Discrete least squares approximation of piecewise-linear functions by trigonometric polynomials Текст научной статьи по специальности «Математика»

Probl. Anal. Issues Anal. Vol. 6(24), No. 2, 2017, pp. 3-24 3

DOI: 10.15393/j3.art.2017.4070

UDC 517.521.2

G. G. Akniyev

DISCRETE LEAST SQUARES APPROXIMATION OF PIECEWISE-LINEAR FUNCTIONS BY TRIGONOMETRIC POLYNOMIALS

Abstract. Let N be a natural number greater than 1. Select N uniformly distributed points tk = 2nk/N (0 < k <

< N — 1) on [0, 2n]. Denote by L„,n(f) = L„,w(f,x) (1 <

< n < N/2) the trigonometric polynomial of order n possessing the least quadratic deviation from f with respect to the system {tk}N="01. In this article approximation of functions by the polynomials Ln,N(f,x) is considered. Special attention is paid to approximation of 2n-periodic functions fi and f2 by the polynomials Ln,N(f, x), where fi(x) = |x| and f2(x) = signx for x G G [—n, n]. For the first function fi we show that instead of the estimation |fi(x) — Ln,N(fi,x)| < clnn/n which follows from the well-known Lebesgue inequality for the polynomials Ln,N (f, x) we found an exact order estimation |fi(x) — Ln,N(fi,x)| < c/n (x G R) which is uniform with respect to 1 < n < N/2. Moreover, we found a local estimation |fi(x) — Ln,N(fi,x)| < c(e)/n2 (|x — nk| > e) which is also uniform with respect to 1 < n <

< N/2. For the second function f2 we found only a local estimation |f2(x) — Ln,N(f2,x)| < c(e)/n (|x — nk| > e) which is uniform with respect to 1 < n < N/2. The proofs of these estimations are based on comparing of approximating properties of discrete and continuous finite Fourier series.

Key words: function approximation, trigonometric polynomials, Fourier series

2010 Mathematical Subject Classification: 41A25

1. Introduction. Let N be a natural number greater then 1. Let {tk lyti—)1 be a set of points on [0, 2n] where tk = 2nk/N. Denote by Ln,N(f) = Ln,N(f, x) (1 < n < [N/2J) a trigonometric polynomial of

©Petrozavodsk State University, 2017

[MglHl

order n possessing the least quadratic deviation from the function f with respect to the system {tk }fc=0 . In other words, the minimum of the sums N=01 |f (tk) — Tn(tk)|2 on the set of trigonometric polynomials Tn of order n is attained when Tn = Ln,N (f). In particular, L|_n/2j,n (f, tk) = = f(tk). It is easy to show (see [12]) that for n < N/2 the polynomial Ln,N (f, x) can be represented as

n 1 N-1

Ln,N(f,x) = £ cVN)(f)e-, cVN)(f) = - £ f(tk)e-ivtk,

v=—n k=0

and for n = N/2 (for an even N)

Ln/2,N (f,x) = Ln/2 —1,N (f,x)+ «N/2 (f )cos N/2(x — u), (1)

where

N1

«n2n)(f) = «NN/2(f) = N E f (tk) cos N/2(tk — u). (2)

N k=0

To read more about function approximation by trigonometric polynomials see [1, 3, 5, 6], [8]-[11], [13]. In this article we obtain estimations for |Ln,N(fi, x) fi(x)| and |Ln,N(f2,x) —f2(x)| as n, N ^ to, where fi(x) = = |x|, f2(x) = signx, x G [—n, n]. The following theorems are proved:

Theorem 1. Let f1 (x) = |x|, x G [—n, n] and n < |_N/2J. The following estimations hold:

|Ln,N(fi,x) — fi(x)| < c/n, x G [—n,n],

|Ln,N(fi,x) — fi(x)| < c(e)/n2, x G AJ(e).

Theorem 2. Let f2(x) = signx, x G [—n, n] and n < |_N/2J. The following estimation holds:

|Ln,N(f, x) — f (x)| < c(e)/n, x G AJ(e).

We begin with some notation. Denote by

ck(f ) = 2n/f (t)6-'''dt, k G Z,

the Fourier coefficients of a function f, and by

n

f(x) = £Ck(f)eikx, Sn(f,x) = £ Ck(f)eikx

keZ k=-n

the Fourier series of a function f and its partial sum of order n, respectively. Denote by AJ(e) the set [—n + e, —e] U [e, n — e], where 0 < e < n/2. By c and c(e) we denote some positive constants that depend only on specified parameters; these constants may be different in different places.

Lemma 1. [12] If the Fourier series of f converges at the points tk = = u + 2kn/N, then the representation

Ln,N (f, x) = Sn (f, x) + Rn,N (f, x),

where

2 ^ n

Rn,N(f, x) = - Dn(x — t) cos ^N(u — t)f (t)dt, (3)

^ — n

holds true when 2n < N.

The following estimation follows from this lemma:

|Ln,N(f,x) — f(x)|<|Sn(f,x) — f(x)| + |Rn,N(f,x)|, n < N/2. (4)

Let us consider the case 2n = N. From (1) and (4)

|Ln,2n(f, x) — f (x)| <

<|Sn—i(f,x) — f (x)| + |Rn—i,2n(f,x)| + |«n2n)(f )|, n = N/2. (5)

From (4) and (5) we can see that estimation of the values |Sn(f1, x) — —fi(x)|, |Sn(f2,x) — f2(x)|, |Rn,N(fi,x)|, |Rn,N(f2,x)|, |«n2n)(fi)|, and |an2n) (f2)| implies an estimation |Ln,N(f, x) — f(x)| for the functions fi and f2.

2. Estimations for |Sn(fi,x) — fi(x)| and |Sn(f2,x) — f2(x)|.

Lemma 2. For the value |Sn(fi,x) — fi(x)|, where fi(x) = |x|, x G G [—n, n], we have the following estimations:

|Sn(fi,x) — fi(x)| < c/n, x G [—n,n], (6)

(fi,x) - fi(x)| < c(e)/n2, x e AJ(e).

(7)

Proof. Using [2, p.443] or [4, p.690], we can get the following representation for f1 (x) on [—n, n]:

From the above we can obtain estimation (6) (using the fact that |cos(2k — 1)x| < 1). Also, if we apply the Abel transformation to the above equation, we get (7). □

Lemma 3. For the value |Sn(f2, x) — f2(x)|, where f2 (x) = sign x, x e e [—n, n] we have the following estimation:

The proof of this lemma is obtained in [7].

3. An estimation for Rn,N(fi,x). The following lemma takes place.

Lemma 4. For Rn,N (f1,x), n < N/2 the following estimations hold:

Note that the estimation for remainder (3) for the function f1 has the following form:

|Sn(f2,x) — f2(x)|< c(e)/n, x e AJ(e).

x e [—n, n],

where

n

Dn (x — t) = 1/2 cos k(x — t).

From above we have

|Rn,N(fi,x)| < |Rn,N(fi,x)| + |Rn,N(fi,x)|, (8)

where

1 ^ n

Rn,N(fi,x) = |t| cos^N(u — t)dt, (9)

, x) = —

n

2 ^ ^ n

Rn,N(fi, x) = — ^^ |t| ^^ cos k(x — t) cos^N(u — t)dt. (10)

, n ^=i—n k = i

Lemma 5. The value (fi,x)| has the following estimation:

(fi,x)| < c/n2, x G [—n, n]. Proof. Using the formula

cos(^Nu — ^Nt) = cos ^Nu cos ^Nt + sin ^Nu sin ^Nt (11) we can rewrite (9) as follows:

2 ^ n Rn,N(fi,x) = — ^^ cos^Nu t cos^Ntdt.

n ^=i 0 Then we calculate the integral using the previous equation:

n (—1)^N — 1 t cos ^Ntdt =-^—.

J (^N)2

0

From these equations we get

T>1 (f \ 2 ^ cos ^Nu ^ i^N

Rn,N (fi,x) = „2 ((—— ^

or

i 0, N = 21,

Rn,N (fi,x) = \ 4 ^ cos(2^- i)Nu a7" _ 07 i 1

' l — nN2 ^ (2^-1)2 , N = 21 + 1

\ ^=i

Now we can estimate | Rn n (fi, x) |:

|Rn,N(fi,x)| < -N2 E

nN2 ^ (2^ — 1)2'

1

It is known that

So, using (12) we obtain

E

n

1 (2^ - 1)2 8

(12)

n n

\Rn,N (/i,x)\ < WT^ <

2N2 " 8n2

□

Lemma 6. The value |Rn n(fi,x)| has the following estimation for x e e [—n, n] :

| Rn,N (fi,x)| <-, x e [—n,n].

n

Before proving Lemma 6, we prove another one: Lemma 7. The following estimations take place:

^cos(2i — 1)x

i=0

<

1

2 |sin x| '

y^ cos 2ix

i=0

<

|sin x|

y sin(2i + 1)x

i=i

<

|sinx| '

k-1

y sin 2ix

i=0

<

|sin x|

Proof. After some simple transforms we obtain the following:

E cos(2i — 1)

- 1)x =

i=i

sin x

y sin x cos(2i — 1)

- 1)x =

i=i

1

2 sin x

y (sin (x — (2i — 1)x) + sin(x + (2i — 1)x) =

i=i

1 \—^ / • n n / 1 \ \ sin 2kx

(sin 2ix — sin 2 (i — 1) x) =

2 sin x

i=i

2 sin x

Likewise, obtain

k

E

i=i

cos 2ix =-V sin x cos 2ix =

sin x

i=i

sin(2k + 1)x — sin x 2 sin x

1

1

1

1

1

k

v^ . cos 2x — cos(2k + 2)x

> sin(2i + 1)x =---—,

2 sin x

i=i

cos x — cos(2k + 1)x

y sin2ix =-.

2 sin x

i=i

Now the proof is complete. □ Proof. (Lemma 6) From (11) and

cos k(x — t) = cos kx cos kt + sin kx sin kt (13)

(10) can be rewritten as follows:

n

4 to n „

Rn,N (f i, x) = — ^^ cos ^Nu ^^ cos kx t cos kt cos ^Ntdt+ n ^=i k=i 0

n

4 to n „

+— ^^ sin ^Nu ^^ sin kx t sin kt sin ^Ntdt.

n ,, — 1 7--1 J

n

^=i k=i

From above and the formulas

cos kt cos ^Nt = 1/2 (cos (^N — k) t + cos (^N + k) t),

sin kt sin ^Nt = 1 /2 (cos (^N — k) t — cos (^N + k) t) we can obtain

2 to

Rn N (fi, x) = — / ^ cos ^Nux

Ai=i /

V

n I n n \

£ cos kx J t cos (^N — k) tdt + J t cos (^N + k) tdt J +

00

n

x

k=i

2 to

+— ^^ sin ^Nux

n / ? ?

x y^ sin kx / t cos (^N — k) tdt — t cos (^N + k) tdt

k=i 0

0

After calculating the integrals, we have 2 TO

Rn N (A,x) = — / cos

n N

X V ((— 1)^N+fc — 1) cos kx -1-2 + -1-2 +

' V(^N — k)2 (,N + k)2 1

2 TO

+— ^^ sin mNux X V ((— 1)^N+k — 1) sin kx(-1-2--1-2 ). (14)

¿T( ) J V (MN — k)2 (MN + k)2' ( )

We can rewrite this as follows:

■ TO n / .. ..

|Rn N (fi,x)| < - V V -2 + -2 ) +

| , (f , )|< n (MN — k)2 (mN + k)2 I

4 oo n

7T EE

2<

n i=ii=iV (mN — k)2 (mn + k) 1

a (MN <nN i=i M2 k=N (1 — lN)

„ to n ^ _ to ^ n . _

12 1 12 1 1 4n 2n

< - > > -o < - > - > -o < - < -.

< n (,,.N - k)2 < n^M^^A n < N < -

□

Lemma 8. The value |Rn N(fi,x)| has the following estimation for x e e AJ (e) : ,

|RU(fi,x)| < = #, x e A1 (e).

1 , 1 -2 sin e -2

Proof. To prove this lemma, we use (14). It is easy to show that for

even N

4 to 4 TO

R2n (fi,x) =--V cos (x) — 4 V sin (x), (15)

where

tn/21 / 1 1 An,N (x) = V cos(2k — 1)x -2 +-2

n ( ) k=i( ) V (MN — (2k — 1))2 (MN + (2k — 1))2

1

1

tn/21

(x) = £ sin(2k - 1)x

k=i

(juN - (2k - 1))2 (uN + (2k - 1))2

And, therefore

.to .to

Kn(fi,x)| = - £ |(x)| + 4 £ |Bn,N(x)| .

I I n 1 1 n 1 1

Consider the values |An,N (x)| and |Bn,N (x)|. Apply the Abel transformation for An,N (x):

(x) =

+

(^N - (2 [n/2] - 1))2 (^N + (2 [n/2] - 1))2

x

[n/2] tn/21-1

x £ cos(2i-1)x- £

i=i

k=i

+

(uN - (2k + 1))2 (^N + (2k + 1))2

1

+

(uN - (2k - 1))2 (^N + (2k - 1))2

^ cos(2i - 1)x.

i=i

From this we get the following estimation:

(x)| <

+

(UN - (2 [n/2] - 1))2 (uN + (2 [n/2] - 1))2

x

x

[n/21

£ cos(2i - 1)

i=i

[n/21-1 + £

k=i

(UN - (2k + 1))2 (uN + (2k + 1))2

1

+

(UN - (2k - 1))2 (uN + (2k - 1))2

y cos(2i - 1)x

i=0

Using Lemma 7, we can rewrite the obtained estimation:

|sin x| \ (uN - (2 [n/2] - 1))2

+

(UN + (2 [n/2] - 1))2 / < (uN)2 |sinx|

10

(16)

1

1

1

1

1

1

1

1

1

1

1

1

2

1

1

Using the similar approach, we get an estimation for |Bn,N (x)|:

8

|Bn,N(x)| < vN|5nx• (17)

Using (15), (16), and (17) we can write

a to a to

|Rn,N(fi,x)| < 4 £ |(x)| + 4 £ |Bn,N(x)| <

<4 V 10 4 ^ 8 < 26 N -21

< n ^ (^N)2 |sinx| + n ^ 3^3N2 |sinx| < N2 |sinx|, = •

" " (18) If N is odd, then we can rewrite (14) as follows:

Rn ,N (fi,x) =

oo

A ^^ A

4 V Cn,N(x) cos(2^ — 1)Nu — - V D^,n(x) sin(2^ — 1)Nu—

.to . to

— 4 V En,N (x) cos 2^Nu —

V En,N (x) cos 2^Nu — 4 V F£,N (x) sin 2^Nu,

where

Ln/2J

(x) = £ cos 2kx I -2 +

k=1 Ln/2J

(x) =

((2^ — 1)N — 2k)2 ((2^ — 1)N + 2k)2

l_n/2j /

D^,n (x) = £ sin 2kx k=1

((2^ — 1)N — 2k)2 ((2^ — 1)N + 2k)2

tn/21 . En,N(x) = £ cos(2k—1)x -2 +

k=1 fn/21

(2^N — (2k — 1))2 (2^N + (2k — 1))2

(x) = V sin(2k—1)x ( -1-2 — .

k= V(2^N — (2k — 1))2 (2^N + (2k — 1))2

1

1

1

1

.00 ^00

4 ^ , 4

And R,n(/i,x)\ <-E \Cn,N(x)\ + -E \Di,N(x)\ +

1 1 7T ' 1 1 7T ' 1 1

A to /1 to

+ - E\En,N(x)\ + - E\FTN(x)\ .

7T 1 1 7T 1 1

Using the Abel transformation, we can easily obtain these estimations: C,N (x)\<,. 10 2 A„ , \D^,N (x)\ < 8

|sin x| N2' 1 " 1 |sin x| N2' (*)|<,. 2 2 A7~2 , (x)|< 8

|sinx| ^2N2' ' ^ ' -9 |sinx| N2 ' From this we have

(/>-x)l< • N = 21 + 1 (19)

From (18) and (19) we have (for every N > 2)

1 2 1 14n 7n 7n i

N( f 1 ,x)l < , .-t < ^ 01 •-r < ^ o 1 .-r, x G A (e).

I ^ N2 |sinx| < 2n2 |sinx| < 2n2 |sine|' w

□

4. An estimation for ai2n)(/1). When 2n = N from (2) we have the following:

.. 2n-1 ^ 2n-1 ✓ , s

ai2n) (/) = ^ E /(tk) cos nk = - £ (-1)'/ u + n- . (20) k=0 k=0 ^ 7

Subtract n from u to get (provided that / is 2n periodic function)

h E1(-i)k / ((«- n)+nk)=in E1(-i)k / (u+^

k=0 v 7 k=0 v

=2nE (-i)k-1/ (u+?)=- bE (-i)k / (u+?) =

27 2^'(-1)'+ +(-D2"f (u + 2")

k=1 v 7

¿17 E^)* /(« + = -an2"' (/ ).

Therefore, if we subtract or add — from u, we get

<e> (/) = <—^ EM)k /((u+3 +

In other words, adding — (l e Z) to u does not change the value of

«2n)(f)

, so we can assume, without loss of generality, that 0 < u < E.

It holds for both

an2n) (fi) and an2n) (f2)

(2n)

Lemma 9. The following estimation takes place:

«n2n)(fi) < , x e [—n, n].

n2

Proof. It is easy to show that

«nf = 2n E (—D"

ni

"=-n

ni

k

--+ u

n

7T

, 0 < u < —.

n

«n2n) (fi) = 2n E (—1)"

" = n

k

--+ u

n

i

2n

(—1)

1

2n

k=-n

n

k

---u

n

ni

Em)

k

--

k=i

1 n k

n

u

+2n £W( ?+») = " =0 n1

+ 2nE ("DM T + -) =

" =0

n k

2n

E(—1)k(-—^ —

"=1

n

1 £ (_1)kY i)+ui =

2n

"=1

n

1 £ (—W £—^—=2n £ (—1)k (n—2u

2n

"=1

n

"=1

Get the following estimation:

«n2n)(fi)

<

1

2n

£ (^Hn — 2«

1 ||

< — --2u

< 2n n

<

7T

2n2

□

1

"

5. An estimation for Rn,N(f2, x). We prove the following lemma. Lemma 10. The following takes place:

|Rn,N(f2,x)|< ^, x e AJ(e).

n

To prove this lemma we estimate |Rn,N(fi,x)| as follows:

|Rn,N (f2,x)| < R,N (f2 ,x)| + |Rn,N (f2 ,x) | , (21)

where

1 to r

#nN (f2,x) = — 7 cos ^N (u — t)sign tdt, (22)

, Tn

n

2 to „ n

Rn,N (f2 ,x) = 2 ^^ X^cos k(x — t) cos ^N(u — t)sign tdt. n ^=i-7r fc=i

First we prove the following lemmas:

Lemma 11. The following takes place:

|Ri,N (f2, x) | < n Proof. Using (11) for (22) we have

1 to n

Ri n (f2,x) = — 7 cos ^N (u — t)sign tdt =

1 to n

y / (cos ^Nu cos ^Nt + sin ^Nu sin ^Nt) sign tdt

(

71 i

n

n

1 to

— > cos ^Nu cos ^Ntsign tdt + sin ^Nu sin ^Ntsign tdt ) =

i /

1 to n 2 to n

= — > sin ^Nu sin ^Ntsign tdt = — > sin ^Nu sin ^Ntdt.

7T i J 7T i J

From above and

sin ^Ntdt = —

cos ^Nt

^N

1 — (— ^N

we have

2 ^

(f — V(1 — (— 1)"N)

N

^=1

sin ^Nm

If N is even, then N(/2,x) = 0, otherwise

(f2,x) = — ^J] ^=1

4 ^ sin (2^ — 1) Nu

nN ^ 2^ — 1

where

Therefore

□

4 ^^ sin(2^ — 1)x

7F

- 2^ — 1

= |/2(x)|< 1-

11

Kn(f2,x)i < N <

Lemma 12. Denote Yn(x) = 5^(1 — (—1)k+m)ak cos kx, where all

In I n

k=1

> 0, the sequence m=1 is monotone, and m = 0,1. Then

lYn(x)| <

|sin x|

Proof. Assume that > ak+1. After applying the Abel transformation we get

Yn(x) = £(1 — (—1)k+m)afc cos kx

k=i

cos kx =

n1

= an

£(1 — (— 1)i+m) cos ix — £ (ak+1 — ak) £(1 — (—1)i+m)

cosix.

i=1

k=1

i=1

n

n

0

c

From the above equation and Lemma 7 we have

lYn(x)| < a

- (-l)i+m)

cos ix

i=1 n — 1

+

+ £ (ak+1 - ak)

k=i

£(1 - (-1)*+m)

cos ix

i=1

<

^ 1 f \ ^ 2an

< --- (an + an - ao) < —

|sin x|

|sin x| |sin x|

□

Lemma 13. If vn(x) = (1 — (— 1) m)ak sin kx, where all ak > 0,

k=i

the sequence ak is monotone, and m = 0,1. Then

|Vn (x)| <

|sin x|

The proof of this lemma is analogous to the proof of Lemma 12. Lemma 14.

y sin kx

k=1

<

1

lsin 2 I

-, Vn G N.

Proof. From

sin kx =-- y^ sin kx

Z—/ Qin x '

k=1

sin x k=1

x

sin —

we can easily obtain the equality

n

y sin kx =

k=1

sin x sin n x sin x '

which gives us the desired estimation ^n=1 sin kx| < , .1 x ,. □

|sin x |

Lemma 15. The following takes place:

c(£)

,n(f2,x)| , x G A1 (e).

1 1 n

c

Proof. From (11) and (13) we have

n

2 to n

R2 (/2,x) = -£ £cos k(x — t) cos aN (u — t)sign tdt

, x) = —

k=1

2 to

y sin ^Nu ^^ cos kx / (cos kt sin aNt) sign tdt+

7T

^=1 k = 1

2 to

— n n

+— ^^ cos ^Nu ^^ sin kx (sin kt cos aNt) sign tdt

7T

^=1 k = 1

— n

n

4 c^o n «

y sin ^Nu ^^ cos kx cos kt sin aNtdt+ n ^=1 k = 1 0

4 oo n «

+— ^^ cos ^Nu ^^ sin kx

n U = 1 k = 1 n

n

4 to n

— cos ^Nu sin kx I sin kt cos aNtdt. n ^=1 fc=1 0

Using the formula sin a cos ft = 1 (sin (a — + sin (a + ft)) and calculating the integrals from the above equation, we get |Rn N(/2,x)| < < | Rn,N (/2 ,x) | + | R^V (/2 ,x) | , where '

RnVv (/2, x) = - V sin ^Nu V (1 — (—1)^v+k) cos kx-^-,

n'N ^ ) ^ ( ) ; (^N )2 — k2,

.to n ,

R^v (/2, x) = - £ cos ^Nu £ (1 — (—1)^v+k) sin kx———.

n ^=1 k=1 (a )

We estimate values |Rn'V(/2, x)| and ^^V(/2,x)| separately. Begin with

|R?Jv (/2,x)|. ' '

Rn''V (/2 ,x) = ^ £ £(1 — (—1)^v+k )cos kx ---„

^ = 1 a k = 1 1 (^v)2— k2

nN ^ a ^ V 7 7 1 - —k2

2 £ sin^ £(1 —(—1)^v+k)cos k^ 1+ k2

m=1 a fc=1 V

nN a v 7 7 V (aN)2 — k2y

7T

We can rewrite it as follows:

T-.2 1 2 ^^ sin ^Nm

Rn,N (f2,x) = --- 2^cos kx +

nN ^ -

2 ^ (—sin -Nu ,

> -—----> (—1)k cos kx+

nN ^ - y

2 ^^ sin -Nu

+--— > - > cos kx-

N - (

k2

- (-N )2 — k2

+ V (—sin -Nu ^ (—1)kk2

^=1

-

k = 1

(-N )2 — k2

Now we have

2

<JV (f ,x)| < ^

E

sin -Nu

-

+

2

+

(—sin -Nu -

k=i

y cos kx +

n

£(—1)k cos kx

k=i

E

2 ^^ sin -Nu , k2

—— > - > cos kx

N -

+

(-N )2 — k2

+

+

2 ^ (—sin -Nu ^ , (—1)k k2 > - > cos kx-

N

-

k=i

(-N )2 — k2

From [4, p. 448] we get

2

7T

E

sin -Nu -

Also, from Lemma 7

< 1 and —

7T

E

^=1

(—sin -Nu

-

cos kx

k=i

<

| sin x|

£(—1)k cos kx

k=i

<

| sin x|

1

1

Using the above estimations we can write

R. N (/ ,x)| <

+

N |sinx| nN

to , T n

sin uNu

> - > cos kx

k2

p=1

u

k=i

(UN)2 - k2

+

+

2

nN

^ (-1)pn sin uNu ^ , (-1)kk2 > - > cos kx-

p=1

u

k=i

(UN)2 - k2

. (24)

To estimate the value

2

sin uNu y^- y^ cos kx

k2

p=1

u

k=i

(UN)2 - k2

make some transformations:

sin uNu

2

sin uNu

> - cos kx

k2

p=1

u

k=i

(UN)2 - k2

2

sin uNu

¿v-3— cos kx

p=1

U

k=1

n 2 - 4 p2

<

<

2 xTO 1

nN E U p=1

cos kx

k=1 -1 /

N2 - 4

A v —

nN ^ u3

n

N2 - ^ ■ !

p2 j=1

y cosjx—

£

k=1

(k + 1)2

z

N 2 - ^ N2 - £ 7=1 p2 p / j=1

cos jx

<

<

2 A 1

E

2n

2

nN u3 \ N2 - ^ p=1 \ p2

-1

+£

k=1

(k + 1)2

p k2

cos jx

j=1

+

N2 _ (W N2 -

cos jx

j=1

<

2 ^ 1

< ——^E

2n2

1

nN |sinx| ^ uM N2 - 4 N2 - -1

<

2

2

2

k

2

k

2

k

<

8 ^ < 4 3nN |sinx| u3 < nN |sinx|

1 1 u=1

Using the similar approach, prove

2

nN

to £

^=1

(-1)^N sin^Nu ^ , (-1)kk2 cos kx

U

k=i

(uN)2 - k2

<

4

N | sin x|

Now we can rewrite (24) as follows

1

|<N (/2 ,x)| < Now consider |Rn.'2v (/2, x) |.

,2 + 8

N | sin x|

N | sin x|

(25)

4

R2,2v(/2, x) = - £cosuNu£ (1 - (-1)^N+k) sin kx ^=1

k

k=1

nN ^ u ^=1

4 to 1

I**(/2-x)|s nN £U2

i toto at

4 £ cosuNn £(1 - (_1)^)sinkx

k2 - (uN)2 k

k=1

^ _ 1 2 N 1

£(1 - (-1)^N+k)sinkx-

N

k

k = 1

1 1

^2N

Using the approach we used for estimating |Rn'N(/2,x)|, prove that

RnN (/2, x) | <

N | sin x|

(26)

From the obtained estimations (25) and (26), and the inequalities n < N/2 and |sine| < |sinx| for x G AJ(e), we have (23). □

From (21) and Lemmas 11 and 15 we have

|Rn,N(/2, x)| < 1 (c + c(e)) = ^, x G AJ(e). n n

So, Lemma 10 is proved.

6. An estimation for ai2n) (/2). We prove a lemma:

c

Lemma 16. The following estimation takes place:

42n) (/2)

c

< -. n

Proof. Using formula (20) we get (for a 2n-periodic function f)

1 2n-1 1 2n-1

ai2n)(f) = 2n E f (tk)cosnk = - £ (-1)kf (u + nk/n). k=0 k=0

As has been mentioned earlier, we can safely assume that 0 < u < n/n; so we write

n — 1

a(?n) (/2) = 1/2n £ (-1)k/2 (nk/n + u), 0 < u < n/n.

n

1 / -1 n-1

ann)(/2) = 2n E (-i)k+1 + E(-i)k+

Vk=-n+1 k=1

+ (-1)nf2 (U - n) + f2 (u) j = 2- (f2(u) - ( l)nf2 (U)) . From this we get

aL2n)(f2)| < 1/n.

□

7. Proofs of Theorems 1 and 2. From (4) and (5) we have |Ln,N (fl,x) - fl (x)| < |Sn (fl,x) - fi(x)| + |Rn,N (fl ,x)|, n < N/2,

lLn,2n (f1,x) - f1 (x)| <

< |Sn-1 (f1,x) - f1 (x)| + |Rn-1,2n(f1,x)| + |«n2n)(f1 )|, n = N/2. From Lemmas 2, 4, 9 we easily get

|Ln,N(f1,x) - f1 (x)| < c/n, x G [-n,n], |Ln,N(f1,x) - f1(x)| < c(e)/n2, x G AJ(e).

Theorem 1 is proved.

Using the same technique, inequalities (4) and (5), and Lemmas 3, 10

and 16, we get

|Ln,N(f,x) - f (x)| < c(e)/n, x G AJ(e).

So, theorem 2 is also proved.

References

[1] Bernshtein S. N. On trigonometric interpolation by the method of least squares. Dokl. Akad. Nauk USSR, 1934, vol. 4, pp. 1-5. (in Russian)

[2] Courant R. Differential and Integral Calculus. New Jersey. : Wiley-Interscience, 1988, vol. 1, 704 p.

[3] Erdos P. Some theorems and remarks on interpolation. Acta Sci. Math. (Szeged), 1950, vol. 12, pp. 11-17.

[4] Fikhtengol'ts G. M. Course of differential and integral calculus. Moscow. : FIZMATLIT, 1969, vol. 1, 656 p. (in Russian)

[5] Kalashnikov M. D. On polynomials of best (quadratic) approximation on a given system of points. Dokl. Akad. Nauk USSR, 1955, vol. 105, pp. 634636. (in Russian)

[6] Krilov V. I. Convergence of algebraic interpolation with respect to the roots of a Chebyshev polynomial for absolutely continuous functions and functions with bounded variation. Dokl. Akad. Nauk USSR, 1956, vol. 107, pp. 362-365. (in Russian)

[7] Magomed-Kasumov M. G. Approximation properties of de la Valle-Poussin means for piecewise smooth functions. Mat. Zametki, 2016, vol. 100, is. 2, pp. 229-244. D0I:10.1134/S000143461607018X

[8] Marcinkiewicz J. Quelques remarques sur l'interpolation. Acta Sci. Math. (Szeged), 1936, vol. 8, pp. 127-130. (in French)

[9] Marcinkiewicz J. Sur la divergence des polynômes d'interpolation. Acta Sci. Math. (Szeged), 1936, vol. 8, pp. 131-135. (in French)

[10] Natanson I. P. On the Convergence of Trigonometrical Interpolation at Equi-Distant Knots. Annals of Mathematics, Second Series, 1944, vol. 45, no. 3, pp. 457-471. D0I:10.2307/1969188.

[11] Nikol'skii S. M. On some methods of approximation by trigonometric sums. Mathematics of the USSR - Izvestiya, 1940, vol. 4, pp. 509-520. (in Russian)

[12] Sarapudinov I. I. On the best approximation and polynomials of the least quadratic deviation. Anal. Math., vol. 9, is. 3. pp. 223-234.

[13] Zygmund A. Trigonometric Series. Cambridge. : Cambridge University Press, 1959, vol. 1, 747 p.

Received October 11, 2017. In revised form, December 13, 2017. Accepted December 15, 2017. Published online December 27, 2017.

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