On the Convergence of the least square method in case of non-uniform grids Текст научной статьи по специальности «Математика»

166

Probl. Anal. Issues Anal. Vol. 8 (26), No3, 2019, pp. 166-186

DOI: 10.15393/j3.art.2019.6410

UDC 517.521

M. S. SULTANAKHMEDOV

ON THE CONVERGENCE OF THE LEAST SQUARE METHOD IN CASE OF NON-UNIFORM GRIDS

Abstract. Let f (t) be a continuous on [-1,1] function, which values are given at the points of arbitrary non-uniform grid QN = = {tj j-1, where nodes tj satisfy the only condition nj ^ tj ^ Vj+i, 0 ^ j ^ N — 1, and nodes Vj are such that —1 = n0 < n1 < V2 < < ■ ■ ■ < Vn-1 < Vn = 1- We investigate approximative properties of the finite Fourier series for f (t) by algebraic polynomials Pn, N (t), that are orthogonal on QN = {tj j-1- Lebesgue-type inequalities for the partial Fourier sums by Pn, N(t) are obtained. Key words: random net, non-uniform grid, orthogonal polynomials, Legendre polynomials, least square method, Fourier series, function approximation

2010 Mathematical Subject Classification: 42C10, 41A10, 33F05

1. Introduction. Let {nj}N=0 be a system of points, such that

— 1 = no < ni < V2 < ■■■ < VN-1 <VN = 1. (1)

We assume Anj = Vj+1 — Vj, 0 ^ j ^ N — 1, AN = ^ max ^ Anj. Now, we construct a grid from the points

Vj ^ tj ^ Vj+1, j = 0,1,..., N — 1, (2)

selected on each segment [nj ,Vj+1]. Without loss of generality, we can consider all the nodes {tj j-1 distinct, because if tj = tj+1 for some j, we can leave only one of them and denote the grid by QN-1.

Consider the space l2 (QN) of discrete functions f : ^ R, where the inner product is given by

N-1 N-1

(f, 9) = E f (tj)g(tj)AVj = An £ f (tj)g(tj)pj. (3)

j=0 j=0

© Petrozavodsk State University, 2019

By Pn,N (t), 0 ^ n ^ N — 1, we denote polynomials that form finite orthonormal system with respect to this inner product:

n-i rn /

i ^ ^ \ _ ^ ^ I 0 n -— ^^

( Pn, N ,Pm, N / / , Pn, N (tj )Pm,N (tj )^Vj — S -, (4)

\ / ^ 1, n — m.

j=0 k

We call polynomials Pn,N (t), 0 ^ n ^ N — 1, the discrete orthonormal Legendre polynomials.

Since the system {Pn,N(t)}n=0 is complete in l2(QN), any function f € l2(^N) can be expanded in a finite Fourier series by this system. Let An,N(f, t) be the partial Fourier sum of order n for the function f — f (t) by the system {A, N}fc=0\ in other words

k=0

N-1

A„,n(f,t) = fkPk,N(t), where fk = ^ f (tj)Pk,N(tj)An. k=0 j=0

The main goal of this article is to study the approximative properties of An,N(f, t) in case when f (t) is continuous on [—1,1] and t G [—1,1]. More precisely, we want to obtain an estimate for the value

|Rn,N(f,t)| = |f (t) - An,N(f,t)| , t G [-1, 1]. (5)

Note that the value |Rn,N (f, t)| for the discrete Legendre polynomials was studied in [2] for the case of tj = nj and was studied in [3] for the case of tj = Vj +2nj+1. But the results obtained there are valid only when n = O(AN1/5) and n = O(AN2/7), respectively, while we managed to get estimates for n = O(AN1/3) and for a more general case when tj is arbitrary on the segment [nj ,nj+1].

To solve this problem, we need some information about discrete Le-gendre polynomials Pk,N (t), as well as discrete Jacobi polynomials P^N (t), which are a generalization of Pk,N (t). This information is based on the properties of classical continuous Legendre and Jacobi polynomials.

2. Some information about Jacobi and Legendre polynomials. The Jacobi polynomials can be written using Rodrigues' formula (see, for example, [4]) as follows:

(_ 1)n i dn

Pna'3 (t) = --^ d- [Ka'3 (t)an(t)},

n w 2nn! Ka'3(t) dtn [ w W}'

where a,ß are arbitrary real numbers, Ka'ß(t) = (1 — t)a(1 + t)ß, a(t) = = 1 — t2. In the case when a,ß > —1, the Jacobi polynomials form an orthogonal system with the weight (t):

Pn'ß (t)Pm ß (t)Ka'ß (t)dt = $

-1

where

h a>ß = 2a+ß+1r(n + a + 1)r(n + ß + 1) n = n!(2n + a + ß + 1)r(n + a + ß + 1) '

and, therefore, h^ x n 1, n = 1, 2,.... For the derivative of Pna'^(t), the following equality holds:

(P^ (t))' = " + P + n + 1 P„-+i1'^+1(t). (6)

We will also need the following weighted estimate

/ _ i \-«-2 / i 2

vn\Pna'^(t)I ^ c(a,fl)( V1—t + n) fv/TTi + n) , (7)

where —1 ^ t ^ 1. An important particular case of Jacobi polynomials with a = ^ = 0 is Legendre polynomials Pn(t), orthogonal on [—1,1] with

the unit weight p(t) = 1. Denote by Pn(t) = ^^n^1 Pn(t), n = 0,1, 2,... the corresponding orthonormal Legendre polynomials. The leading coefficient of polynomial Pn(t) can be written as

(2n)! /2n + 1

n

(n!)22n V 2

3. Discrete Jacobi and Legendre polynomials. We will use the integral analogue of the Markov inequality for estimating the derivative of an algebraic polynomial (see [5,6]), which for r =1 has the following form:

1 1

J Wm(t)|dt ^ cMm2^ |qm(t)|dt, (9)

-1 -1

1

where qm(t) is an arbitrary algebraic polynomial of degree m. For every m, denote by xm the minimum of constants c(m) that satisfy inequality (9) i.e.,

I |qm(t)|dt -1

Xm = sup -1-,

qm m2 / |qm(t)|dt -1

where the upper bound is taken by polynomials qm(t) of degree at most m and not equal to zero identically. In work [5] by N. K. Bari, it is shown that x = sup xm < ro. Given this fact, we derive from (9):

1 1

J Wm(t)|dt ^ Xm2 J |qm(t)|dt. (10)

-1 -1

Let {P^'n (t)}n—o be polynomials that form a finite orthonormal system with respect to the inner product

N-1

<PnaN ,P:;N > = E Pna'N (tj )P:;N (j ^ (j

j—o

We call these polynomials discrete orthonormal Jacobi polynomials.

In the case when the grid QN consists of equidistant nodes tj = — 1 + Nty , the asymptotic properties and weighted estimates for the polynomials orthogonal on QN were first studied in the papers by 1.1. Sharapudinov (see [7]). Later, 1.1. Sharapudinov [8-10] and A. A. Nur-magomedov [11], [12] studied the asymptotic properties of polynomials that are orthogonal on nonuniform grids of the real axis. In particular, in [12] the author investigated the asymptotic properties of the discrete Jacobi polynomials P™'N(t) (a and ^ are integers), orthogonal on nonuniform grid ttN with tj = Vj +2nj+1, 0 ^ j ^ N — 1.

In our work [13], we investigated asymptotic properties of these polynomials in the general case of random tj (a, ^ are still integers). When

n = O(AN3) and n,N ^ ro we obtained asymptotic formula

Pna;N (t) = PnaJi (t) + <N (t),

0, n = m,

1, n = m.

here Praa'v(t) is a normed Jacobi polynomial, and van 'N (t) is the remainder, for which the following estimate is established:

<'N (cos

/ , „ - an x(2n + a r x1 i 1 n 2 V^ Yn 1 ^ # ^ 2: ^ c(a,^Y) ---2—< ,__2

1 - ANx2(2n+a+( n«+27AN, o ^ ^ ^ 7n-1,

where x is the smallest of the constants in the Markov integral inequality for estimating the derivative of an algebraic polynomial. Here and further in the text, c, c(a), ), c(a,^,... ,7) are positive constants depend-

ing only on the specified parameters, which, generally speaking, may be different in different places. For the sake of simplicity, these estimates are given for the segment [0,1]; they apply to [— 1, 0] in the similar way.

In the article, the indicated asymptotic formula is directly used to study the value |Rn,N (f, t)|.

4. Auxiliary statements. In this section, we collect some of the statements that will be needed in the future.

Lemma 1. Let f (t) be a function, absolutely continuous on [— 1,1];

in }N

0 and {tj },=Q be systems of nodes that satisfy (1) and (2), res' pectively. Then

b

J f (t)dt = £ f (tj)An; + rN(f)

a a^tj

for every segment [a, b] C [-1,1], where

b

|rN(f)| ^ An f |f'(t)|dt.

a

Proof of this lemma can be found in [13].

From Lemma 1 the next statement also follows:

Lemma 2. Let {nj }N=0 and {tj j-1 be systems of nodes that satisfy (1) and (2), respectively. Then the following inequality holds for an absolutely continuous on [-1,1] monotonous non-negative function f (x):

b

£ f (tj)Anj ^J f (t)dt + An|f (b) - f (a)|.

a^tj ^b

Lemma 3. For the leading coefficients of the discrete Legendre polynomials, the inequality

kn, N

^ 1

11)

kn+1, N

holds; here kn,N and kn+1;N are the leading coefficients of the polynomials Pn. n and Pn+i,N, respectively.

Proof. Following [14], let us consider the expression

N-1

y^ Pn+1,N (tj)tjPn,N (tj)A% = j=0

k

N-i

N-i

n, N

E p2+l, N (tj) + E Qn,N (tj ) =

k

n, N

7 /, - n+i, N J ) ' / -b n, N \-J/ J

kn+i,N . „ . „ k

j=0 j=0

n+i, N

On the other hand,

n, N

N -i

k

n+i, N

j=0

^ I Pn+i, N (tj ) tj Pn, N (tj ) A% ^

Ni

max {|tj|} V |Pn+1,N(tj) pPn.N(tj) AVj. " j=0

Applying the Cauchy-Bunyakovsky inequality, we finally get

kn. N

Ni

kn+i,N -

m^N^i{|tj|l E j^n+i,N(tj)

j=0

A j X

' N -i

XI E | Pn, N (tj )| An j=0

j

max ||t7|j ^ 1. 0<KN -i

This completes the proof. □

The following lemma establishes the relation between polynomials of degrees n and n + 1 .

Lemma 4. For An

k

n, N

k

n+i, N

the following equalities hold:

(i - (t) =

= aAn(tJ^^ - PW(tJ, (12)

\ V P«,N(1) V pJ™+!;N(1) J

(1+ t)pP°;1(t)

= -P., N(th/ Pn+1'N (-1) - 4+1, N (t) J PP"N( ) . (13)

"ra,N(-1) V ^n+1'N(-1)

Proof. Consider the polynomial (t), given by the equality

(1 - t)Qn(t) = Pn+1'N(1)pPn,N(t) - pP„'N(1)pPn+1'N(t). (14)

From its definition, we have

N-1

£ Qn(tj)A,n(tj)(1 - tj)Anj = 0, 0 ^ k ^ n - 1. (15) j=0

Let Mi (t) be an arbitrary polynomial of degree l ^ n - 1. Since each polynomial Pk'N(t) has degree k, it is obvious that M(t) can be represented as their linear combination:

Mi(t) = £ 4 A,n (t).

k=0

Then, from (15) we get

N-!

£ Qn(tj)Mi(tj)(1 - tj)A% = 0, j=0

i. e., polynomials Q0(t),..., QN-!(t) form an orthogonal system with the weight k!' 0 (t) = 1 — t on the grid . Hence,

Qn(t)= YnPlN(t), Yn > 0. (16)

To find Yn, taking into account (14), consider the expression

N -1

Hn = E Qn(tj)(1 — tj)AVj = (17)

j—0

N — 1 r

= 4+1, N (1) E Pn,N (tj )Qn(tj )AVj = Pn+l,N (1)^-^ , (18)

j—0

-n, N

where kn, -n,N are the leading coefficients of polynomials Qn(t) and Pn,N (t), respectively.

In addition, notice that kn = Pn,N(1)-n+1)N from (14), and, therefore Hn = Pn+1, N (1) Pn, N (1) -n±^. (19)

On the other hand, we get, from (16) and (17),

N-1 2

Hn = Yn E (tj)) (1 — tj)AVj = 7n. (20)

j—0

Comparing (19) with (20), we derive

Yn =\l Pn+1, N (1) Pn, N (1) . (21)

-n, N

Returning to equality (14) and using (16), we have

(1 — t)Yn Pn1: N (t) = Pn+1, N (1)Pn,N (t) — Pn, N (1)Pn+1,N (t).

This equality, together with (21), gives us (12). Similarly, we derive equality (13). □

Next, let us agree on the following notation:

n

Kn, N (x,y) = E Pk,N (x) Pk,N (y). (22)

k—0

Then, using the Christoffel-Darboux formula and Lemma 4, we can also prove the following assertion.

Lemma 5. The following equality holds:

= N

Pni 'n (x)P,N (y) - — 0 (y)Pn,N (x)

y - x

y - x

(23)

where

C,

n, N

N Pn+1, N

(1)

pn+1, N Pn, N (1)

The weighted estimate for the discrete Legendre polynomials, obtained by the author in [13], takes the following form:

Theorem A. Let us put 4ANxn2 < 1; then there is a constant a > 0, such that

P,

n, n(t)| « c(a) (l + BTn^) (V1 - t2 +

n)-2 «

n

« c(a) (l + Byn3An) <

(1 + t) (1 -1)

i

n2,

i " 4

- 1 « t « -1 + an

2

- 1 + an-2 « t « 0

i " 4 ,

1

n2 ,

0 « t « 1 - an

2

(24)

1 - an-2 « t « 1,

where

B

3 - 4Anxn2 1 - 16AN x2n4

5. Approximative properties of the Fourier sums by PnaN (t). Suppose we are given the values of some continuous on [-1,1] function f (t) at the points of the grid . Our main goal is to estimate the value

|Rn,N(f,t)| = |f (t) - A„,n(f,t)| , t G [-1, 1].

Denote by the Hilbert space of all polynomials of degree n and by

En(f ) = min max |f (t) - p,(t)| Pn ePn te [-1,1]

the best approximation for the function f (t) by polynomials of degree at most n.

2

It is easy to show that An, N(pn, t) = pn(t) for any polynomial pn G Pn Hence, using the Lebesgue-type inequality, we get

|Rn,N(f,t)| = |f(t) — An,N(f,t)| 4 En(f) [1 + Ln,N(t)] ,

(25)

where

N1

Ln,N (t) = £ |Kn ,N(tj,t)| Anj (26) j—0

^ 1 N-1

is the Lebesgue function for <Pn,N> and Kn,N (tj, t) is the kernel

n—0

from (22).

Thus, it is necessary to study the Lebesgue function Ln,N(t). Theorem 1. There exists a real number y > 0, such that for 2 4 n 4

— 1/3

4 yAn ' and 0 < e < 1 the following estimates hold:

max Ln N(t) 4 c(y)n2,

-14t41

Ln,N(t) 4 c(y) ln n, —1 + e 4 t 4 1 — e.

Proof. We consider only the case t G [0,1], because for t G [—1,0] the proof is quite similar.

Let us start with t G [0,1 — 4n-2]. We divide the sum on the right-hand side of (26) according to the following scheme:

Ln,N (t) =

E + E + E + E

-1<tj 4-2 -1 4j ^qi qi^j ^qa q2<tj 41

|Kn , N (tj, t) | Anj

= A1 + A2 + A3 + A4, where q1 = t — ^1-t2, q2 = t + ^1-t2.

I1 n IT-2 n

1. To estimate A1, we use the Christoffel-Darboux formula and Lemma 3, as well as the fact that |t — tj| ^ 2 for t G [0,1 — 4n-2] and tj G [—1, — 2]. We have

A1 4 2 E (|4+1,n(t)Pn,n(tj)| + |Pn,n(t)Pn+1,n(tj)|) Anj.

Again, we divide the sum into two parts: denote by A11 the sum over — 1 4 tj 4 —1 + 4n-2, and by A12 that over —1 + 4n-2 4 tj 4 — 2.

~14tj 4-2

Using the weighted estimate (24) and Lemma 2, we obtain

An < cn

1

A12 < c (1 - t)-4 ,

(27)

which means that

A1 < c (1 - t)-4 . (28)

2. We proceed to the estimation of A2, for which we again use the Christoffel-Darboux formula and the transformation (23) from Lemma 5. We have

A2 < C,

n, N

£ 1 -

-1 <tj <91

t tj

P^N (tj )Pn,N (t)

Anj +

+ £

1 -1

t - tj

-1 <tj <91 ^

Consider, firstly, A21:

JP1'N (t)PPn, N (tj ) Anj = Cn, N [A21 + A22]

A21 < c(1 -1)-1 £ j

- 2 <tj <91 j

pP1;N(tj) Anj = c(1 -1)-4A211,

where

A

211

£ + £

-1 <j <0 0<tj <91

1 - tj t - t,"

P^ri,'N (tj )

Anj <

< c

£ 1-r Anj + £

-1 <j <0

0<tj

(1 - tj ) 4 t-ti

Anj

<

V^ (1 - tj) 4 .

< c E ^ j) Anj.

-1 <tj <91

t tj

Due to the obvious inequality (1 - tj) 4 < (1 - t) 4 + (t - tj)1, we can rewrite

A21 < c (1 -1)-4 £

-1 <tj <91

(1 - t) 4 + (t - tj )1 t tj t tj

Anj

=c

£

An

j

1 t - t j -1 <tj <91

-1)-4 E

Anj

-1 <tj <91

(t - tj)

a21) + (1 — t)-4 a21

(29)

Using Lemma 2, the theorem condition of ANn3 4 Y3 and the fact that (1 — t2)-2 4 n at t G [0,1 — 4n-2], we get the estimates

A2i) 4 c ln n, A221 4 c,

wherefrom

A21 4 c (lnn + (1 — t)-4) . (30)

We proceed to studying A22. Applying the weighted estimate for P'n ,N (t) we derive

Pn,N (tj )

i

A22 4 c(1 — t)4 E t — t, AV3.

- 2 4tj 4qi 3

Before applying the weighted estimate for Pn,N(tj), note that (1 + tj)-1 4 4 31 (1 — tj)-1 and (1 — tj)-1 4 (1 — t)-1 for — 2 4 tj 4 0. Then

2j

(1 — tj )"

A22 4 c(1 — t)1 J] (11—tj) Anj 4

- 2 4j 4q1 j

4 c E At~ 4 c(a)ln n. (31)

- 2 4tj 4q1 j

Substituting (30)-(31) in (29), we finally get

A2 4 Cn,N [A21 + A22] 4 c ((1 — t)-1 +ln n) .

(32)

3. To estimate A3, we do not apply any transformations, but substitute the weighted estimates directly in (26):

A3 4 c (n + 1{ E Anj 1 4

(1— t) 1 qj (1 — tj ) 1

(n + 1) W2 — W1 . f , 4 c-i--r 4 c(a). (33)

(1 — t)4 (1 — W2)1

c

4. The study of the last part of the sum is similar to the study of A2:

A4 ^ N

£ 1 — tj

tj t

Pu!;N(tj)pn,N(t) Anj+

+£

1t

tj - t 92€tj € ! j

92€tj € !

¿UN(t)pPn,N(tj) Anj = Cn,N [A4! + A42]. (34)

In turn, A41 can also be represented in the form of several sums

A4! € c(1 - t)-4 x

x

£ + £ + £

92€tj € €tj €!—u-2 !-n-2<ij €!

1 - tj tj - t

4!;N(tj) Anj

= c(1 - t)-4

A(!) + A(2) + A(3) A41 + A41 + A41

41

41

(35)

Using Lemma 2 and the weighted estimates, we can obtain the following estimates for these terms:

a4? € c [(1 -1) 4 A4!! + A4121 = c [(1 -1)1 in n +1

a42) € cn 1,

a43! € cn 1.

Returning to inequality (35) and using (36)—(37), we derive

A4 1 € c

in n + (1 - t) 4 ^n 2 + 1 j € c (1 - t) 4 +ln n

(36)

(37)

(38)

Similarly, the second term from (34) is estimated as

A42 = c(1-t)4 x

x

£ + £ + £

92 € ^ ^ €tj € !—n—2 ! -n-2€ij € !

Pu, N (tj )

tj - t

Anj

/ s 2

c(1 - t) 4

A !) + A(2) + A(3)

A42 + A42 + A42

42

(39)

Applying Lemma 2 and a series of transformations, we obtain estimates for these parts:

A( ! €

42 € cn2, A422 € cn2, A432 € cn2.

Then

A4 4 c [A41 + A42] 4 c (1 — t) 4 +ln n

(40)

Finally, all the estimates (28), (32), (33) and (40) in total allow us to display for t G [0,1 — 4n-2]

Ln, N (t) = A1 + A2 + A3 + A4 4 c (1 — t)1 +ln n

(41)

Now we consider the behavior of the Lebesgue function Ln,N (t) for t G [1 — 4n-2,1]. Let us represent the Lebesgue function in the following form:

Ln, N (t)

E + E + E ] |Kn,n(t3,t)|An

-14j 4-1 -14tj 4q1 q14tj 41

= /1 + /2 + /3.

1. The estimation of /1 is similar to the estimation of A1:

(42)

/1 4 2

E + E

-14tj 4-1+4n-2 -1+4n-24tj 4-2

Pn+1, N (t) P'n, N (tj )

+

+

P'n, N (t)Pn+1,N (tj )|) Anj = 2(/n + /12).

Using the weighted estimates for the discrete Legendre polynomials, we get for /11 and /12 the following inequalities:

/11 4 cn 1, /12 4 cn

Therefore,

/1 4 cn2.

(43)

2. To estimate /2, we use Lemma 5 and the Christoffel - Darboux formula again:

/2 4 Cn, N

E

1t

1 8 t — tj

-14j41-5 J

P^N (tj )Pn,N (t)

Anj+

+E

1t

1 81 — tj -14j 41-"

P^N(t)Pn,N (tj ) Anj = Cn, N [/21 + /22] .

Consider, firstly, /21:

/2i 4 cn2

+

1 — tj t-t,-

(tj) Anj 4

4c

(1 — tj)

£ t — f -14j 41-j

-Anj.

Due to the obvious inequality (1 — tj)4 4 (1 — t)4 + (t — tj)4, we can

rewrite

/2i 4 cn2

(1 — t'1 , E .rj + ,E,

- 2 4tj 41-- 2 4tj 41-n2

Anj

(t — tj)^

= cn2

(1 — t)4 /21) + /2?

(44)

Using Lemma 1, we obtain for these new parts the estimates

/21) 4 c ln n,

/(2) 4 c J21 4 c,

and finally

/21 4

cn2 (n 1 ln n + 4 cn 1 .

(45)

Let us start with /22. Using the weighted estimates and the fact that

(1 + tj) 4 4 31 (1 — tj) 1 4 cn2 for — 1 4 tj 4 0, we derive

r 1 v^ (1 — tj)- A

/22 4 cn-2 E V , j Anj 4

-14tj 41-

2 4tj41-n2

8 t — t j

_1 1 4 cn 2n2

E

Anj

1 8 t — tj -14tj 41-n2 j

4 c ln n.

(46)

Combining (45) and (46), we finally get

/2 4 Cn,n [/21 + /22] 4 c (n2 + lnn) 4 cn1. 3. For the last part, we just use (24):

/3 4 EE |^k,N (t)Pk,N (tj )| Anj 4

1-n-24j 41 k=0

8

- 2 4tj 40 04tj 41-

< c £ 2k1A^ < cn2 £ A^- < c(a). (48

1-ra-2^ij <1 fc=0 1-n"2<ij <1

Combining estimates (43), (47) and (48), we finally get

(t) = h + /2 + Is < c(n1 +lnn) , t G [1 - 4n-2,1]. (49

Note that for t G [1 — 4n-2,1] the expressions (1 — t2)4 and n1 are of the same order. Hence, from (41) and (49) we deduce the assertion of the theorem. □

Returning to (25), we also get the following statement from Theorem 1: Theorem 2. The estimate

K,N(t)| < c(y) e„(/)

ln n + V1 — t2 + -

1

1\ -11

n

— 1/3

holds for the remainder Rn,N(t), where 2 < n < yAn ' , 7 > 0, and En(/) is the best approximation for the function / (t) by polynomials of degree at most n.

6. Some applications. Once again, let /(t) be a continuous on [—1,1] function, which is measured at the nodes of some arbitrary grid = {tj j-1, satisfying (1)-(2). We denote these measurements by yj = /(tj) + 0, 0 < j < N — 1. Here £j are observation errors, which are independent random variables satisfying the following conditions:

E&] = 0, E]= jj, 0 < j < N — 1, (50)

where E [X] is the expected value of a random variable X. It is required to approximately restore / (t) at the point t G [—1,1] using discrete information {yj j-1. To solve this problem, we introduce an algebraic polynomial (t) that minimizes the sum

N-1

J..., ara) = £ (yj — p„(tj))2 pj j=0

on the set of all polynomials pn(t) = a0+a1t+.. .+antn of degree n < N —1, where pj are positive weight factors.

The question is, how precise Sn,N(t) approximates the original function f (t) at t G [-1,1], i. e., it is required to estimate the value (f (t) — Sn,N(t))2. Since this value depends on random errors £0,... , £N-i, a more accurate formulation of the problem is to estimate its average value

Jn,N(f,t) = E [(f(t) — Sn,N(t))2] . (51)

In [1] this problem was studied for the uniform grid tj = — 1 + N2 , Pj = 1, 0 4 j 4 N — 1. In this article, we consider a more general case when the nodes tj form a non-uniform grid = {tj j-1 C [—1,1], and weights Pj satisfy certain natural conditions.

More precisely, the values of o2-, appearing in (50), and corresponding weights Pj are defined for a given real o using equalities

2 2 AN ( I \2

o2 = 0 A", pj = (o/o") = AN". (52)

It is well-known (see [15]) that polynomials Sn,N (t) minimizing the value (51), can be represented as

n N -1

Sn,N(t) = EPkPk,n(t), where pk = ^ y,-Pk,N(tj)A^j. k=0 j=0

Let An,N (f, t) be the partial Fourier sum of order n for the original (noise-

f P ^ N-1

less) function f = f (t) by the system ^ Pk N f , i. e.,

I ' J k=0

n N-1

An ,n(f,t) = E fkPk,n(t), where f = ^ f (tj)Pk,N(tj)A^j. k=0 2=0

From (50) it follows that An,N(f,t) = E [Sn,n(t)]. In fact,

n n

E [Sn, N (t)] = E yk Pk,N (t)] = E E [Pk ] Pk, N (t), k=0 k=0

where

N -1

E [Pk] = e[ E(f (tj) + &)Pk,N(tj)A%

j=0

N-1

= £ E [f (tj)+ 0] Pfc;N(tj= f. j=0

In addition, it can be shown that

n

Jn,N(f,t) = (f (t) - An,N(f,t))2 + a2An £ (A,n

k=0

= <n (f,t) + Dn,N (t). (53)

To do this, consider the expression

n

An, N (f,t) = Sn,N (f,t) - An, N (f,t) = £ (yfc - fk) Pfc,N (t) =

k=0

n n N-1

= £ IkPfc, n(t) = ££ oPfc, n(t)Pk,N(tj)Anj. k=0 k=0 j=0

We will need the expected value of this value:

n

E [An,N(f,t)] = £ E [Ik] Pk,N(t) = k=0

n N -1

= £ £ E [Ij] Pk,N(t)Pk,N(tj)Anj = 0, (54)

k=0 j=0

and its square:

E [An, n(f,t)] = E

£ IkPk,N (t) £ IiPi,n (t)

k=0

J=0

= E [IkIl] Pk,N(t)Pl,N(t). k=0 1=0

Due to (50) and (4), we know that

E

N-1 N-1

IkIi] = £ £ E [IiIj] Pk,N(ti)P,N(tj)AniA^j i=0 j=0

N-1

E (o2A j -k,N(tj)-A,n(tj)A^j = o2An,

(55)

j=0

wherefrom

E [An,N(f,t)] = o2An E (A,n(t)) .

k=0

Then, taking into account (54) and (55), we have

Jn,N (f,t) = E [(f (t) — Sn, N (f,t))2] =

= E [(f(t) — An,N(f,t) + An,N(f,t))2] = (f(t) — An,N(f,t))2 + +2 (f (t) — An,N(f,t)) E [An,n(f,t)] + E [An,n(f,t)] =

n 2

= (f (t) — An,N(f, t))2 + 02An E (^,N(t)) = <N(f, t) + Dn,N(t)

k=0

Thus, the original objective of estimating the deviation of partial sums by discrete Legendre polynomials Pn,N (t) from the desired function f (t) comes to estimating these two values: R N(f, t) and Dn, N(t).

The estimate for Rn,N (f, t) is given in Theorem 2. Let us consider

n / " \ 2

the value Dn, N(t) = o2AN Y1 (-pk,N(t) ) . Using weighted estimates (24)

k=0

obtained in Theorem A, we have

n __2

,n(t) 4 o2An E (c(a) (1 + Bv/k3AN) k'

4

k=0

4 c(a)o2(n2AN) (1 + B^n3A^ 4 c(a)o2(n5An)2.

So, the value Dn,N(t) tends to zero when n = O(AN1/3). Finally, we conclude with the following statement.

Theorem 3. Let f (t) be a continuous on [—1,1] function given by its measurements y2- = f (tj) + £2-, j = 0,1,..., N — 1, in the nodes of the grid nN, which satisfy (1)-(2), where £2- are independent random mistakes of observation that satisfy (50)-(52). Then, for 2 4 n 4 yAN1/3, Y > 0, the following estimate holds:

Jn,N(f,t) 4 C(a,Y,o) En(f)

ln n + ( V1 — t2 + 1

n

+ A3

2

where En(f ) is the best approximation for the function f (t) by polynomials of degree at most n.

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Received June 3, 2019. In revised form, October 18, 2019. Accepted October 22, 2019. Published online November 9, 2019.

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