∆ m a k := a k − ma k +1 + m ( m − · a k +2 − ... + ( − m a k + m > , m ∈ Z + , k ∈ N , (38)lim k →∞ a k = 0 , (39) ∞ X k = n a k < ∞ . (40)2 Then the function-sum (37) has exactly n simple zeros in the interval (0 , π ) ,which are alternately located inside of respective intervals (cid:18) π − γ n − , π − γn (cid:19) , (cid:18) π − γ n − , π − γn (cid:19) , ..., (cid:18) (2 n − π − γ n − , nπ − γn (cid:19) , (cid:18) ( n + 1) π − γn , (2 n + 1) π − γ n − (cid:19) , ..., (cid:18) (2 n − π − γn , (4 n − π − γ n − (cid:19) , (cid:18) nπ − γ n − , π (cid:19) . The condition (38) of Proposition 1 can be written in the form( − m ∆ m a k > , k ∈ Z + , m ∈ N , (41)(so called condition of absolutely monotonicity of the sequence a k ), where thedifference operator ∆ m is defined by induction with a help of equalities∆ a k = a k , ∆ a k = a k +1 − a k , ∆ a k = ∆ (∆ a k ) , ..., ∆ m a k = ∆ (∆ m − a k ) = m X v =0 mv ! ( − m + v a k + v . To apply Proposition 1 to the sequence a k = e − αk r , α > r ∈ (0 ,
1) it isenough to be sure that (41) holds, because the verification of (39) and (40) istrivial. For α > r ∈ (0 ,
1) the function ψ ( t ) = ψ ( α, r, t ) = e − αt r is absolutelymonotonic, namely, the condition holds( − m ψ ( m ) ( t ) > , m ∈ N , t > . It follows from the fact that the function ψ ( t ) is a superposition of absolutelymonotonic function exp ( − t ) and positive function g ( t ) = g ( α, r, t ) = αt r , α > r ∈ (0 , § a k = e − αk r satisfies the condition (38)–(40) of Proposi-tion 1, and herefrom the following statement holds. Corollary 1.
Let α > , r ∈ (0 , , β ∈ [0 , and n ∈ N . Then, on [0 , π ) the function P ( n ) α,r,β ( t ) has exactly n simple zeros, which are alternately locatedinside of respective intervals (cid:18) βπ n − , π − (1 − β ) π n (cid:19) , (cid:18) (2 + β ) π n − , π − (1 − β ) π n (cid:19) , ..., (cid:18) (2 n − β ) π n − , nπ − (1 − β ) π n (cid:19) , (cid:18) ( n + 1) π − (1 − β ) π n , (2 n + β ) π n − (cid:19) , ..., (cid:18) (2 n − π − (1 − β ) π n , (4( n −
1) + β ) π n − (cid:19) , (cid:18) nπ − (1 − β ) π n − , π (cid:19) . In the same way as it was done in the works [9]–[13] we consider the classes C α,rβ C ( ε ) of 2 π –periodic functions f of the form (3), where ϕ = f α,rβ belongs tothe class C ( ε ), where as earlier ε = { ε ν } ∞ ν =0 is monotonically decreasing to zerothe sequence of nonnegative numbers.The following statement gives an example that the inequality (7) is bestpossible not only on the set C α,rβ C , but also on such important subsets C α,rβ C ( ε ).of the set C α,rβ C . Theorem 2.
Let α > , r ∈ (0 , , β ∈ R and ε = { ε ν } ∞ ν =0 is an arbitrary mono-tonically decreasing to zero the sequence of nonnegative real numbers. Then, forarbitrary class C α,rβ C ( ε ) and all numbers n ≥ n ( α, r ) the equalities hold E n ( C α,rβ C ( ε )) C = sup f ∈ C α,rβ C ( ε ) k f ( · ) − S n − ( f, · ) k C = e − αn r (cid:18) π ln n − r αr + γ n (cid:19) ε n , (42) where | γ n | ≤ π .Proof. Let f ∈ C α,rβ C ( ε ). Then, the function ϕ = f α,rβ is continuous and E n ( f α,rβ ) C ≤ ε n . Then, taking into account (7), we obtain that for n ≥ n ( α, r ) k ρ n ( f ; · ) k C ≤ e − αn r (cid:18) π ln n − r αr + γ n (cid:19) ε n ∀ f ∈ C α,rβ C ( ε ) , (43)where | γ n | ≤ π .On other hand, from Theorem 1 it follows that for the function F ( x ), whichis constructed for the fucntion ϕ = f α,rβ ∈ C ( ε )), and such that E n ( f α,rβ ) = ε n ,the inequality (43) becomes an equality for n ≥ n ( α, r ). Herefrom we get (42).Theorem 2 is proved.41. L. Fejer (1910) Lebesguesche konstanten und divergente Fourierreihen, J.Reine Angew Math. V. 138, 22–53.2. N.I. Akhiezer (1965) Lectures on approximation theory. Mir, Moscow.3. P.V. Galkin (1971) Estimate for Lebesgue constants, Trudy MIAN SSSR.109, 3–5 . [Proc. Steklov Inst. Math.] 109, 1–4.4. V. K. Dzyadyk (1977) Introduction to the theory of uniform approximationof functions by polynomials [in Russian], Nauka, Moscow.5. A.I. Stepanets (2005) Methods of Approximation Theory. VSP: Leiden,Boston.6. V.V. Zhuk and G.I. Natanson (1983) Trigonometrical Fourier series andelements of approximation theory, Izdat. Leningr. Univ. (in Russian).7. G.I. Natanson (1986) An estimate for Lebesgue constants of de la Vallee-Poussin sums, in ”Geometric questions in the theory of functions and sets”,Kalinin State Univ., Kalinin. (in Russian).8. I. A. Shakirov (2018) On two-sided estimate for norm of Fourier operator,Ufimsk. Mat. Zh., 10:1, 96–117; Ufa Math. J., 10:1, 94–114.9. A.I. Stepanets (1989) On the Lebesgue inequality on classes of ( ψ, β )-differentiable functions, Ukr. Math. J. 41:4, 435–443.10. A.I. Stepanets, A.S. Serdyuk, A.L. Shidlich (2009) On relationship betweenclasses of ( ψ, β )–differentiable functions and Gevrey classes, Ukr. Math.J. 61:1, 171-177.11. K. I. Oskolkov (1975) Lebesgue’s inequality in a uniform metric and ona set of full measure, Mat. Zametki, 18:4, 515–526; Math. Notes, 18:4,895–902.12. A.I. Stepanets, A.S. Serdyuk (2000) Lebesgue inequalities for Poisson in-tegrals, Ukr. Math. J. 52:6, 798-808.513. A.P. Musienko, A.S. Serdyuk (2013) Lebesgue-type inequalities for thede la Vallee-Poussin sums on sets of entire functions Ukr. Math. J. 65:5,709–722; translation from Ukr. Mat. Zh. 65:5, 642–653.14. A.P. Musienko, A.S. Serdyuk (2013) Lebesgue-type inequalities for the dela Valee-Poussin sums on sets of analytic functions Ukr. Math. J. 65:4575-592; translation from Ukr. Mat. Zh. 65:4, 522–537.15. A.S. Serdyuk and T.A. Stepanyuk (2019) Asymptotically best possibleLebesgue-type inequalities for the Fourier sums on sets of generalized Pois-son integrals arXiv:1908.09517 https://arxiv.org/abs/1908.09517.16. Serdyuk A. S., Stepanyuk T. A. (2018) Lebesgue–type inequalities forthe Fourier sums on classes of generalized Poisson integrals vol. 68, No 2,Bulletin de la societe des sciences et des lettres de Lodz, 45–52.17. A.S. Serdyuk, T.A. Stepanyuk (2017) Approximations by Fourier sums ofclasses of generalized Poisson integrals in metrics of spaces L ss