INNER PRODUCT AND GEGENBAUER POLYNOMIALS IN SOBOLEV SPACE Текст научной статьи по специальности «Математика»

Том 27, № 138

2022

© M. A. Boudref, 2022

DOI 10.20310/2686-9667-2022-27-138-150-163

м9

OPEN fil ACCESS

Inner product and Gegenbauer polynomials in Sobolev space

Mohamed Ahmed BOUDREF

University of Bouira, 10000 Drissi Yahia Bouira St., Bouira, Algeria

Abstract. In this paper we consider the system of functions G^n(x) (r G N, n = 0,1,...) which is orthogonal with respect to the Sobolev-type inner product on (-1,1) and generated by orthogonal Gegenbauer polynomials. The main goal of this work is to study some properties related to the system {^k ,r(x)}k>0 of the functions generated by the orthogonal system {Gran(x)} of Gegenbauer functions. We study the conditions on a function f (x) given in a generalized Gegenbauer orthogonal system for it to be expandable into a generalized mixed Fourier series of the form

r-1 / \ k ^

f(x) ~ £f+ £C?ik(f)<ifc(x),

k=0 k=r

as well as the convergence of this Fourier series. The second result of this paper is the proof of a recurrence formula for the system {^k,r(x)}k>0. We also discuss the asymptotic properties of these functions, and this represents the latter result of our contribution.

Keywords: inner product, Sobolev space, Gegenbauer polynomials

Acknowledgements: The work was carried out at the Faculty of Sciences and Applied Sciences and at the LIMPAF Research Laboratory of the University of Bouira, Algeria.

Mathematics Subject Classification: 42C10.

For citation: Boudref M.A. Skalyarnoye proizvedeniye i mnogochleny Gegenbauera v prost-ranstve Soboleva [Inner product and Gegenbauer polynomials in Sobolev space]. Vestnik ros-siyskikh universitetov. Matematika - Russian Universities Reports. Mathematics, 2022, vol. 27, no. 138, pp. 150-163. DOI 10.20310/2686-9667-2022-27-138-150-163.

© Будреф М.А., 2022

DOI 10.20310/2686-9667-2022-27-138-150-163 УДК 517.518.36

Скалярное произведение и многочлены Гегенбауэра в пространстве Соболева

Мохамед Ахмед БУДРЕФ

Университет Буира, Алжир, г. Буира, ул. Дрисси Яхья Буира 10000

Аннотация. В данной работе рассматривается система функций Ga„(x) (r G N, n = 0,1,...), которые ортогональны относительно скалярного произведения соболевского типа на (-1,1) и порождены ортогональными полиномами Гегенбауэра. Основной целью данной работы является изучение некоторых свойств, связанных с системой r(x)}k>o функций, порожденных ортогональной системой jGa„(x)} функций Гегенбауэра. Исследуются условия на функцию f (x) , заданную в обобщенной ортогональной системе Гегенбауэра, которые гарантируют ее разложимость в обобщенный смешанный ряд Фурье вида

r— 1

f (x) ~ £ f(k)(-1) frac(x + 1)kk! + £ C^(f *(x),

k=0 k=r

и изучается сходимость этого ряда Фурье. Второй результат этой статьи состоит в доказательстве рекуррентной формулы для системы j^k,r(x)}k>0. Мы также обсуждаем асимптотические свойства этих функций, что составляет заключительный результат нашей работы.

Ключевые слова: скалярное произведение, пространство Соболева, многочлены Геген-бауэра

Благодарности: Работа выполнена на Факультете естественных и прикладных наук и в Исследовательской лаборатории LIMPAF Университета Буира, Алжир.

Для цитирования: Будреф М.А. Скалярное произведение и многочлены Гегенбауэра в пространстве Соболева // Вестник российских университетов. Математика. 2022. Т. 27. № 138. С. 150-163. DOI 10.20310/2686-9667-2022-27-138-150-163. (In Engl., Abstr. in Russian)

Introduction

Consider an orthogonal system k(x)}^0 on (a,b) with p(x) as a weight function, and let r G N. We construct a new orthogonal system (pk,r(x)}^0 following the Sobolev-type inner product:

r-1 r-b

(f,9)s = £ f{v)(a)g{v)(b)+ f(r)(t)g(r)(t)p(t)dt. (0.1)

v=0 Ja

Quite a few authors have presented this type of construction, see, for example, the works of R. M. Gadzhimirzaev ( [1]) and 1.1. Sharapudinov ( [2-6]) on the construction of mixed Fourier series. The author in his works presents some particular cases of systems generated by classes of orthogonal functions, namely Jacobi, Legendre, Chebychev, Laguerre, and Haar.

Gegenbauer polynomials are widely used in several fields, they are of a particular interest in applications. It is clear that these polynomials present a special case of those of Jacobi for particular values of parameters. In this work, we will reconstruct the orthogonal system Wk,r(x)}^0 generated by the Gegenbauer polynomials using the approach which is different from the one used by I. Sharapudinov.

Denote by Lpp(a,b) the space of measurable functions f (x), x G (a,b), with

, b

/ f (r)(t)g(r)(t)p(t)dt<

a

When p(x) = 1, we write Lpp(a,b) = Lp(a,b). It is clear that Lpp(a,b) is the Banach space with the norm

i

fb

p

- I f I / 1 x II

p,p

p

lf(x)\pp(x)dxJ .

We can define the functions of the system (<pk,r(x)}^0 as follows [5]

1

I k>0

fb

Pr,r+k(x) — -—^—T (x - t)r 1^k(t)dt, k - 0,1, 2, (r - 1)! Ja

(x — a)k

Pr,k (x) — ,, , k -0,1, 2,... ,r - 1. k!

(0.2)

From (0.2) for x G (a,b), we have

<pr-v,k-v(x), if 0 < v < r — 1, r < k,

(v)( ) I '-fik-v (x), if v — r < k,

^r,k pr-v,k-v (x), if v < k<r,

0, if k < v < r.

Denote by W1Lp^ab) the Sobolev weighted space. This space consists of all r — 1 times continuously differentiable on the interval [a,b] functions f (x) such that f (r-1\x) is absolutely continuous on [a,b] and f (r\x) G Lpp(a,b).

For p —2, we define in W^(ab) the inner product by (0.1). We can set the norm for any

function f g Wpab) by

W, — V f,f )S ,

Lj(a,b) V

which allows us to deduce that \wl2(ab), ll'llWr2 j is the Banach space, and (Wir2(a b), ("> ')sj is the Hilbert space.

The system ,k(x)}fc>0 is said to be Sobolev-orthogonal according to the inner product (0.1) generated by the orthonormal system (x)}fc>0 .

1. Main concepts: some properties of Gegenbauer polynomials

Let ^On(x) be the Sobolev-orthogonal polynomials according to the inner product

r-i „ i (f,g) = £/(v)(-i)g(v)(-i) W /(r)(t)g(r)(t)w(t)dt,

v=0

I-1

where w(t) = (1 - t2)a 2 .

Here are some properties of the Gegenbauer polynomials:

• Gegenbauer polynomials are given by [7]

ga(x) = (2a)n p (a-2,a-1 )(x) yn(x) = , 1 ) pn (x),

(« + 2) n

where p!" ' ß)(x) is the Jacobi polynomial, (a)n = a (a + 1) (a + 2)... (a + n — 1). We can have the following formula

0«(r) = r (a + D r(2a + n) (a-1 ,a-2) 9n = r (2a) r (a + n + 2) Pn ^

with

90a (x) = 1, 9a (x) = 2ax.

• Recurrence formula:

g£(x) = n (2x (a + n - 1) g£-i(x) - (n + 2a - 2) gan-2(x)) Orthogonality formula:

I ga(x)gm(x)(i - x2)a 1 dx — ínmha,

where

= 21-2ar (n + 2a) n n!(n + a) r2 (a)'

Let us put

Ga(x) = \j ~(r9a(x),

where

J2\a-1

W(x) = (1 — X2)a-2 .

Ga(x) are the Gegenbauer functions [7, p. 776]. The system {Ga(x)} is orthogonal on (—1,1),

i. e.

r i

gjj ( x ) gg mm ( x ) dx —

Some values:

Hx)ar2(a) Ga(x) = 2ax /w(x)(l+ a)r2(a) Go (x) = ^ n2i-2ar(a) , Gl (x) = 2a^ n2i-2ar(1 + a) •

1.1. Orthogonal Sobolev functions generated by the Gegenbauer functions GO(x) Definition 1.1. For r G N, we define the functions ^Ok(x) ( k = 0,1,... ) by

(x + 1)

k

<k(x) =-k-' k = 0' 1'--r - 1

1 r

<k+r(x) = (^y J (x - t)r-1Ga(i)dt, k = 0,1,....

We will calculate the functions (x) for any k G N and x G [-1,1].

Theorem 1.1 (Fisrt aim result). For a > —1, we have the following relations:

<r+n(x)

= —22r\/1 + f<+1,r+n(x) + (x) — (1 — ^ +

2 (x) = -2ry r(j1(^c2)a) <r+2(x) + 2x^ r(T(+a2)a) <r+i(x) - ^ r(]1('+a2)a) <r+2(x).

3. ^i+n(x) = + 2x</1+n^o>ra(x) - (l - V/1+n^a,n-i(x).

' y n ' V n V n + a/y n '

To prove this theorem, we have the following lemma:

Lemma 1.1. [8, p. 80-83 ] Here are the formulas for the derivations of the Gegenbauer functions:

d

!■ dx9"(x) = 2a<-i(x). d 2

2. (1 — x2)-ga (x) = {(n + 2a — 1) (n + 2a) g£-i(x) — n(n + 1)<+i(x)}

dx n + a

= —nx< (x) + (n + a — 1)g"-i(x)

= (n + 2a)xg^ (x) — (n + 1)g^+i(x).

3 nga (x) = xdx9ra (x) dx9n-i(x). d

4. dx [9l+i(x) - gO-i(x)] = 2(n + a)gn(x) = 2a [g^l+i(x) - gO^x)] •

Proof. (of theorem 1.1): 1. Firstly, its clear that:

ex

<P0n(x) = Gan(x), rf^x) = 1, vh(x) = J i Ga0(t)dt.

We have

where

By Lemma 1.1,

tf,r+n(x) = J^TYyJ (x - t)r-1Ga(t)dt, (1.1)

Ga (t) = ^W? ga (t).

2x(a + n — 1) a n + 2a — 2

gn-1(x) n - n

ga (x) = -ga-i(x)----ga-2(x). (1.2)

Then (1.1) becomes

tfrr+n(x) = 2a + n ^ r VW)t (x — t)r-1 gnn-,(t)dt

y/Wf)(x — t)r-1 ga _2(t)dt

(r — 1)\n^hz J-l a + n — 1 (r — 1)\n^K J-1

2(a + n — ^^ fx ^t (x — t)r-1 ga_im

(r — 1)\n^/h* J-ly/K-i

(r— 1)\nv/ha J-iy/K-2

2(a + n — 1)^-1

x

(r — 1)\n^/K J-i

n-1 1 tGa _1(t)(x — t)r-1 dt

(a + n — 1U/ha 2 fx r 1

v ) v n-2 I Ga_2(t)(x — t)r-1 dt.

(r — 1)\n,fhha J-i After simplification of the terms without integral signs, we obtain

tf,r+«(x) = 2xj1 + ajr—yJ ^(x — t)r-1Ga-i(t)dt

Put

Still to be calculated

-11 - nra—d) f + av—ryjjx—t)r-lGa-2(t)dt-

1x

tf,r+n-2(x) = (—[yj ((x — t)r-1Ga-2 (t)dt.

1x

j=-—— t(x — t)r-i Ga-i(t)dt. (r — 1)\ J-1

We have

1 r

J = ———J (t — x + x)(x — t)r-iGa-i(t)di

1 ix r r

(x — t)rG?_i(t)dt + 7-— (x — t)r-iG?_i(t)dt

(r — ' n-iW (r — 1)\J-i

— r^r+i,r+ra(x) + x^r,r+ra-i(x)

Thus

^r ,r+n(x)

= —W1+ a^a+i,r+n(x)+2^/1+ a^a,r+n-i(x) —(1 — + 1 10\/1+a^a,r+n-2(x). V n ' V n ' \ n + a — 1) / V n '

2■ We use the following relation:

1 fx 11 fx

(x) = (^T y ^x — t)r-iGa(t)dt = (r — 1);^hg J 1 V^O?(t)(x — t)r-idt.

By lemma 1.1 (formula 4), we have

g? (t) = 2(-±1x g? (t) — 2 g? (t),

Q' Q'

so

ir (x) = Ka+I U n ^ W(x—— -^/V !f ga —^

= -T^l/il f t(x — i)'-iG?(t)dt — -t^ttJ h| f (x — i)'-iG?(t)dt.

a(r — 1); v h? J-i a(r — 1); V h? J-i

Put

1x

^a,r+2(x) = ^T^ J (x — t)r-iG|(t)dt.

Let us calculate

1

H = " ,, I t(x — t)r-iG?(t)dt (r — 1); J-i

1 (t — x + x) (x — t)r-i G?(t)dt

Then

hence

(r — 1)! J-i

= (7^1)! y (x — t)rG?(t)dt +(^Tyi(x — t)r-iG|(t)dt

= —r^r+i,r+2(x) + x^r,r+i(x).

^ (x) = a(a + 11))^( —r^?+i,r+2(x) + x<r+i(x)) — 2 J (x) ,

T(1 + 2a) ( r (x)+ x (x)) r(2 + 2a) (x)

r(2a) l—r^r+i,r+2(x) + x^r,r+i(x)J ^ y r(2a) ^r,r+2(x).

So we obtain the second formula.

3. It is sufficient to replace r = 1 in the first formula, since it represents a special case. □

x

1.2. Problem of mixed Fourier series

Let f G WL2 ^. If this function is given in the generalized Gegenbauer orthogonal system WarM}™=0 , then

f (x) -£ C«k(f )tftk(x). (1.3) k=0

This Fourier series will have the form

i 1 \k x

f (x) - £ f (k)(-l)([x+T~ + £ C^k(f Wkc(x), (1.4)

k=0 k=r

with

ca,k (f ) = u = £ f {r)(t)^k(t)dt

= £ f(r)(t)Gak-r(t)dt, k = r,r + 1,..., called the Fourier coefficient. For r = 0, we have

f (x) - £ C0k(fWlk(x) - £ fo,k(x),

k=r k=r

with

fo,k = £ f (t)Gak (t)dt. (1.5)

In this section, we will give the expressions of (1.4) and (1.5) taking into account the expression of G°a (t). Also we will prove the convergence of the series (1.3). The following result is similar to the one given in [5].

Theorem 1.2. For a > —1, the system of functions {^an(x)} generated by the Gegenbauer functions Ga (x) and given by the formula

1 r

<n+r(x) = jr^Tiy J (x — t)r-1Ga(t)dt, n > 0,

is complete in W^^ and orthonormal via Sobolev's inner product

r-1 r-1

(f,g) = £ f(v)(—i) g(v)(—i) + / f(r)(t)g(r) (t)w(t)dt.

v=0

1-1

It follows from the two formulas

1

(r - 1)\J _i

= (x - t)r-1Gan(t)dt, n > 0,

(x + 1)n

van(x) = -—, n = o,i,...r-1,

' n!

that for all x G (-1,1),

ÍVa-v,n-v(x), if 0 < V < r - 1, r < n Gn-v(x), if v = r < n,

Va-vn-v (x), if v < n<r, 0, if n < v < r,

with Von(x) = Ga(x).

1.2.1. Study of the convergence of the series (1.4)

Let f G W£2(-1>1), then f(p) G Lp with

f(p)(x) - £ G^ (f(p)) Ga(x),

k=0

where

Gp,fc (f(P)) = f(p)(t)Ga(t)dt, for all k > 0.

1

We will prove the following result:

Theorem 1.3 (Second aim result). For a > 0, x G (—1, A], (A < 1), and f G W£P with < p < 4, the Fourier series

p-i / , ro

f(x) - £f(k)(—+ £Cp^k(f№(x) k=0 k=p

converges uniformly to the function f.

P r o o f. We note the following partial sums:

Then

p—1 / , -|\k n

Span (f,x) = £ f (k)(—1)(x+rL + £ ^k (f )^a.k (x), k=0 k=p

n

sa (f (p),x) = £ <cp,k (f(p)) Ga (x).

k=0

If(x) — Sp«n+p (f,x)|

1

p+n

(r — 1)\J —i

(x — t)p—1f(p)(t)dt — £ Gp«k(f )<k(x)

k=p

with

so

Hence,

1 /"x

<k+p(x) = TT^ I (x—t)p—1Ga(t)dt,

<k(x) =

(r — 1)!J —1

1

(r — 1)\J—1

(x — t)p—1Ga_p (t)dt.

|f(x) Sp,n+p (f,x) 1 i

p+n

(r - 1)! —1 1

(x — t)p—1f(p)(t)dt — (^rECpk(f) / > — t)p—1 Ga—p(t)dt

k=p

1

(r - 1)!

p+n

1

(x — t)p—1 f(p) — £ Cpk(f)Ga—p(t) dt

k=p

4

3

x

x

with

r+n

y^ Cr,k (f )Gk-r (t) — (/(r),x)

k=r

Then

1

|f (X) - S*n+r (/,X)\ —

(r - 1)!

(x - t)r-1 (f(r)(t) - Sa (f(r),t)) dt

i-i

< (x - t)r-1\f (r)(t) - sn (f(r),t)\dt.

Using Holder's inequality, we get:

i i 1 Í i'X , „ \ 4 / i'x \ , , , N )|P \ p

|f (x) - Sr"n+r (f,x)\ < ^{J (X - t)(r-1)q dtj (J ^ \f (r)(t) - sa / (r),t) \P dt

with 1 + 1 — 1. p q

P + 1 p q

Calculate

x , - (1 + X)q(r-1)+2

J — [X (x - t)(r-1)qdt — (-1)q(r-1) f (x - t)q(r-1)dt — (1 + X')q{[ 11+2 for X G (-1,A].

J-i J-1 q(r-1) + 1

1

\f (x) - S?tf+r (f,x)\ ( (1T ) 4 II f (r)(x) - S* ' ^

1 f(i + A)q(r-1)+2

Then

JT,n+r (J,x)\ ^ (r — 1)^ q(r — 1) + 1 I \\J (x) Sn f ,x)\Ilp

Since

||f (r)(x) — SS (f(r),x)||LP ^ 0 as n

it results that

If (x) — (f,x)H 0

uniformly on (—1,A\. □

— 2 < a < 3 Tf f - Wr

convergence of the Fourier series

r-1 I 1 \k x

Theorem 1.4. Suppose that — ^ < a < ^. If f G WL2(-11), then we have the uniform

r-1 (x + 1)k ^ f(x) - £ f(k)(-1)(~i;rL + £C*k(fWr,k(x)

k\

k=0 k=r

on (—1, 1) to the function f.

2. Asymptotic forms of the functions pai+n(x)

We say that

/x 1 nx

aa (t)dt=^haJ 1^W(f)gan (t)dt,

where

^ 1 , a 21-2ar(1 + 2a)

w(t) = (1 — t2)a-1, ha = n---~2.

n\(n + a) [r (a)]

Then

^,i+n(x)

J-

(1 — t2)2 -1 g" (t)dt.

Integrating by parts and using the first formula of Lemma 1.1, we get:

x) =

u

(1 — t2) 2-4 du = (f — (1 — t2) 2-4-i( — 2t)dt

so

^,i+n(x)

2 (a — 1)^

^?,i+n(x) =

dv = g? (t)dt

(1 — x2)2 -1 gn+i(x) +

V = 2(a-i) g|+l(t)

i

(a— 1)^ J-i

t(1— t2)2 - 4-ig?+i(t)dt

u = (1 — t2)2-9 du = — 2t (1 — t2)2-4 Qa — f) dt

dv = gn+i(t)dt v = g?^22 (t)

Then

^?,i+n(x)

where

2 (a — 1) v/h"

(1 — x2) 2 - 4 g|+i(x) +

(1 — x2) 2 4

R" (x)

fa _ i\ f a _ 5\ r x

V 2 47 V 2 47

(a — 1) (a — 2) Via J-i

2 (a — 1) (a — 2)

t (1 — t2)2-4 ga+22(t)dt.

gra +-2 (x) + Ra (x),

(2.1)

Theorem 2.1 (Third aim result). Suppose that a > |. Then the following asymptotic formula holds

<i+Jx)

(1 — x2) 2 4

2 (a — 1)^

gra+i (x) +

(1 — x2) 2 4

2 (a — 1) (a — 2)^

gra+2(x) + (x),

where Ra (x) is given by (2.1) and satisfies the estimate

R" (x) = o( -

Proof. To estimate the remainder Ra (x) , we must consider two cases. First case: — 1 < x < —1 + ^. Here we have:

ga+^t) - (—1)

t r(2a + n — 2) r (2a — 4) r (n + 3)

(1 + o (x + 1))

then

|Ra(x)| <

<

<

(2 — i)(2 — f) r(2a + n — 2)

(a — 1) (a — 2) yi«

(a _ iA (a _ 5A

V 2 4) V 2 4)

(a — 1)(a —2) V?

(a _ iA (a _ 5A

V 2 47 V 2 47

(a 1) (a

r (2a — 4) r (n + 3) r(2a + n — 2)

-i+i 1 n

-i

|t (1 — t2) 2 4 |dt

r (2a — 4) r (n + 3)

r(2a + n — 2) r (2a — 4) r (n + 3)

1 + o ( |n

1 + o (

n

-i+J

n

-i

-i+n

n

i

|t (1 — t2) 2 4 |dt

a _ 9 1 + t2|2 4dt.

x

i

x

1

1

We let

It is easy to see that

= n-

2 r(1 + 2a) n!(n + a) [r (a)]2

'-i

1 +t212 4 dt< 1 + - 2 + —

n

n2

11

- = o -nn

Then

R(x)|< o(n)

fa _ fa _

V 2 4) V 2 4)

(a - 1) (a - 2) yfhñ

r(2a + n — 2)

r (2a — 4) r (n + 3)

1 + o (

n

o| 1 n

Second case: — 1 + x < 1. We have: " — —

fa _ fa _ f\

V 2 4) V 2 4/

iRa (x)i <

i

< ^a,n

(a — 1) (a — 2) v/h"

t (1 — t2)2 - 4 ga+22(t)

t (1 — t2)2 - 4 gn+2(t)

dt

-i+1 1 n

i

dt +

'-i+n

n

t (1—t2)2 - 4 ga+22(t)

dt

with

fa _ fa _

V 2 4) V 2 4/

(a — 1) (a — 2) v/h"

So

iRa (x)i < ^

,-i+1

i

t (1 — t2)2 - 4 ga+2(t)

dt + w0

'-i+J

t (1— t2) 2-4 g"+22(t)

dt

< O - +

'-!+n

t (1 — t2)2 - 4 ga+2(t)

dt.

We say that for each t = cos 9 , 0 < $ < 9 < n — the asymptotic representation is [9, p. 318]

(a -1 ,a - 2)

(cos 9)

cos{(n + a) 0 — f}

i-( • e)a ( gV«

/in (sin 2J (cos 2

+ O 3

where P"a,ß)(t) is a Jacobi polynomial.

Since

then

<(cos 0)

ga (t) = r (a + 2( r(2a + n) (a - 2, a -1) (t) gn (t)= r(2a)r (a + n + i) (t)'

r (a + 2) r (2a + n)cos{(n + a) 0 — 2} /1

r(2a)r (a + n + 2) ^ (sin §)a (cos §r + 0 U2

First, find an asymptotic estimate for

r (2a + n) r (a + n + 2

24

x

x

x

We see that

h-a T(Z + a)

r (az + b) - V2ne-az (az)az+b-2 , |arg z| < n, a> 0.

r(z + b)

1 +

(b — a)(a + b — 1) 1

2z

+ Uz*Ca-b ^ + b — 1)2 — a + b — 1] +

So

r (2a + n) r (a + n + *)

5

- 3 i a — -

na-2 2na-2 V 2

11

— +-

o I -n

9

since a > -.

51

3a — - +- .

2) 12na-2

„U 2 1 C2

3[3a — 5) — a + 3 I +

Taking into account the fact that |1 — t2| < 1 for —1 + n < t < x < 1, it follows that:

K(x)|< + ^

'-i+1 1 n

|t| |ga+22(t)| dt,

|Ra(x)|< M 1

n

+

r (a + 2) ^o

T(2a)y/nn 1 (sin I) (cos

par cos(x) ' ar cos( —1+1)

-n

n

cos <j (n + a) 9 — —

sin 9d9 + o ( — n2

Now, using the properties of the trigonometric functions, we get

idovm^ / 1V r (a + 1) w«,n

KM < o - + o - ^ ,2' )a .-

1 nj \nj Y(2a)^/nn |(Sin 2) [cos f)

r (a + D

< ^ n)+ r(2a)Vnn Ksin f)a (cos f)

r (a + D

< o| n) + T(2a)Vnn Ksin 2)a (cos 2)a

o I 1 n

par cos(x) / 1

9d9+of —

' ar cos(— 1+ 1)

n2

[ 9d9 + o( -1

lar cos(-1+1) \n

n2 ar cos2 (— 1 + n) ~2

+ ol

n2

So, we have the desired estimate.

□

x

2

1

References

[1] R. M. Gadzhimirzaev, "Sobolev-orthonormal system of functions generated by the system of Laguerre functions", Probl. Anal. Issues Anal., 8(26):1 (2019), 32-46.

[2] 1.1. Sharapudinov, "Approximation of functions of variable smoothness by Fourier-Legendre sums", Sb. Math., 191:5 (2000), 759-777.

[3] I. Sharapudinov, Mixed Series of Orthogonal Polynomials, Daghestan Sientific Centre Press, Makhachkala, 2004.

[4] I. I. Sharapudinov, "Approximation properties of mixed series in terms of Legendre polynomials on the classes Wr ", Sb. Math., 197:3 (2006), 433-452.

[5] I.I. Sharapudinov, "Sobolev orthogonal systems of functions associated with an orthogonal system", Izv. Math., 82:1 (2018), 212-244.

[6] I.I. Sharapudinov, T.I. Sharapudinov, "Polynomials orthogonal in the Sobolev sens, generated by Chebychev polynomials orthogonal on a mesh", Russian Math. (Iz. VUZ), 61:8 (2017), 59-70.

[7] M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, USA, 1964.

[8] G. Szegö, Orthogonal Plynomials. V. 23, American Mathematical Society, Providence, Rhode Island, 1975.

[9] A. F. Nikiforov, V. B. Uvarov, Special Functions of Mathematical Physics, BirkMuser Veriag Basel, Springer Basel AG., 1988.

Information about the author

Mohamed Ahmed Boudref, PhD

of Mathematics, Director of the LIMPAF Mathematics and Computer Science Laboratory, Lecturer of the High Mathematics Department. University of Bouira, Algeria. E-mail: [email protected]

Информация об авторе

Будреф Мохамед Ахмед, кандидат физико-математических наук, директор лаборатории математики и компьютерных наук LIMPAF, преподаватель кафедры высшей математики. Университет Буира, Алжир. E-mail: [email protected]

Received 17.02.2022 Reviewed 26.05.2022 Accepted for press 09.06.2022

Поступила в редакцию 17.02.2022 г. Поступила после рецензирования 26.05.2022 г. Принята к публикации 09.06.2022 г.