The L1 norm of the generalized de la Vallee Poussin kernel
TTHE L NORMS OF DE LA VALL´EE POUSSIN KERNELS
HARSH MEHTA
Abstract.
Charles de la Vall´ee Poussin defined two different kernels that bear hisname. This paper considers the ones are a linear combinations of two Fej´er kernels,which are known as the delayed means. We show that the L norms are constant infamilies of delayed means, and determine the exact value of the L norm for some ofthem. Introduction
This paper studies properties of certain summability kernels for Fourier series, the de laVall´ee Poussin kernels defined below. Let f ( x ) be a periodic function on R of period 1,of finite L -norm in [0 , f ( x ) ∼ (cid:88) k ∈ Z ˆ f ( k ) e ( kx ) , where e ( x ) := e πix . Define the partial sum of its Fourier series S n ( f, x ) := n (cid:88) k = − n (cid:98) f ( k ) e ( kx ) (1.1)The Fej´er mean (of period 1) with parameter n is σ n ( f, x ) := 1 n + 1 n (cid:88) k =0 S k ( f, x ) = n (cid:88) k = − n (cid:18) − | k | n + 1 (cid:19) (cid:98) f ( k ) e ( kx ) . (1.2)This mean is given as singular integral of convolution type σ n ( f, x ) = (cid:90) f ( x − u ) K n ( u ) du, where the Fej´er kernel K n ( x ), rescaled for functions of period 1, is K n ( x ) := ∆ n +1 ( x ) = 1 n + 1 (cid:16) sin π ( n + 1) x sin πx (cid:17) . (1.3)In 1918 de la Vall´ee Poussin [17] introduced the delayed means (also called de la Vall´eePoussin sums ([5], [10]), as σ n,p ( f, x ) := 1 p n + p − (cid:88) k = n S n ( f, x ) = n + pp σ n + p − ( f, x ) − np σ n − ( f, x ) , (1.4)Here we follow the notation for these means used in Zygmund [21, Chap. III.1, (1.30)],after rescaling the periodicity interval from [0 , π ] to [0 , . The means above are given
Date : November 2, 2013.2010
Mathematics Subject Classification.
Primary: 42A05.
Key words and phrases. de la Vall´ee Poussin kernel. a r X i v : . [ m a t h . C A ] N ov HARSH MEHTA by the convolution integral σ n,p ( f, x ) = (cid:90) f ( x − u ) K n,p ( u ) du, (1.5)in which K n,p ( x ) is the de la Vall´ee Poussin kernel function [17] with parameters ( n, p ).This is found to be K n,p ( x ) = n + pp ∆ n + p ( t ) − np ∆ n ( t ) (1.6)= 1 p (cid:16) (sin π ( n + p ) x ) − (sin πnx ) (sin πx ) (cid:17) . (1.7)Taking n = 0 and p = n we obtain K ,n ( x ) = ∆ n ( x ) = K n − ( x ), the Fej´er kernel with ashifted parameter. All of these kernels have (cid:90) K n,p ( x ) dx = 1 . (1.8)For each parameter set ( n, p ) there is a family F n,p := { K nN,pN ( x ) : N ≥ } ofkernel functions indexed by the positive integer parameter N ≥
1. This family formsa summation kernel in the sense of Walker [20, (8.1), (8.23)], or a finite θ -factor in thesense of Butzer and Nessel [2, Sec. 1.2.5]. These authors define a general finite θ -factor to be an infinite family of data { θ N ( j ) : N ≥ , j ∈ Z } with S θ ( f, x ) = m ( N ) (cid:88) j = − m ( N ) θ N ( j ) ˆ f ( j ) e ( jx )and with a function m ( N ) → ∞ as N → ∞ . For the de la Vall´ee Poussin kernel withparameters ( n, p ) the associated θ -factor takes m ( N ) = N and sets θ N ( j ) := v n,p ( jN ) , where v n,p ( x ) being a compactly supported piecewise linear function on R given by v n,p ( u ) := | u | ≤ n, n + p −| u | p if n ≤ | u | ≤ n + p, | u | ≥ n + p Special cases of function are pictured in Figure 1 and Figure 2.For later convenience we make a linear change of variables in the parameters, setting N = gcd( n, n + p ) and writing N r = n and N s = n + p , so that 0 ≤ r < s and r and s are relatively prime. (Thus n = N r and p = N ( s − r ).) In the new parameters the de laVall´ee Poussin kernels (1.6) become V rN,sN ( x ) := K rN, ( s − r ) N ( x ) = s ∆ sN ( x ) − r ∆ rN ( s ) s − r . (1.9)In consequence V rN,sN ( x ) = (sin sN πx ) − (sin rN πx ) ( s − r ) N (sin πx ) (1.10)= sN − (cid:88) n = − sN +1 v r,s − r ( n/N ) e ( nx ) (1.11) HE DE LA VALL´EE POUSSIN KERNAL 3
Figure 1. v n,p ( x ) with ( n, p ) = (1 , N = 1 and ( r, s ) = (1 , Figure 2. v n,p ( x ) with ( n, p ) = (2 , N = 2 and ( r, s ) = (1 , u v r,s − r ( u ) = | u | ≤ r ) ,s − | u | s − r ( r ≤ | u | ≤ s ) , | u | > s ) . (1.12) HARSH MEHTA
A particularly well known case of his summability kernel [18, Sect. 29] occurs for r =1 , s = 2, and is K N,N ( x ) = V N, N ( x ) = 2∆ N ( x ) − ∆ N ( x ) (1.13)= (sin 2 N πx ) − (sin N πx ) N (sin πx ) (1.14)= (cid:88) | n |≤ N e ( nx ) + (cid:88) N< | n | < N (cid:0) − | n | /N (cid:1) e ( nx ) . (1.15)The shown for N = 1 in Figure 3 and for N = 2 in Figure 4. Figure 3. de la Vall´ee Poussin kernel K , ( x ) = V , ( x ) on [0 , L -norm of V rN,sN ( x ). From (1.8) and (1.9) we have the easybounds 1 ≤ || V rN,sN ( x ) || L ( T ) = (cid:90) | V rN,sN ( x ) | dx ≤ s + rs − r . Our main result is the observation that the L -norms of the kernels in these families areindependent of the kernel family parameter N . Theorem 1.1.
Let r and s be fixed integers with ≤ r < s and ( r, s ) = 1 . Then allmembers of the kernel family G r,s = { V rN,sN : N ≥ } have the same L -norm. That is, (cid:107) V rN,sN (cid:107) L ( T ) = (cid:107) V r,s (cid:107) L ( T ) = (cid:90) | V r,s ( x ) | dx. This result is surprising because of the oscillatory nature of these functions, whichhave increasing numbers of sign changes as N increases, visible in Figure 3 and Figure 4;nevertheless both functions have the same L -norm on [0 ,
1] by Theorem 1.1.For individual values of r and s , on taking N = 1, the value can in principle beexplicitly determined. For the special case ( r, s ) = ( n, n + 1) we observe that the kernel HE DE LA VALL´EE POUSSIN KERNAL 5
Figure 4. de la Vall´ee Poussin kernel K , ( x ) = V , ( x ) on [0 , V n,n +1 ( x ) coincides with the Dirichlet kernel D n ( x ) and therefore obtain the followingwell known answer. Theorem 1.2.
For n ≥ we have V n,n +1 ( x ) = D n ( x ) = sin π (2 n + 1) x sin πx . Here D n ( x ) is the Dirichlet kernel for period functions. In particular, (cid:107) V n,n +1 (cid:107) L ( T ) = (cid:107) D n (cid:107) L ( T ) = L n , where L n is the n -th Lebesgue constant. The Lebesgue constants L n solve the extremal problem of giving the supremum ofthe n -th partial sum | S n ( f, x ) | of the Fourier series of f ( x ) where f ( x ) is any periodiccontinuous function of period 1 having | f ( x ) | ≤ L n = 2 π (cid:90) π | sin(( n + 1) / θ / θ | dθ. (1.16)It is well known that L = 13 + 2 √ π = 1 . . . . and L = 15 + (cid:112) − √ π (cid:16) √ (cid:17) = 1 . . . . Combining these two results we obtain for the original kernel of de la Vall´ee Poussin[17, p. 801 top] and [18, Sect. 26], the following answer.
HARSH MEHTA
Corollary 1.3.
Let V N, N ( x ) = 2∆ N ( x ) − ∆ N ( x ) be the de la Vall´ee Poussin kernel.Then (cid:90) | V N, N ( x ) | dx = L = 13 + 2 √ π = 1 . . . . (1.17) for all N . We remark that de la Vall´ee Poussin introduced the delayed to study pointwise ap-proximation to Fourier series in his 1919 book on approximation [18, Chap. II, Sec. 27,29]. These families of kernels form approximate identities, i.e. there is a constant M such that for each N ≥ (cid:90) | K nN,pN ( x ) | dx ≤ M, (cid:90) K nN,pN ( x ) dx = 1 , while for each δ >
0, lim N →∞ (cid:90) δ ≤| u |≤ | K nN,pN ( x ) | dx = 0 . In particular, for continuous functions f ( x ) on the torus T = R / Z one has pointwiseconvergence lim N →∞ σ nN,pN ( f, x ) = f ( x ) . The issue of how fast the approximations converge to a function in given classes has beenmuch studied, with work up to the 1950’s described in Timan [15, Sect. 8.4.4]. Furtherwork includes Efimov [5], [6], Teljakovski [16], Dahmen [4], Stechkin [13], and Serdyuket al. [11], [10], [12]. 2.
Proof of Theorem 1.1
Let r and s be integers with 0 ≤ r < s and ( r, s ) = 1. The parameters r, s will remainfixed throughout this section, we simplify the notation by omitting mention of r and s when naming functions. In particular, V N means V rN,sN .From (1.10) one obtains using trigonometric addition formulas * that V N ( x ) = (cid:0) sin π ( s + r ) N x (cid:1)(cid:0) sin π ( s − r ) N x (cid:1) ( s − r ) N (sin πx ) . (2.1)Hence V N has zeros at points of the form { a/ (( s + r ) N ) : 0 ≤ a ≤ ( s + r ) N − } andalso at points of the form { b/ (( s − r ) N ) : 0 ≤ b ≤ ( s − r ) N − } , excluding pointswhere ( s + r ) N | a or ( s − r ) N | b . Thus there are ( s + r ) N − s − r ) N − sN − V N is a trigonometric polynomial of degree sN −
1, we know that it could haveat most 2 sN − a = ( s + r ) n and b = ( s − r ) n We can see that V N hasa double zero at n/N , for 0 < n < N , taking a = ( s + r ) n and b = ( s − r ) n above.If r and s are of opposite parity, then gcd( s + r, s − r ) = 1 implies there are no otherdouble zeros. If however r and s are both odd, then gcd( s + r, s − r ) = 2, and bytaking a = n ( s + r ) / b = n ( s − r ) / V N has double zeros at n/ (2 N ) for0 < n < N . * The addition formulas yield (cid:0) sin( a + b ) x (cid:1)(cid:0) sin( a − b ) x (cid:1) = (sin ax ) (cos bx ) − (sin bx ) (cos ax ) . Theright side then equals (sin ax ) − (sin bx ) , after adding (sin ax ) (sin bx ) to both the opposing terms. HE DE LA VALL´EE POUSSIN KERNAL 7
As an example, in Figure 4 we have r + s = 3, and V , ( x ) has 5 zeros of the first kindand 1 zero of the second kind; the five zeros of first kind are located at x = j , 1 ≤ j ≤ x = , so there is a double zero at x = . To compute (cid:82) | V N ( x ) | dx , we break the interval [0 ,
1] into subintervals running fromone simple zero of V N to the next and then summing the area enclosed between eachinterval. In order to know what sign to attach to each interval (to obtain | V N ( x ) | ) weneed to consider whether V N is increasing or decreasing at a simple zero. Proof of Theorem 1.1.
We define the function F N ( x ) by the equality V N ( x ) = F N ( x ) sin( π ( s + r ) N x ), so F N ( x ) = sin( π ( s − r ) N x )( s − r ) N (sin πx ) using (2.1). Then for integer a with a ( s + r ) N (cid:54)∈ Z , we have V (cid:48) N (cid:16) a ( s + r ) N (cid:17) = ( − a F N (cid:16) a ( s + r ) N (cid:17) π ( s + r ) N .
Hence sgn V (cid:48) N (cid:16) a ( s + r ) N (cid:17) = ( − a sgn (cid:16) sin π ( s − r ) as + r (cid:17) (2.2)= ( − a sgn (cid:16) sin (cid:16) πa − πars + r (cid:17)(cid:17) = − sgn (cid:16) sin 2 πars + r (cid:17) . Accordingly, we set ε ( a ) := sgn (cid:16) sin 2 πars + r (cid:17) . (2.3)We note that ε ( a + s + r ) = sgn (cid:16) sin 2 π ( a + s + r ) rs + r (cid:17) = sgn (cid:16) sin 2 πars + r (cid:17) = ε ( a ) . Hence the values ε ( a ) are periodic with period s + r .We define the function G N ( x ) so that V N ( x ) = G N ( x ) sin( π ( s − r ) N x ), so G N ( x ) = sin( π ( s + r ) N x ))( s − r ) N (sin πx ) , using (2.1). Then for integer b with b ( s − r ) N / ∈ Z , one has V (cid:48) N (cid:16) b ( s − r ) N (cid:17) = ( − b G N (cid:16) b ( s − r ) N (cid:17) π ( s − r ) N .
Hence sgn V (cid:48) N (cid:16) b ( s − r ) N (cid:17) = ( − b sgn (cid:16) sin π ( s + r ) bs − r (cid:17) (2.4)= ( − b sgn (cid:0) sin (cid:16) πb + 2 πbrs − r (cid:17)(cid:17) = sgn (cid:16) sin 2 πbrs − r (cid:17) . Accordingly, we set δ ( b ) := − sgn (cid:16) sin 2 πbrs − r (cid:17) . (2.5)We note that δ ( b + s − r ) = − sgn (cid:16) sin 2 π ( b + s − r ) rs − r (cid:17) = − sgn (cid:16) sin 2 πbrs − r (cid:17) = δ ( b ) . Hence the values δ ( b ) are periodic with period s − r . HARSH MEHTA
Next set W N ( x ) := x + (cid:88) n (cid:54) =0 v r,s − r ( n/N )2 πin e ( nx ) , (2.6)and note that W (cid:48) N ( x ) = V N ( x ) by (1.11). Suppose that x k − , x k , x k +1 are three consec-utive simple zeros of V N in (0 , x k = a/ (( s + r ) N ). If V (cid:48) N ( x k ) < V N ( x ) > x k − < x < x k and V N ( x ) < x k < x < x k +1 . These intervalscontribute to the integral an amount (cid:0) W N ( x k ) − W N ( x k − ) (cid:1) − ( W N ( x k +1 ) − W N ( x k ) (cid:1) = − W N ( x k − ) + 2 W N ( x k ) − W N ( x k +1 ) . In this situation ε ( a ) = 1, so the point x k contributes 2 ε ( a ) W N ( x k ). If V (cid:48) N ( x k ) > x k is still 2 ε ( a ) W N ( x k ). Now supposethat x k is a double zero, and that V N ( x ) > (cid:0) W N ( x k ) − W N ( x k − ) (cid:1) + (cid:0) W N ( x k +1 ) − W N ( x k ) (cid:1) = W N ( x k +1 ) − W N ( x k − ) . In this case, x k makes no contribution, but ε ( a ) = 0, so the contribution is still2 ε ( a ) W N ( x k ). If V N ( x ) < x k is still 2 ε ( a ) W N ( x k ). Similarly, if x k = b/ (( s − r ) N ), then thecontribution of x k is 2 δ ( b ) W N ( x k ).The interval [0 , / (( s + r ) N )] contributes W N (1 / (( s + r ) N )) − W N (0). The first termhere is half of the contribution made by the point 1 / (( s + r ) N ), since ε (1) = 1. Thecontribution made by the interval [1 − / (( s + r ) N ) ,
1] is W N (1) − W N (1 − / (( s + r ) N )).The latter term is half the contribution made by the point 1 − / (( s + r ) N ), since ε (( s + r ) N −
1) = −
1. We note that W N (1) − W N (0) = 1. Hence we conclude that (cid:90) | V N ( x ) | dx = 1 + 2 ( s + r ) N (cid:88) a =1 ε ( a ) W N (cid:16) a ( s + r ) N (cid:17) (2.7)+ 2 ( s − r ) N (cid:88) b =1 δ ( b ) W N (cid:16) b ( s − r ) N (cid:17) . Here the terms a = ( r + s ) N and b = ( s − r ) N ought not to be included in the above,since V N (1) (cid:54) = 0. However, ε (( s + r ) N ) = 0 and δ (( s − r ) N ) = 0, so no harm is done.We write W N ( x ) = x + X N ( x ) and evaluate the contributions of the two terms sepa-rately to the right side of (2.7). The contribution of the linear term x to the sum (2.7)is 2 ( s + r ) N (cid:88) a =1 ε ( a ) a ( s + r ) N + 2 ( s − r ) N (cid:88) b =1 δ ( b ) b ( s − r ) N .
Since the ε ( a ) are periodic with period s + r , and the δ ( b ) are periodic with period s − r ,the above is= 2 N − (cid:88) n =0 (cid:18) s + r (cid:88) a =1 ε ( a ) (cid:16) a ( s + r ) N + nN (cid:17) + s − r (cid:88) b =1 δ ( b ) (cid:16) b ( s − r ) N + nN (cid:17)(cid:19) . On reversing the order of the double sums, we see that this is= 2 s + r (cid:88) a =1 ε ( a ) (cid:16) as + r + N − (cid:17) + 2 s − r (cid:88) b =1 δ ( b ) (cid:16) bs − r + N − (cid:17) . (2.8) HE DE LA VALL´EE POUSSIN KERNAL 9
Since V N (0) = V N (1) = s + r >
0, in the interval [0 ,
1] we pass from positive values tonegative the same number of times that we pass from negative values to positive. Thatis, 0 = ( s + r ) N (cid:88) a =1 ε ( a ) + ( s − r ) N (cid:88) b =1 δ ( b ) = N s + r (cid:88) a =1 ε ( a ) + N s − r (cid:88) b =1 δ ( b ) . Hence the expression (2.8) is= 2 s + r s + r (cid:88) a =1 ε ( a ) a + 2 s − r s − r (cid:88) b =1 δ ( b ) b . (2.9)The contribution of X N ( x ) to the right side of (2.7) is2 ( s + r ) N (cid:88) a =1 ε ( a ) X N (cid:16) a ( s + r ) N (cid:17) + 2 ( s − r ) N (cid:88) b =1 δ ( b ) X N (cid:16) b ( s − r ) N (cid:17) . (2.10)Here the sum over a is (cid:88) n (cid:54) =0 v r,s − r ( n/N ) πin ( s + r ) N (cid:88) a =1 ε ( a ) e (cid:16) an ( s + r ) N (cid:17) . (2.11)where v r,s − r ( n/N ) are given in (1.12). Since the ε ( a ) have period s + r , we know by thetheory of the Discrete Fourier Transform that there exist numbers (cid:98) ε ( k ) such that ε ( a ) = s + r (cid:88) k =1 (cid:98) ε ( k ) e (cid:16) aks + r (cid:17) (2.12)holds for all integer a . Hence the expression (2.11) is= (cid:88) n (cid:54) =0 v r,s − r ( n/N ) πin s + r (cid:88) k =1 (cid:98) ε ( k ) ( s + r ) N (cid:88) a =1 e (cid:18) a ( n + kN )( s + r ) N (cid:19) . Here the innermost sum is ( s + r ) N if n ≡ − kN (mod ( s + r ) N ), and is 0 otherwise.We write n = − kN + m ( r + s ) N . Then the above is= ( s + r ) s + r (cid:88) k =1 (cid:98) ε ( k ) (cid:88) m ∈ Z ( s + r ) m (cid:54) = k v r,s − r ( − k + m ( s + r )) πi ( − k + m ( s + r )) . (2.13)We note that if 1 ≤ k ≤ s + r , then v ( − k + m ( s + r )) = 0 if m > m <
0. Thus thesum over m can be restricted to just m = 0 ,
1. However, the main point of the above isthat it is independent of N .The sum over b in (2.10) is (cid:88) n (cid:54) =0 v r,s − r ( n/N ) πin ( s − r ) N (cid:88) b =1 δ ( b ) e (cid:16) bn ( s − r ) N (cid:17) . (2.14)The δ ( b ) have period s − r , so let numbers (cid:98) δ ( n ) be determined so that δ ( b ) = s − r (cid:88) k =1 (cid:98) δ ( k ) e (cid:16) kbs − r (cid:17) (2.15) for all b . Hence the expression (2.14) is= (cid:88) n (cid:54) =0 v r,s − r ( n/N ) πin s − r (cid:88) k =1 (cid:98) δ ( k ) ( s − r ) N (cid:88) b =1 e (cid:16) b ( n + kN )( s − r ) N (cid:17) . The innermost sum is ( s − r ) N if n ≡ − kN (mod ( s − r ) N ), and is 0 otherwise. Wewrite n = − kN + m ( s − r ) N . Then the above is( s − r ) s − r (cid:88) k =1 (cid:98) δ ( k ) (cid:88) m ∈ Z ( s − r ) m (cid:54) = k v r,s − r ( − k + m ( s − r )) πi ( − k + m ( s − r )) . (2.16)This formula and (2.13) serve to evaluate the expression (2.10). On combining thisevaluation with (2.9) in (2.7), we conclude that (cid:90) | V N ( x ) | dx = 1 + 2 s + r s + r (cid:88) a =1 ε ( a ) a + 2 s − r s − r (cid:88) b =1 δ ( b ) b + ( s + r ) s + r (cid:88) k =1 (cid:98) ε ( k ) (cid:88) m ( s + r ) m (cid:54) = k v r,s − r ( − k + m ( s + r )) πi ( − k + m ( s + r )) (2.17)+ ( s − r ) s − r (cid:88) k =1 (cid:98) δ ( k ) (cid:88) m ( s − r ) m (cid:54) = k v r,s − r ( − k + m ( s − r )) πi ( − k + m ( s − r )) . Since this value is independent of N , the proof is complete. (cid:3) Proofs of Theorem 1.2 and Corollary 1.3
Proof of Theorem 1.2. . Recall from (2.1) that for ( r, s ) = 1, V r,s ( x ) = (cid:0) sin π ( s + r ) x (cid:1)(cid:0) sin π ( s − r ) x (cid:1) ( s − r )(sin πx ) . (3.1)In the special case s − r = 1, which corresponds to ( r, s ) = ( n, n + 1) we obtain thesimplification V n,n +1 ( x ) = (cid:0) sin π (2 n + 1) x (cid:1)(cid:0) sin πx (cid:1) (sin πx ) = sin π (2 n + 1) x sin πx = D n ( x )The right hand side is exactly the Dirichlet kernel D n ( x ), rescaled to the interval [0 , L n = (cid:107) D n ( x ) (cid:107) L ( T ) = (cid:90) | D n ( x ) | dx, which on rescaling to the usual interval [0 , π ] recovers the usual definition (1.16). (cid:3) We give explicit computations yielding Corollary 1.3.
Proof of Corollary 1.3.
The result follows on combining Theorems 1.1 and 1.2. For theexplicit value we have V , ( x ) = 1 + 2 cos 2 πx = sin 3 πx sin πx HE DE LA VALL´EE POUSSIN KERNAL 11 by (1.15) and (2.1). Hence (cid:90) | V , ( x ) | dx = (cid:104) x + sin 2 πxπ (cid:12)(cid:12)(cid:12) / − (cid:104) x + sin 2 πxπ (cid:12)(cid:12)(cid:12) / / + (cid:104) x + sin 2 πxπ (cid:12)(cid:12)(cid:12) / = 13 + 2 √ π . Alternatively, one can argue from (2.17). We find that ε (1) = 1, ε (2) = − ε (3) = 0, (cid:98) ε (1) = − i/ √ (cid:98) ε (2) = i/ √ (cid:98) ε (3) = 0, and δ (1) = (cid:98) δ (1) = 0. The result is the same. (cid:3) Concluding Remarks
We showed that the L -norm of V rN,sN ( x ) = (sin π ( s + r ) N x ) (sin π ( s − r ) N x )( s − r ) N sin πx is independent of N . If we let A + r,s,N resp. A − r,s,N denote the positive and negative areasof the graph then we have shown A + r,s,N − A − r,s,N = (cid:107) V r,s (cid:107) L ( T ) is independent of N . Since A + r,s,N + A − r,s,N = 1 we have that both these areas areindependent of N , with A + r,s,N = 12 (cid:16) (cid:107) V r,s (cid:107) L ( T ) (cid:17) . The effect of increasing N is does not change the area, but shifts its location. As N increases most area (both positive and negative) is concentrated near integer values of x . One can show that | V rN,sN ( x ) | ≤ √ N for 1 √ N ≤ x ≤ − √ N . Acknowledgments
The author thanks H. L. Montgomery for suggesting the project of improving thebounds for L norms of de la Vallee Poussin kernels as part of an REU program atthe University of Michigan in summer 2012. He thanks H. L. Montgomery and J. C.Lagarias for references and for editorial assistance with exposition. The author thanksP. Nevai for helpful comments and references. P. Nevai observed that Theorem 1.1 canbe deduced from results which are contained in his unpublished 1969 manuscript (Nevai[9, Lemma 3]). References [1] S. N. Bernstein,
Sur l’ordre de la meilleure approximation des fonctions continues par les poly-nomes de degr´e donne,
Academie Royale de Belgique, Cl. Sci. M´em. Coll. in quarto, Ser. 2 (1922), No. 1, 104 pages. [This paper submitted in 1912 to Concors Annuel .][2] P. L. Butzer and R. Nessel,
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