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22
ISSN 2686-9667. Вестник российских университетов. Математика
Том 28, № 142
2023
SCIENTIFIC ARTICLES © M. A. Boudref, 2023
https://doi.org/10.20310/2686-9667-2023-28-142-155-168
OPEN /Tj ACCESS
Hermite functions and inner product in Sobolev space
Mohamed Ahmed BOUDREF
University of Bouira, 10000, Drissi Yahia Bouira St., Bouira, Algeria
Abstract. Let us consider the orthogonal Hermite system {^2n(x)}n>0 of even index defined on (-œ, œ), where
e 2 \J (2n)!n 12n
by H2n(x) we denote a Hermite polynomial of degree 2n. In this paper, we consider a generalized system {^r,2n(x)} with r > 0, n > 0 which is orthogonal with respect to the Sobolev type inner product on (—<x>, x>), i.e.
r—1 f w
(f,g) = lim V f(fc)(t)g(fc)(t) + / f(r)(x)g(r)(x)p(x)dx t0>
with p(x) = e-xX, and generated by {<£>2n(x)}n>0 . The main goal of this work is to study some properties related to the system {^r,2n(x)}n>0,
(x — a)n n!
фг n(x) = -:-, n = 0,1, 2, .. ., r — 1,
n!
1 fb
фт,т+п(х) = 7--TT (x — t)r-1(fn(t)dt, n = 0,1, 2,....
(r — 1)! Ja
We study the conditions on a function f (x), given in a generalized Hermite orthogonal system, for it to be expandable into a generalized mixed Fourier series as well as the convergence of this Fourier series. The second result of the paper is the proof of a recurrent formula for the system {^r,2n(x)}n>0 . We also discuss the asymptotic properties of these functions, and this concludes our contribution.
Keywords: inner product, Sobolev space, Hermite polynomials Mathematics Subject Classification: 42C10.
For citation: Boudref M.A. Hermite functions and inner product in Sobolev space. Vestnik rossiyskikh universitetov. Matematika = Russian Universities Reports. Mathematics, 28:142 (2023), 155-168. https://doi.org/10.20310/2686-9667-2023-28-142-155-168.
НАУЧНАЯ СТАТЬЯ © Будреф MA., 2023
https://doi.org/10.20310/2686-9667-2023-28-142-155-168 ЮУ
УДК 517.518.36
Функции Эрмита и скалярное произведение в пространстве Соболева
Мохамед Ахмед БУДРЕФ
Университет Буира, Алжир, г. Буира, ул. Дрисси Яхья Буира, 10000
Аннотация. Рассмотрим ортогональную систему Эрмита {^2п(х)}п>о четного индекса, определенную на (-то, то) формулой
e x2
^2"(Х) = ^п 12" Н2П (Х),
где через Н2П(х) обозначен полином Эрмита степени 2п. В данной работе рассматривается обобщенная система {фг,2п(х)} с г > 0, п > 0, ортогональная относительно скалярного произведения Соболевского типа на (-то, то)
/,д) = 11т V /(к)(1)я(к)(г)+ /(г)(х)д(г)(х)р(х)ах
с р(х) = е-х , и порожденная системой {^2п(х)}п>о . Основной целью работы является изучение некоторых свойств, связанных с системой {"0г,2п(х)}п>о ,
(х — а) п!
фг n(x) =---, n = 0,1, 2,..., r — 1,
n!
1 fb
^r,r+n(x) = 7-— (x — t)r-1<^>„(t)dt, n = 0,1, 2,....
(r — 1)! Ja
Изучаются условия на функцию f (x), заданную в обобщенной ортогональной системе Эрмита, достаточные для ее разложения в обобщенный смешанный ряд Фурье, а также сходимость этого ряда Фурье. Второй результат статьи — доказательство рекуррентной формулы для системы {^r,2n(x)}n>0 . Также обсуждаются асимптотические свойства этих функций, что составляет заключительную часть работы.
Ключевые слова: скалярное произведение, пространство Соболева, многочлены Эрмита
Для цитирования: Будреф М.А. Функции Эрмита и скалярное произведение в пространстве Соболева // Вестник российских университетов. Математика. 2023. Т. 28. № 142. С. 155-168. https://doi.org/10.20310/2686-9667-2023-28-142-155-168. (In Engl., Abstr. in Russian)
Introduction
Consider an orthogonal system {^n(x)}n>0 on (—to, to) with p(x) as a weight function, and let r > 0. We construct a new orthogonal system {^r,n(x)}n>0 by following the Sobolev type inner product:
r—1 n ro
{f,g)s = t lim J] f (v)(t)g(v)(t)+ / f (r)(x)g(r)(x)p(x)dx. (0.1)
Quite a few authors have presented this type of construction. For example, 1.1. Sharapu-dinov, in his works [1-4] on the construction of mixed Fourier series, considered some particular cases of systems generated by classes of orthogonal functions, namely, those of Jacobi, Legendre, Chebychev, Laguerre, and Haar. The case of the Gegenbauer system is covered in [5].
Hermite polynomials are widely used in mechanics and physics, and are of particular interest for applications. In this work, we reconstruct the orthogonal system {^r,n(x)}n>0 generated by Hermite polynomials of even index, and this approach is different from that used by M. A. Boud-ref.
Denote by Lpp(a,b) the space of measurable functions f (x), x G (a,b), such that
, b
/ If (x)|pp(x)dx < to.
J a
It is clear that Lp(a, b) is a Banach space with the norm
p,p
|f (x)|p p(x)dx .
(0.2)
When p(x) = 1, we write Lpp(a,b) = Lp(a,b).
We can define functions of the system {^r,n(x)}n>0 for r > 0 as follows [4]:
(x - a)n
Vvn(x) = --, n = 0,1, 2,..., r - 1,
n!
1 fb
^r,r+n(x) = -(x - t)r-Vn(t)dt, n = 0,1, 2,....
(r - 1)! a From (0.2), we have for a. e. x G (a, b)
^r-v,n-v (x), if 0 < v < r — 1, r < n,
Pn-v (x), if v = r < n,
^r-v,n-v(x), if v < n < r,
0, if n < v < r.
Denote by W£p(ab) the Sobolev weighted space. This space consists of all functions f (x) which are r — 1 times continuously differentiable on the closed interval [a, b] and such that f (r-1)(x) is absolutely continuous on [a, b] and f(r) (x) G Lp(a, b).
For p =2, we define in W£2(ab) the inner product by (0.1). For any function f G (ab), we can set the norm by
vfx,
yr,n
(x)
W r2
which allows us to deduce that {wrL2(ab), j is a Banach space, and (ab), (■,-)s^
is a Hilbert space.
The system {^r,n(x)}n>0 is said to be Sobolev-orthogonal with respect to the inner product (0.1) generated by the system {<£n(x)}n>0 .
b
1. Some properties of Hermite polynomials
Let Vv,ra(x) be Sobolev-orthogonal polynomials with respect to the inner product
r-1 „00
(f,g) = t lim J] f(v)(t)g(v)(t) + / f(r) (x)g(r) (x)w(x)dx, t^—œ *—^ 1
v=0 " —
where w(x) = e-x2.
Here are some properties of Hermite polynomials:
• The Hermite polynomials are given by [6]
dn dxn
where
H0(x) = 1, Hi(x) = 2x, H2(x) = 4x2 — 2.
Hn(x) = (—1)nex2d-n (e—x") , x G R,
Recurrence formula [7, p. 106]:
Hn+i(x) = 2xHn(x) - 2nHn—i(x), Hn+1(x) = 2(u + 1)Hn(x), Hn(x) = 2xHn—i(x) - Hn — 1(x).
Orthogonality formula:
fœ _ 2
e x Hn(x)Hm(x)dx = SnmCn, where Cn = 2nu\\/n.
J—œ
A Hermite function is defined by
<^n(x) = e 2 Hn(x).
We define a Hermite function of even index by
e 2
^2n(x) = - 1-H2n(x),
v/(2u)\n 12n
where H2n(x) is a Hermite polynomial of even index, in particular, H2n(0) = (-1)n^ , H2n+l(0) = (-1)n (2U + 2)\
u\ ' (u + 1)
Relation to Legendre functions:
H2n(x) = (—1)n22nL—1 (x2) .
So if
L—1 (x2) = x—V2dxn (x2(—1 +">e—='') ,
then
H2n(x) = (-1)n22nx — (x2(—1 +n)e—x2)
Asymptotic formula for (x) [8, p. 594]:
= ^2n(0)
It is clear that
5
x 2
—+ //4ñr19-(x)
COS V4n
where — 1 < 0n(x) < 1.
lim
n—^^o
5
(_ \ x 2
V4n + 1x) + 0n(x)
/ A/4n + 1
So, the asymptotic formula for H2n(x) is
H2n(x) = (—1)n2n ■ l ■ 3 ■ 5 ■ ... ■ (2n — 1)e2x
= 0, for each x.
cos f—4n + 1x^ + of —t= V ) \ y n
The system {^2n(x)}n> 1 is orthogonal on (-œ, œ), i. e.
/•œ
Some particular cases:
e 2
<Po(x) = Ho(x), ^2(x)
e 2
V2 /—n
H2(x).
1.1. Sobolev-orthogonal functions generated by Hermite functions ^2n(x)
Definition 1.1. For r > 0, we define the functions ^r,2n(x) (n = 0,1,...) by
(x)2n
^r2n(x) = -tt , n = 0, 1,...,r— 1,
V ; (2n)!' ' ' ' '
1 r
^r,r+2n(x) = -7T7 (x — t)r-1^2n(t)dt, n > 0.
(r — -i
We will calculate the functions r+2n(x) for any n G N and x G (—to, to) . Theorem 1.1 (Fisrt aim result). For n > 1 and r > 0, we have the following relations:
,2 2 /2n — 1
1. ^r, r+2n(x) = — rW--,r+2n(x) + xW--"^r,r+2n-1 (x) — \ —-^r,r+2n-2 (x).
2n 1
2n 1
2n
2. ^r+1,r+1+2n(x) = -^r,r+2n(x) + 1 ^V-1,r+2n-l(x) + ^^ ^r,r+2n-1 (x). r r r
3. For n = 0 :
(x)
——2 I ( ^ , x—/2
= ^r+1,r+2(x) + --T= ^r,r+1 (x).
1 — v^
1 — V2'
To prove this theorem, we need the following lemma.
2
2
X
X
Lemma 1.1. The formulas for the derivations of the Hermite functions of even index are:
V2 I 1
= 1 x^2n-1(x) _ ^ 1 - 2 (x) , (1.1)
P2n(x) = —x^2n(x) + 2Vn^2n-l(x). (1.2)
Proof. First, for the Hermite functions, we put for n > 1
22 __
e 2 e 2
P2n(x) = /777-^7 1 ~ H2n(x), P2n -1(x) = , - i n i H2n - 1(x),
\J (2n)!n 4 2n \J (2n — 1)!n 4 2n- 2
e 2
P2n-2 (x) = --H2n-2(x).
a/ (2n — 2)!n 4 2n-1
Use these formulas to prove (1.1), (1.2). a) We have (see [7, p. 106])
Hn+i(x) = 2xHn(x) — 2nHn-i(x).
e 2
Multiplying both sides by — 1—, we get
v/(2n)!n i 2n
o
2xe 2 e 2
P2n(x) = - —-#2n-l(x) — 2(2n — 1)- 1-H2n-2(x)
a/ (2n)!n 12n V(2n)!n i 2n
n 2 -, x e -1 H2n-l(x) _ J2n-1 6 ^^--H2n-2(x),
2n _ 1 y/(2n _ 1)!n42n-2 w V 2n ^/(2n _ 2)!n42™-1
and so we have (1.1).
b) For the second relation, we differentiate
e 2
(x) = - I-H2n(x)
A/ (2n)!n 12n
with respect to x, and get
/ e 2 e 2 ,
^2n(x) = -x ,-—— H2n(x) H--, !-H2n(x).
Using the fact that H2n(x) = 4nH2n-1(x), we get
x2
x2
x2
/ e 2 e 2 ,
^2n(x) = _x , —-#2«.(x) H--. !— H2n(x)
03
t \ I 2 2 ne 2 u i \ = _x^2n(x) + -- !-—"2n-1 (x) ,
V^V(2n _ 1)!n42n 2
SO ^2n(x) = _x^2n(x) + 2^n^2n-l(x). □
x2
x2
Proof. (of Theorem 1.1) 1. First of all, it is clear that
/x
^o(t)dt.
We have
t2
1 r e- v
^r,r+2n(x) = 7-7TT (x — t)r-1^2n(t)dt, where ^2n(t) = n-- !-#2n(i). (1.3)
(r — 1)! J-x V(2n)!n42n
Then, from Lemma 1.1 and (1.3), it follows that
2ra—1
2 ..... V"2—" 'x
^r,r+2n(x) =(r - - 1 t(x - t)r 1^2n(t)dt - (r-Tl) J - t)r 1^2n—2(t)dt
2 / 2
2n — 1 I - - r—1 V 2n — 1 I . , r_i
t (x - t)r ^2n— 1(t)dt -— / (x - t)r ^2n— 2(t)dt.
(r — 1)U - ^ ' ^ ^ (r — 1)U-In the second term of this expression, we have
1 /*x
^r,r+n-2(x) = 7-7T7 (x — t)r-1^2n-2(t)dt.
(r — 1)! J
Let us calculate
. 1 x
J =7-— t(x — i)r-V2n-i(i)di.
(r — 1)! J
We get
1
J = 7-— (t - x + x)(x - t)r—1 ^2n—1(t)dt
(r - JJ! J—
race
1 fx x r
(x - t)r^2n—1(t)dt + 7-— / (x - t)r—1^2n—1(t)dt
(r - ' r2n— (r - 1)U—
- r^r+1,r+2ra(x) + x^r,r+2ra—1(x).
Then
so
,2 2 /2n - 1
^r,r+2n(x) = -r^/ 2n _ 1 ^r+1,r+2n(x) + xJ ^r,r+2n— 1(x) - W ^^^r,r+2n—2(x).
2. By Lemma 1.1 (formula (1.2)), we have
-x^2n(t) = ^2n(x) - 2//ñ^2ra—1 (x) ,
1 x 1 x (T^n J t(x - t)r—1W(t)dt =j—1yJ (x - t)r—1W(t)dt
2, /n rx
V 1 (x - t)r—1^2n—1(t)dt.
(r - 1)U—c
Let us calculate
1 fx 1 fx H = (—[yj t(x — t)r-1p2n(t)dt, H = (—[yj (x — t)r-1^'2n(t)di,
2n
(r — -ro
1
(r — 1)U-c
First,
H3 = 7-7T7 (x — t)r-1p2n-1(t)dt.
1 x 1 x
H1 = 7-—r t(x — t)r-1^2n(t)dt = 7--- (t — x + x) (x — t)r-1 P2n(t)dt
(r — 1)! -ro (r — 1)! -1
r x x
=---/ (x — t)rP2n(t)dt + 7-—7/ (x — t)r P2n(t)dt = —r^r+1,r+2n(x) + x^V,r+2»(x).
r! -1 (r — 1)! -1
For H2 , integrating by parts we get
1 x 1 i'x
H2 = (r — 1)! [(x — t)r-1 P2n(t)] !ro + (r — 2)^qo (x — t)r-2 P2n(t)dt.
Since
-t2
lim (x — t)r-1 P2n(t) = lim (x — t)r-1 —. -,— = 0,
t^-rov y v/(2n)!n 12n
then
1x
H2 = ( _ 2)! / (x — t)r-2 P2n(t)dt = ^r-1,r+2n-1(x). (r 2)! J-ro
Regarding H3, we have
1 fx
H3 = 7-7T7 (x — t)r-1p2n-1(t)dt = ^r,r+2n-1(x).
(r — 1)! J-ro
Finally,
^r,r+1+2n(x) = ;^^r,r+2n(x) + 1^r-1,r+2n-1(x) + r ^r,r+2n-1(x).
So we obtain the second formula.
3. For this part, it is easy to see that for n = 0, p0(x) = ^v/2^1(x) + v/2^2(x). Then
1 fx
^r,r(x) = ( _ / (x — t)r-1p0(t)dt (r — 1)! -ro
, , V2 [* 1
t(x — t)r-1p1(t)dt + v / (x — t)r-1p0(t)dt
(r — 1)! ' r1W (r — 1)U-
—fx , . w .\r-1 „ , v^ / / 1
(x — t — x)(x — t)r-1p1(t)dt + ^—- / (x — t)r-1p0(t)dt,
(r — 1)!./-roV ' W (r — 1)U-
so
—/"x , iNr ,, i v^x 1 , 1
xx
^r,r (x) = ( _ / (x — t)r p1(t)dt+( _ / (x — t)r-1p1(t)dt (r — 1)! -ro (r — 1)! -ro
ro
x
V2 / , „r-1
(x — t)r-1p0(t)dt,
(r — 1)U-ro
^r,r (x) = — V^0r+1,r+2(x) + (x) + V^V,r (x).
Then ^r,r (x) = — ^^^r+1,r+2(x) + . x^^T^r,r+1(x). □
1 — 2 1 — 2
x
1.2. Problem of mixed Fourier series
Let f G (—œœ). If this function is given in the generalized Hermite orthogonal system {^r,2n(x)}„>o, then
oo
f (x) - ^ 0,fc^,2fc(x). (1.4)
oo
(x) ~ V Cr,k(x
fc=0
This Fourier series will have the form:
r—1 2fc œ
x ■ - ' (x)
f (x) - JmE f ^W + E Cr,fc^(x), (1.5)
fc=0 ( )! fc=r
with
/OO POG
f (fc)(t)^r,2fc (t)dt = f (fc)(tW—r (t)dt
-œ J —œ
called the Fourier coefficient. For r = 0, we have
oo oo
f (x) - ^ Co,fc^0,2fc(x) - ^ fo,fc^0,2fc(x) (1.6)
fc=r fc=r
with
/o,2fc = f (t)^2fc (t)dt.
' —oo
In this section, we will give the expressions of (1.5) and (1.6) taking into account the expression of <^2n(t). Also we will prove the convergence of the series (1.4). The following result is similar to the one given in [5].
Theorem 1.2. For n > 0, r > 0, the system of functions {^r,2n(x)} generated by Hermite functions ^2n(x) by the formula
1 PX
^r,r+2n(x) = ( _ ,), / (x - t)r—1^2n(t)dt, n > 0, (r 1)- J —œ
is complete in (—œ œ) and orthonormal via Sobolev's inner product
1 i-œ
(f,g) = tlim £ f(v)(t)g(v)(t)+ / f(r)(t)g(r)(t)w(i)di.
t^ — œ ^—* I
It follows from the formulas
t^ — œ ■
v=0
1
^r,r+2n(x) = 7--TT (x — t)r—1^2n(t)dt, n > 0,
(r — 1)! J—œ
x2ra
^r,2n(x) = ——r, n = 0, 1,...,r — 1, (2n)!
that for all x G (—œ, œ),
^r—v,2n—v(x), 0 < V < r — 1, r < 2n,
^2n—v(x), V = r < 2n, (x)T"' = ,
^r—v,2n—v(x), v < 2n < r,
0, 2n < v < r,
(^r,2n(x))(v) = <
with ^0,2n(x) = ^2n(x).
1.3. Study of the convergence of the series (1.5)
Let f G ( ), then f(r) G Lp with
fOO
^ ^^ (Cr,fc(x), where Cr,fc = f (r)(t)pk(t)dt, for all k > 0. fc=0 ^-ro
Theorem 1.3 (Second aim result). For x G [A,B] (A < B < to) and f G WL, where
3 < p < 4, the Fourier series
r-1 2fc ro
f (x) - £ Jim f(fc) (t) (f^ + £ Cr,k^,2* (x)
fc=0 ^ ^ fc=r
converges uniformly to the function f.
Proof. We note the following partial sums:
r-1 2fc n n
Sr,n (f,x) = £ tnmro f(k)(t)^ + £ Cr,fc^r,2fc(x), s;,n (f(r),x) = £ Cr,^2fc(x).
Then
with
so
fc=0 v ' fc=r fc=0
1 i'x r+n
If (x) — Sr,r+n (f,x) | = --— (x — t)r-1f (r)(t)dt — V Cr,fc^r,2fc(x)
(r — 1)! JA fc=r
1 fx
^r,r+2fc(x) = 7-— (x — t)r-1p2n(t)dt,
(r — 1)! Ja
1 fx
^r,2fc(x) = 7--T7 (x — t)r-1^2fc-r(t)dt.
(r — 1)! ./A
Hence, |f(x)
Sr,r+n (f, x) 1 "
1 fx 1 r+n fx
1 I , 1 ^ I , 1
-- (x — t)r-1f (r)(t)dt — ?--£ CrJ (x — t)r-1^2fc-r(t)dt
(r — 1)! A (r — 1)! J A
•x / r+n \
^ (x — t)r-^f (r)(t) — £ Cr,fc^fc-r (t) J dt
(r — 1)!
with
£Cr,fc P2fc-r (t) = s;,n (f (r),x).
Then
r+n
r
fc=r
1x
If (x) — Sr,r+n (f,x)| < (^—^ J^ (x — t)r-1 If (r)(t) — s;,n (f(r),t) I dt.
Using Holder's inequality, we get:
i i 1 ( ^ t \q ( (x I/A ^ , . )| p \ P
If (x) — Sr,r+n (f,x)| < ^ (x — t)(r-1)q dtj yj^ If (r)(t) — s;,n (f(r),t) |p dt
with 1 + 1 = 1.
p q
Calculate
'A
(x - t)(r-1)qdt = (-1)q(r-1)-
_1) (t - x)q(r-1)+1
q(r - 1) + 1
A
= (-1)q(r-1) (A - x)q(r-1)+1 = (B - A)q(r-1)+1
q(r - 1) + 1 q(r - 1) + 1
< TO.
Then
If(x) - Sr,r+n (f,x)| < ((Bq(r A1)(r+1)1+1) 1 11/(r)(x) - S;,n (/(r),x)I
LP
and since ||f(r)(x) — S*n (f(r),x) || ^ 0 as n ^ to, it results that |f (x) — Sr,r+n (f, x)| ^ 0,
uniformly on [A, B].
□
2. Asymptotic forms of the functions ^11+2n(x)
We say that
^1,1+2n(x) = ^2n(t)dt =
2n\J(2n)!n4 7-c
e 2 H2n(t)dt.
Integrating by parts and using the first formula of Lemma 1.1, we will have:
^1,1+2n(x)
u = e 2
du = — te 2 dt
dv = H2n(t)dt V = 2(2>1+1) H2n+1(t)
so
^1,1+2n(x)
e 2
v/(2ñ)ln42n 2 (2n + 1)
H2n+1(x) +
\J (2n)!n 12n
te 2 H2n+1(t)dt,
^1,1+2n(x)
u = e 2
du = I e 2 — t2e 2 ) dt
dv = H2n+1(t)dt V = 2(2^+2) H2n+2(t)
Then
^1,1+2n(x)
where
_ X_
e 2
y^nyin 12n+1 (2n + 2)
#2n+1(x) -
_ X_
xe 2
v/(2n)!n12n+1 (2n + 2)
H2n+1(x) + Rn(x),
Rn(x)
2n+3(2n + 1)(n + 1^v/(2 n)!n4 J-
(1 - t^ 2 H2n+2(t)dt.
(2.1)
Theorem 2.1 (Third aim result). The following asymptotic formula holds:
^1,1+2n(x)
_ X_
e 2
-#2n+1(x) -
_ X_
xe 2
y/(2n)Tff12n+1 (2n + 2) +w 12n+1 (2n + 2)
where Rn(x) is given by (2.1) and satisfies the estimate Rn(x) = o .
H2n+1(x) + Rn(x),
X
X
X
X
1
2
2
X
X
1
1
2
1
1
X
2
1
1
1
Proof. By the relationship between Hermite and Laguerre functions
we have
where
H2n(x) = CnL- 2 (x2), where C, = (-1)n22n,
Г _ «2 _i
Rn(x) = ПпСП+W (1 - t2) 2 L-+2i(t2)di,
J—те
1
(2n + 1)(n + 1)^(2n)!n 4 2n+3 Introducing a new variable u = t2, we get
¿0! is a Laguerre function.
Rn(x) =
nraCra+1 2
1 — u
e 2 L—_21 (u)du.
To estimate the residue Rn(x), we must consider two cases:
1. First case: 0 < x2 < n. For this case, we can use the weight estimate [9,10]
e—x |La(x)| < c(a)A^(x), a> -1,
and
na nra,
П,2 4~ — 1—1
1_1 x 2 4 ,
e 4
Щ + |x - n,
0 < x < ^
"n
— < x < — < x < 2
f- < x <
< x
2 ^)
where = = 4n + 2a + 2. So
e 2
L—2(x)
<
c -
4(n
then
|Rn(x)| <
nraCra+1 2
1 — u
u
lc( ^ I du<nnCn+1
4n + 5
<
2 2
—f= + v u I du u
21
+ 77-
4n +1 3 (4n +1)3
So R(x) = o (П
2. Second case: - < x2 < w. We have
n
|Rn(x)| <
nraCra+1
20
For this case, we use the following estimate:
1u
u
L— 2 (u)
du.
x a/ ч а Г(п + a + 1) T / /llT si \ / a 3 S e—xx2¿a(x) = N — ---J«(2(Nx)2 ) + o(na — 4),
n!
N = n +
a
2
x > 0, a > 1.
T/n ^ 2 \ 2 ( an n \ , _
J«(x)=( —) cos (x - "У - 4J+ o(x
x —+00.
2
x
0
2
x
0
2
x
We have
D , . ПпСга+1 r fn 1 - u -1 1 - u -2 , , ,
Rn(x) = —-—i e 2 —— Lra 2 (u)du + e 2 —L„ 2 (u)du 2 L ./n Vu 11 л/u
so
|вд|< o( n)+^
< o( 1) + nn^n+i
< KnJ 2
e 2 | 1 1 L- 2 (u)du
e-2 -1 ^^ L- 2 (u)du Vu
+
n«,Cra±1
2
e 2 v^üL- 2 (u)d-
u
Calculate
A
2u
e-2 _2
u
L-2(u)du
u 4 u 4 e 2 L- 2 (u)du
u 4 N4-
i Г (n + 1) /
<
fX
N 4
1Г (n + 1)
(n + 1)!
X
mmfJ-1 (2v/—u)+4n)jdu
2
/*X
+
u-1 n4 м j- 1 (2^nu) ^u
I u- 4 J-1 (^2V Nujdu
+ o.i
n
o ( — ) u 4 du 1 \ n
u 4 du
On the other hand, we have
J_ 1(x)
—
nV Nu 1
4 u 4
cos
cos
u + T--7 + o
4 4
u + o
1
(Nu)
So
u 4 J_ 1
u du
u 2 cos
u ) du +--o
u
(Nu)
Set a new variable as
t = v—u
t
2
2t
u = —, du = 2^dt, у — < t < xVN.
Then
and
For
x/N
e <-T |cos 2t| dt + o- <-г
A—3 ./^N w vn—4 VÑ
x/N
dt + o ( — n
= o -
A
2u e-2 _2
u
L-2(u)du
< »(n).
B
e 2 ^/uL- 2 (u)du
n
2
X
n
2
X
2
X
n
2
X
n
2
2
X
X
1
1
e
2
in the same way, we get
B < o(n).
Then
Rn(x) < o(-) + o(-) + o(-) = o(-Vnz Vnz \n/ Vn
Finally, we have the desired estimate. □
References
[1] 1.1. Sharapudinov, "Approximation of functions of variable smoothness by Fourier-Legendre sums", Sb. Math., 191:5 (2000), 759-777.
[2] I. Sharapudinov, Mixed Series of Orthogonal Polynomials, Daghestan Sientific Centre Press, Makhachkala, 2004.
[3] 1.1. Sharapudinov, "Approximation properties of mixed series in terms of Legendre polynomials on the classes Wr ", Sb. Math., 197:3 (2006), 433-452.
[4] I.I. Sharapudinov, "Sobolev orthogonal systems of functions associated with an orthogonal system", Izv. Math., 82:1 (2018), 212-244.
[5] M.A. Boudref, "Inner product and Gegenbauer polynomials in Sobolev space", Russian Universities Reports. Mathematics, 27:138 (2022), 150-163.
[6] M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 1-st ed., Dover Publications, USA, 1964.
[7] G. Szego, Orthogonal Plynomials. V. 23, American Mathematical Society, Providence, Rhode Island, 1975, 432 pp.
[8] V. Smirnov, Higher Mathematics Courses. V. III, Mir Publ., Moscow, 1972 (In French).
[9] R. Askey, S. Wainger, "Mean convergence of expansions in Laguerre and Hermite series", American Journal of Mathematics, 87 (1965), 698-708.
[10] B. Muckenhoupt, "Mean convergence of Hermite and Laguerre series. II", Transactions of the American Mathematical Society, 147:2 (1970), 433-460.
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, доктор философии по математике, директор лаборатории математики и компьютерных наук LIMPAF, преподаватель кафедры высшей математики. Университет Буира, Алжир. E-mail: [email protected]
Received 08.02.2023 Reviewed 26.05.2023 Accepted for press 09.06.2023
Поступила в редакцию 08.02.2023 г. Поступила после рецензирования 26.05.2023 г. Принята к публикации 09.06.2023 г.
