Order estimates of best orthogonal trigonometric approximations of classes of infinitely differentiable functions
aa r X i v : . [ m a t h . C A ] D ec Order estimates of best orthogonaltrigonometric approximations of classes ofinfinitely differentiable functions
Tetiana A. Stepanyuk
Abstract
In this paper we establish exact order estimates for the best uniformorthogonal trigonometric approximations of the classes of 2 π -periodic functions,whose ( ψ , β ) –derivatives belong to unit balls of spaces L p , 1 ≤ p < ∞ , in the case,when the sequence ψ ( k ) tends to zero faster, than any power function, but slowerthan geometric progression. Similar estimates are also established in the L s -metric,1 < s ≤ ∞ for the classes of differentiable functions, which ( ψ , β ) –derivatives be-long to unit ball of space L . Let L p , 1 ≤ p < ∞ , be the space of 2 π –periodic functions f summable to thepower p on [ , π ) , with the norm k f k p = (cid:16) π R | f ( t ) | p dt (cid:17) p ; L ∞ be the space of 2 π –periodic functions f , which are Lebesque measurable and essentially bounded withthe norm k f k ∞ = ess sup t | f ( t ) | .Let f : R → R be the function from L , whose Fourier series is given by ∞ ∑ k = − ∞ ˆ f ( k ) e ikx , where ˆ f ( k ) = π π R − π f ( t ) e − ikt dt are the Fourier coefficients of the function f , ψ ( k ) is an arbitrary fixed sequence of real numbers and β is a fixed real number. Then, if Institute of Analysis and Number Theory Kopernikusgasse 24/II 8010, Graz, Austria, Graz Uni-versity of TechnologyInstitute of Mathematics of Ukrainian National Academy of Sciences, 3, Tereshchenkivska st.,01601, Kyiv-4, Ukrainee-mail: tania$_{-}[email protected] the series ∑ k ∈ Z \{ } ˆ f ( k ) ψ ( | k | ) e i ( kx + βπ sign k ) is the Fourier series of some function ϕ from L , then this function is called the ( ψ , β ) –derivative of the function f and is denoted by f ψβ . A set of functions f ,whose ( ψ , β ) –derivatives exist, is denoted by L ψβ (see [13]).Let B p : = (cid:8) ϕ ∈ L p : || ϕ || p ≤ , ϕ ⊥ (cid:9) , ≤ p ≤ ∞ . If f ∈ L ψβ , and, at the same time f ψβ ∈ B p , then we say that the function f belongs tothe class L ψβ , p .By M we denote the set of all convex (downward) continuous functions ψ ( t ) , t ≥ , such that lim t → ∞ ψ ( t ) =
0. Assume that the sequence ψ ( k ) , k ∈ N , specifyingthe class L ψβ , p , ≤ p ≤ ∞ , is the restriction of the functions ψ ( t ) from M to the setof natural numbers.Following Stepanets (see, e.g., [13]), by using the characteristic µ ( ψ ; t ) of func-tions ψ from ∈ M of the form µ ( t ) = µ ( ψ ; t ) : = t η ( t ) − t , (1)where η ( t ) = η ( ψ ; t ) : = ψ − ( ψ ( t ) / ) , ψ − is the function inverse to ψ , we selectthe following subsets of the set M : M + ∞ = { ψ ∈ M : µ ( ψ ; t ) ↑ ∞ } . M ′′ ∞ = (cid:8) ψ ∈ M + ∞ : ∃ K > η ( ψ ; t ) − t ≥ K t ≥ (cid:9) . The functions ψ r , α ( t ) = exp ( − α t r ) are typical representatives of the set M + ∞ .Moreover, if r ∈ ( , ] , then ψ r , α ∈ M ′′ ∞ . The classes L ψβ , p , generated by the functions ψ = ψ r , α are denoted by L α , r β , p .For functions f from classes L ψβ , p we consider: L s –norms of deviations of thefunctions f from their partial Fourier sums of order n −
1, i.e., the quantities k ρ n ( f ; · ) k s = k f ( · ) − S n − ( f ; · ) k s , ≤ s ≤ ∞ , (2)where S n − ( f ; x ) = n − ∑ k = − n + ˆ f ( k ) e ikx ;and the best orthogonal trigonometric approximations of the functions f in metricof space L s , i.e., the quantities of the form rder estimates of best orthogonal trigonometric approximations 3 e ⊥ m ( f ) s = inf γ m k f ( · ) − S γ m ( f ; · ) k s , ≤ s ≤ ∞ , (3)where γ m , m ∈ N , is an arbitrary collection of m integer numbers, and S γ m ( f ; x ) = ∑ k ∈ γ m ˆ f ( k ) e ikx . We set E n ( L ψβ , p ) s = sup f ∈ L ψβ , p k ρ n ( f ; · ) k s , ≤ p , s ≤ ∞ , (4) e ⊥ n ( L ψβ , p ) s = sup f ∈ L ψβ , p e ⊥ n ( f ) s , ≤ p , s ≤ ∞ . (5)The following inequalities follow from given above definitions (4) and (5) e ⊥ n ( L ψβ , p ) s ≤ e ⊥ n − ( L ψβ , p ) s ≤ E n ( L ψβ , p ) s , ≤ p , s ≤ ∞ . (6)In the case when ψ ( k ) = k − r , r >
0, the classes L ψβ , p , 1 ≤ p ≤ ∞ , β ∈ R are well-known Weyl–Nagy classes W r β , p . For these classes, the order estimates of quantities e ⊥ n ( L ψβ , p ) s are known for 1 < p , s < ∞ (see [4], [5]), for 1 ≤ p < ∞ , s = ∞ , r > p andalso for p =
1, 1 < s < ∞ , r > s ′ , s + s ′ = ψ ( k ) tends to zero not faster than some power function, or-der estimates for quantities (5) were established in [1], [9], [11] and [12]. In thecase, when ψ ( k ) tends to zero not slower than geometric progression, exact orderestimates for e ⊥ n ( L ψβ , p ) s were found in [10] for all 1 ≤ p , s ≤ ∞ .Our aim is to establish the exact-order estimates of e ⊥ n ( L ψβ , p ) ∞ , 1 ≤ p < ∞ , and e ⊥ n ( L ψβ , ) s , 1 < s < ∞ , in the case, when ψ decreases faster than any power function,but slower than geometric progression ( ψ ∈ M ′′ ∞ ). L ψβ , p , < p < ∞ , in the uniform metric We write a n ≍ b n to mean that there exist positive constants C and C indepen-dent of n such that C a n ≤ b n ≤ C a n for all n . Theorem 1.
Let < p < ∞ , ψ ∈ M ′′ ∞ and the function ψ ( t ) | ψ ′ ( t ) | ↑ ∞ as t → ∞ . Then,for all β ∈ R the following order estimates holde ⊥ n − ( L ψβ , p ) ∞ ≍ e ⊥ n ( L ψβ , p ) ∞ ≍ ψ ( n )( η ( n ) − n ) p . (7) Proof.
According to Theorem 1 from [8] under conditions ψ ∈ M + ∞ , β ∈ R ,1 ≤ p < ∞ , for n ∈ N , such that η ( n ) − n ≥ a > , µ ( n ) ≥ b > T. A. Stepanyuk mate is true E n ( L ψβ , p ) ∞ ≤ K a , b ( p ) − p ψ ( n )( η ( n ) − n ) p , (8)where K a , b = π max (cid:26) bb − + a , π (cid:27) . Using inequalities (6) and (8), we obtain e ⊥ n ( L ψβ , p ) ∞ ≤ e ⊥ n − ( L ψβ , p ) ∞ ≤ K a , b ( p ) − p ψ ( n )( η ( n ) − n ) p . (9)Let us find the lower estimate for the quantity e ⊥ n ( L ψβ , p ) ∞ . With this purpose weconstruct the function f ∗ p , n ( t ) = f ∗ p , n ( ψ ; t ) : = λ p ψ ( n )( η ( n ) − n ) p ′ (cid:18) ψ ( ) ψ ( n )++ n − ∑ k = ψ ( k ) ψ ( n − k ) cos kt + n ∑ k = n ψ ( k ) cos kt (cid:19) , p + p ′ = . (10)Let us show that f ∗ p , n ∈ L ψβ , p . The definition of ( ψ , β ) –derivative yields ( f ∗ p , n ( t )) ψβ = λ p ψ ( n )( η ( n ) − n ) p ′ (cid:18) n − ∑ k = ψ ( n − k ) cos (cid:16) kt + βπ (cid:17) + n ∑ k = n ψ ( k ) cos (cid:16) kt + βπ (cid:17)(cid:19) . (11)Obviously (cid:12)(cid:12) ( f ∗ p , n ( t )) ψβ (cid:12)(cid:12) ≤ λ p ψ ( n )( η ( n ) − n ) p ′ (cid:18) n − ∑ k = ψ ( n − k ) + n ∑ k = n ψ ( k ) (cid:19) << λ p ψ ( n )( η ( n ) − n ) p ′ n ∑ k = n ψ ( k ) ≤ λ p ψ ( n )( η ( n ) − n ) p ′ (cid:18) ψ ( n ) + ∞ Z n ψ ( u ) du (cid:19) . (12)To estimate the integral from the right part of formula (12), we use the followingstatement [7, p. 500]. Proposition 1. If ψ ∈ M + ∞ , then for arbitrary m ∈ N , such that µ ( ψ , m ) > thefollowing condition holds ∞ Z m ψ ( u ) du ≤ − µ ( m ) ψ ( m )( η ( m ) − m ) . (13) rder estimates of best orthogonal trigonometric approximations 5 Formulas (12) and (13) imply that (cid:12)(cid:12) ( f ∗ p , n ( t )) ψβ (cid:12)(cid:12) ≤ λ p ψ ( n )( η ( n ) − n ) p ′ (cid:18) ψ ( n ) + bb − ψ ( n )( η ( n ) − n ) (cid:19) << λ p bb − ( η ( n ) − n ) p . (14)We denote D k , β ( t ) : =
12 cos βπ + k ∑ j = cos (cid:16) jt + βπ (cid:17) . (15)Applying Abel transform, we have n − ∑ k = ψ ( n − k ) cos (cid:16) kt + βπ (cid:17) = n − ∑ k = ( ψ ( n − k + ) − ψ ( n − k )) D k , β ( t )+ ψ ( n + ) D n − , β ( t ) − ψ ( n − )
12 cos βπ n ∑ k = n ψ ( k ) cos (cid:16) kt + βπ (cid:17) = n − ∑ k = n ( ψ ( k ) − ψ ( k + )) D k , β ( t )+ ψ ( n ) D n , β ( t ) − ψ ( n ) D n − , β ( t ) . (17)Since N − ∑ k = sin ( γ + kt ) = sin (cid:16) γ + N − t (cid:17) sin Ny t (18)(see, e.g., [2, p.43]), for N = k + γ = ( β − ) π , the following inequality holds | D k , β ( t ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) cos (cid:0) kt + βπ (cid:1) sin k + t sin t −
12 cos βπ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin (cid:0) ( k + ) t + βπ (cid:1) − cos t sin βπ t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ π t , < | t | ≤ π . (19)According to (11), (16), (17) and (19), we obtain T. A. Stepanyuk (cid:12)(cid:12) ( f ∗ p , n ( t )) ψβ (cid:12)(cid:12) ≤ λ p ψ ( n )( η ( n ) − n ) p ′ π | t | (cid:18) n − ∑ k = | ψ ( n − k ) − ψ ( n − k − ) | + ψ ( n + )+ ψ ( n − ) + n − ∑ k = n | ψ ( k ) − ψ ( k + ) | + ψ ( n ) + ψ ( n ) (cid:19) = λ p ψ ( n )( η ( n ) − n ) p ′ π | t | ( ψ ( n + ) + ψ ( n )) ≤ πλ p ( η ( n ) − n ) p ′ | t | . (20)So, (14) and (20) imply (cid:13)(cid:13) ( f ∗ p , n ( t )) ψβ (cid:13)(cid:13) p ≤ λ p max n bb − , π o(cid:18) Z | t |≤ η ( n ) − n ( η ( n ) − n ) dt + ( η ( n ) − n ) pp ′ Z η ( n ) − n ≤| t |≤ π dt | t | p (cid:19) p ≤ λ p max n bb − , π o(cid:16) + p − (cid:17) p = λ p max n bb − , π o ( p ′ ) p . Hence, for λ p = ( p ′ ) p max n bb − , π o the embedding f ∗ p , n ∈ L ψβ , p is true.Let us consider the quantity I : = inf γ n (cid:12)(cid:12)(cid:12)(cid:12) π Z − π ( f ∗ p , n ( t ) − S γ n ( f ∗ p , n ; t )) V n ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12) , (21)where V n are de la Vall´ee-Poisson kernels of the form V m ( t ) : = + m ∑ k = cos kt + m − ∑ k = m + (cid:16) − k m (cid:17) cos kt , m ∈ N . (22)Proposition A1.1 from [3] implies I ≤ inf γ n k f ∗ p ( t ) − S γ n ( f ∗ p , n ; t ) k ∞ k V n k = e ⊥ n ( f ∗ p , n ) ∞ k V n k . (23)Since (see, e.g., [14, p.247]) k V m k ≤ π , m ∈ N , (24)from (23) and (24) we can write down the estimate rder estimates of best orthogonal trigonometric approximations 7 e ⊥ n ( f ∗ p , n ) ∞ ≥ π I . (25)Notice, that f ∗ p , n ( t ) − S γ n ( f ∗ p , n ; t )= λ p ψ ( n )( η ( n ) − n ) p ′ (cid:18) ∑ | k |≤ n − , k / ∈ γ n ψ ( | k | ) ψ ( n − | k | ) e ikt + ∑ n ≤| k |≤ n , k / ∈ γ n ψ ( | k | ) e ikt (cid:19) , (26)where ψ ( ) : = ψ ( ) Whereas π Z − π e ikt e imt dt = (cid:26) , k + m = , π , k + m = , k , m ∈ Z , (27)and taking into account (22), we obtain π Z − π ( f ∗ p , n ( t ) − S γ n ( f ∗ p , n ; t )) V n ( t ) dt (28) = λ p ψ ( n )( η ( n ) − n ) p ′ π Z − π (cid:18) ∑ ≤ k ≤ n − , k / ∈ γ n ψ ( k ) ψ ( n − k ) e ikt + ∑ − n + ≤ k ≤− , k / ∈ γ n ψ ( | k | ) ψ ( n − | k | ) e ikt + ∑ n ≤ k ≤ n , k / ∈ γ n ψ ( k ) e ikt + ∑ − n ≤ k ≤− n , k / ∈ γ n ψ ( | k | ) e ikt (cid:19) ×× (cid:16) ∑ ≤ k ≤ n e ikt + ∑ − n ≤ k ≤− e ikt + ∑ n + ≤| k |≤ n − (cid:16) − | k | n (cid:17) e ikt (cid:17) dt (29) = λ p π ψ ( n )( η ( n ) − n ) p ′ (cid:18) ∑ | k |≤ n − , k / ∈ γ n ψ ( | k | ) ψ ( n − | k | ) + ∑ n ≤| k |≤ n , k / ∈ γ n ψ ( | k | ) (cid:19) . (30)The function φ n ( t ) : = ψ ( t ) ψ ( n − t ) decreases for t ∈ [ , n ] . Indeed φ ′ n ( t ) = | ψ ′ ( t ) || ψ ′ ( n − t ) | (cid:16) ψ ( t ) | ψ ′ ( t ) | − ψ ( n − t ) | ψ ′ ( n − t ) | (cid:17) ≤ , because ψ ( t ) | ψ ′ ( t ) | ↑ ∞ for large n .Thus, the monotonicity of function φ n ( t ) and (30) imply T. A. Stepanyuk I = πλ p ψ ( n )( η ( n ) − n ) p ′ (cid:18) ψ ( n ) + ∑ n + ≤| k |≤ n ψ ( | k | ) (cid:19) > πλ p ψ ( n )( η ( n ) − n ) p ′ n ∑ k = n ψ ( k ) ≥ πλ p ψ ( n )( η ( n ) − n ) p ′ η ( n ) Z n ψ ( t ) dt > πλ p ψ ( n )( η ( n ) − n ) p ′ ψ ( η ( n ))( η ( n ) − n ) = πλ p ψ ( n )( η ( n ) − n ) p . (31)By considering (25) and (31) we can write e ⊥ n ( L ψβ , p ) ∞ ≥ e ⊥ n ( f ∗ p , n ) ∞ ≥ π I ≥ λ p ψ ( n )( η ( n ) − n ) p . (32)Theorem 1 is proved. Remark 1.
Let ψ ∈ M + ∞ , β ∈ R , 1 < p < ∞ , p + p ′ =
1, and the function ψ ( t ) | ψ ′ ( t ) | ↑ ∞ for t → ∞ . Then for n ∈ N the following estimates hold K b , p ψ ( n )( η ( n ) − n ) p ≤ e ⊥ n ( L ψβ , p ) ∞ ≤ e ⊥ n − ( L ψβ , p ) ∞ ≤ K a , b , p ψ ( n )( η ( n ) − n ) p , (33)where K a , b , p = π max (cid:26) bb − + a , π (cid:27) ( p ) p ′ . (34) K b , p =
148 max n bb − , π o ( p ′ ) p . (35) L ψβ , in the uniform metric Theorem 2.
Let ψ ∈ M + ∞ . Then for all β ∈ R order estimates are truee ⊥ n − ( L ψβ , ) ∞ ≍ e ⊥ n ( L ψβ , ) ∞ ≍ ψ ( n )( η ( n ) − n ) . (36) Proof.
According to formula (48) from [8] under conditions ψ ∈ M , ∞ ∑ k = ψ ( k ) < ∞ , β ∈ R , for all n ∈ N the following estimate holds E n ( L ψβ , ) ∞ ≤ π ∞ ∑ k = n ψ ( k ) . (37)Using Proposition 1, we have rder estimates of best orthogonal trigonometric approximations 9 e ⊥ n ( L ψβ , ) ∞ ≤ e ⊥ n − ( L ψβ , ) ∞ ≤ E n ( L ψβ , ) ∞ ≤ π ∞ ∑ k = n ψ ( k ) ≤ π (cid:18) ψ ( n ) + ∞ Z n ψ ( u ) du (cid:19) ≤ ψ ( n ) π (cid:18) + bb − ( η ( n ) − n ) (cid:19) . (38)Let us find the lower estimate for the quantity e ⊥ n ( L ψβ , ) ∞ .We consider the quantity I : = inf γ n (cid:12)(cid:12)(cid:12)(cid:12) π Z − π ( f ∗ n ( t ) − S γ n ( f ∗ n ; t )) V n ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12) , (39)where V m are de la Vall´ee-Poisson kernels of the form (22), and f ∗ m ( t ) = f ∗ m ( ψ ; t ) : = π m (cid:16) ψ ( )+ m ∑ k = k ψ ( k ) cos kt + m ∑ k = m + ( m + − k ) ψ ( k ) cos kt (cid:17) . (40)In [14, p. 263–265] it was shown that k ( f ∗ m ) ψβ k ≤
1, i.e., f ∗ m belongs to the class L ψβ , for all m ∈ N .Using Proposition A1.1 from [3] and inequality (24), we have I ≤ inf γ n k f ∗ n ( t ) − S γ n ( f ∗ n ; t ) k ∞ k V n k ≤ π e ⊥ n ( f ∗ n ) ∞ . (41)Assuming again ψ ( ) : = ψ ( ) , from (22) and (40), we derive I = π n inf γ n (cid:12)(cid:12)(cid:12)(cid:12) π Z − π (cid:16) ∑ | k |≤ n , k / ∈ γ n | k | ψ ( | k | ) e ikt + ∑ n + ≤| k |≤ n , k / ∈ γ n ( n + − | k | ) ψ ( | k | ) e ikt (cid:17) ×× (cid:16) ∑ | k |≤ n e ikt + ∑ n + ≤|| k ≤ n − (cid:16) − | k | n (cid:17) e ikt (cid:17) dt (cid:12)(cid:12)(cid:12)(cid:12) = n inf γ n (cid:18) ∑ | k |≤ n , k / ∈ γ n | k | ψ ( | k | ) + ∑ n + ≤| k |≤ n , k / ∈ γ n (cid:16) − | k | n (cid:17) ψ ( | k | ) (cid:19) > n inf γ n ∑ | k |≤ n , k / ∈ γ n | k | ψ ( | k | ) = n (cid:18) n ψ ( n ) + n ∑ k = n + k ψ ( k ) (cid:19) > n ∑ k = n ψ ( k ) > η ( n ) Z n ψ ( t ) dt > ψ ( n )( η ( n ) − n ) . (42)Formulas (41) and (42) imply e ⊥ n ( L ψβ , ) ∞ ≥ e ⊥ n ( f ∗ n ) ∞ ≥ π I > πψ ( n )( η ( n ) − n ) . Theorem 2 is proved.
Remark 2.
Let ψ ∈ M + ∞ and β ∈ R . Then for n ∈ N , such that µ ( n ) ≥ b > πψ ( n )( η ( n ) − n ) ≤ e ⊥ n ( L ψβ , ) ∞ ≤ e ⊥ n − ( L ψβ , ) ∞ ≤ π (cid:16) b + bb − (cid:17) ψ ( n )( η ( n ) − n ) . (43) Corollary 1.
Let r ∈ ( , ) , α > , ≤ p < ∞ and β ∈ R . Then for all n ∈ N thefollowing estimates are truee ⊥ n ( L α , r β , p ) ∞ ≍ exp ( − α n r ) n − rp . (44) L ψβ , in the metric of spaces L s , < s < ∞ Theorem 3.
Let < s < ∞ , ψ ∈ M ′′ ∞ and function ψ ( t ) | ψ ′ ( t ) | ↑ ∞ as t → ∞ . Then for all β ∈ R order estimates holde ⊥ n − ( L ψβ , ) s ≍ e ⊥ n ( L ψβ , ) s ≍ ψ ( n )( η ( n ) − n ) s ′ , s + s ′ = . (45) Proof.
According to Theorem 2 from [8] under conditions ψ ∈ M + ∞ , β ∈ R ,1 < s ≤ ∞ for n ∈ N , such that η ( n ) − n ≥ a > , µ ( n ) ≥ b > E n ( L ψβ , ) s ≤ K a , b ( s ′ ) s ψ ( n )( η ( n ) − n ) s ′ . (46)Using inequalities (6) and (46), we get e ⊥ n ( L ψβ , ) s ≤ e ⊥ n − ( L ψβ , ) s ≤ K a , b , s ′ (cid:0) s ′ (cid:1) s ψ ( n )( η ( n ) − n ) s ′ . (47)Let us find the lower estimate of the quantity e ⊥ n ( L ψβ , ) s .We consider the quantity I : = inf γ n (cid:12)(cid:12)(cid:12)(cid:12) π Z − π ( f ∗∗ n ( t ) − S γ n ( f ∗∗ n ; t )) f ∗ s ′ , n ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12) , (48)where f ∗∗ m ( t ) = π V m ( t ) , rder estimates of best orthogonal trigonometric approximations 11 and f ∗ s ′ , n is defined by formula (10).On the basis of Proposition A1.1 from [3] we derive I ≤ inf γ n k f ∗∗ n ( t ) − S γ n ( f ∗∗ n ; t ) k s k f ∗ s ′ k s ′ ≤ e ⊥ n ( f ∗∗ n ) s . (49)On other hand, using formulas (27), we write I = λ s ′ πψ ( n )( η ( n ) − n ) s inf γ n (cid:12)(cid:12)(cid:12)(cid:12) π Z − π (cid:16) ∑ | k |≤ n , k / ∈ γ n e ikt + ∑ n + ≤| k |≤ n − , k / ∈ γ n (cid:16) − | k | n (cid:17) e ikt (cid:17) ×× (cid:18) ∑ | k |≤ n − ψ ( | k | ) ψ ( n − | k | ) e ikt + ∑ n ≤| k |≤ n ψ ( | k | ) e ikt (cid:19) dt (cid:12)(cid:12)(cid:12)(cid:12) = λ s ′ ψ ( n )( η ( n ) − n ) s inf γ n (cid:16) ∑ | k |≤ n − , k / ∈ γ n ψ ( | k | ) ψ ( n − | k | ) + ∑ n ≤| k |≤ n , k / ∈ γ n ψ ( | k | ) (cid:17) = λ s ′ ψ ( n )( η ( n ) − n ) s (cid:16) ψ ( n ) + n ∑ k = n + ψ ( k ) (cid:17) > λ πψ ( n )( η ( n ) − n ) s n ∑ k = n ψ ( k ) > λ s ′ ψ ( n )( η ( n ) − n ) s η ( n ) Z n ψ ( t ) dt > λ s ′ ψ ( n )( η ( n ) − n ) s ′ . (50)Hence, formulas (49) and (50) imply e ⊥ n ( L ψβ , ) s ≥ e ⊥ n ( f ∗∗ s ′ ) s ≥ I ≥ λ s ′ ψ ( n )( η ( n ) − n ) s ′ . (51)Theorem 3 is proved.Note, that functions1) e − α t r t γ , α > , r ∈ ( , ] , γ ∈ R ;2) e − α t r ln ( t + K ) , α > , r ∈ ( , ] , K > e − ψ , which satisfy the conditions ofTheorem 1 and Theorem 3. Remark 3.
Let ψ ∈ M + ∞ , β ∈ R , 1 ≤ p < ∞ and function ψ ( t ) | ψ ′ ( t ) | ↑ ∞ as t → ∞ . Thenfor all n ∈ N , such tthe following estimates are true K b , s ′ ψ ( n )( η ( n ) − n ) s ′ ≤ e ⊥ n ( L ψβ , ) s ≤ e ⊥ n − ( L ψβ , ) s ≤ K a , b , s ′ ψ ( n )( η ( n ) − n ) s ′ , (52)where K a , b , s ′ and K b , s ′ are defined by formulas (34) and (35) respectively. Corollary 2.
Let r ∈ ( , ) , α > , < s < ∞ and β ∈ R . Then for all n ∈ N thefollowing estimates are truee ⊥ n ( L α , r β , ) s ≍ exp ( − α n r ) n − rs ′ , s + s ′ = . (53) Acknowledgements
The author is supported by the Austrian Science Fund FWF project F5503 (partof the Special Research Program (SFB) “Quasi-Monte Carlo Methods: Theory andApplications”)
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