ESTIMATES FOR SOBOLEV-ORTHONORMAL FUNCTIONS AND GENERATED BY LAGUERRE FUNCTIONS Текст научной статьи по специальности «Математика»

DOI: 10.15393/j3.art.2021.9210

UDC 517.15

R. M. GADZHIMIRZAEY

ESTIMATES FOR SOBOLEV-ORTHONORMAL FUNCTIONS AND GENERATED BY LAGUERRE

FUNCTIONS

Abstract. In this paper, we consider the system of functions Aan(x) (n = 0,1,...), a > -1, r G N, orthonormal with respect to a Sobolev-type inner product and generated by the system of La-guerre functions. Using asymptotic formulas for the Laguerre polynomials, we obtain estimates for functions Aan(x), x G [0, œ).

Keywords: Laguerre functions, Sobolev-type inner product, Sobolev-orthonormal functions

2010 Mathematical Subject Classification: 26D15

1. Introduction. Recently, the theory of polynomials orthogonal with respect to a Sobolev-type inner product has been intensively developed [2], [4], [5], [10], [11], [14], [16], [17]. In particular, this is due to the fact that Fourier series by Sobolev orthogonal polynomials (Fourier-Sobolev series) is a useful tool for solving initial-value problems for ordinary differential equations [17]. Note that the most representative results in this theory are associated with the following inner product:

r-1 b

(f,9)s = Y, f (v)(°)g(v)(°) + f (r)(x)g(r)(x)w(x)dx, (1)

v=0 J

a

where w(x) is the weight function. One of the methods for constructing systems of functions orthogonal with respect to the inner product (1) was developed in the works of Sharapudinov I. I. [16], [17]. A system of functions

xn

A" (x) = —-, n = 0,1,..., r — 1, ' n!

© Petrozavodsk State University, 2021

A0,r+n(x) = J—tyj(x - t)r-1 n = 0,1,..., (2)

0

orthonormal on [0, <x>) with respect to the inner product

r-1 ~

(f,9)s = £ f(v)(0)g(v)(0) + f (r)(x)g(r)(x)dx V=0 0

and generated by the system of Laguerre functions L^(x) = e xl/xl/ L^(x)

v

_

n r(n+

L^(x) we denote the Laguerre polynomial of degree n. In the same paper, asymptotic properties were investigated and estimates were obtained for ^1,1+n(x), x e [0,w]:

1

was introduced in [7] using this method. Here h^ = r(j?(+'++)1), and by

Al,l+n(x) = S (V ! X 1 V

where u is a fixed positive real number, v = 4n + 2a + 2. In this paper, we obtain estimates for A0,r+n(x), x G [0, œ). The following theorem holds:

Theorem 1. Let a > — 1, r G N. Then the following estimates hold:

O (vf xr+a ) , 0 ^ x ^ 1, A^+Jx) H of^), ! < X ^ v - v1/3,

\(v—x) 5/

O (vr—!), v - v1/3 < x.

Remark. Estimates for the function Al,1+n(x) were obtained in [8].

To prove this theorem, we need some properties of Laguerre polynomials given in the next section.

2. Some properties of Laguerre polynomials. Let a be a real number. Then the following relations hold for Laguerre polynomials:

Rodrigues' formula [18]:

1 dn La(x) = - x-aex — (xn+ae-x).

x

Orthogonality relations [18]:

La(x)Lm(x)p(x)dx = SnmK, a > -1,

where p(x) = e Xxa, 5nm is the Kronecker delta.

• Equalities for derivatives [18]:

dr

dxr La+r = (—1)r La(x), (3)

(—x)r

L_r (x) = — — - —— Ln_r(x). (4)

n(n — 1) ■ ■ ■ (n — r + 1)

• Equality [15, p.623, formula 6]:

y ^(x)La(V\ = (a,x)r(a,y), 0 <x ^ y, (5)

¿-(a + 1)fc(k + 1) (xy)a ' ; V

fc=Q

where 7(a,x) and r(a,y) are incomplete gamma functions defined by

x X

Y(a,x)= / ta-1e-tdt, r(a,x)= / ta-1e-tdt.

Asymptotic formulas [3], [12]:

i) Let a > —1, 0 ^ x ^ bv, 0 < b < 1, n > n0; then

1 x 1

\ r(n + a + 1)2a - 2 e 2 é 2 r ^/x 2 ~ . ,,\i ..

La(x) = ( , a 1 i,i,,Ja(vé) + q-rJ«W0) . (6)

n! V 2 2 x 2 + 2 (é') 2 L V„ 2 /j

1

3

' V 2

ii) Let a > —1, av ^ x, a > 0, n > n0; then

(—1)nn 2 2 5 NN+6 e x i" 2^ / Ai(—v 2 0)\i _

La (x)= , a + 1 N, Ai(—v3 0) + O ^-^ . (7)

n! xa+1 eN(—0') 1 L V x /J

In (6) and (7) above N = n + (a + 1)/2, v = 4n + 2a + 2, t = x/v, 1

é = é(t) = ^ ( vi - t2 + arcsin(vt) ) , 0 ^ t < 1,

Ja(x) is the Bessel function of the first kind, for which the following asymptotic formula holds [18, p.15, formula 1.71.7]:

Ja (x)

2 \ 2 ( an n . , _„ .

— cos ( x-----— ) + O (x 2 ),x ^

nx / V 2 4 /

Ja(«)

ua, 0 < U € 1,

u

-1/2 i

< u.

0 = 0(t)

4(arccos v^ - \/t(1 - t)) t(t — 1) — arcoshv^t)

2/3

2/3

0 < t € 1,

1 < t,

arcosh(t) = ln(t + Vt2 — 1),

Ai(u) and Bi(u) are Airy functions; if u > 0 the following estimates hold [9, pp. 508-511]:

|Ai(—u)| = O (u-1/4) , |Bi(—u)| = O (u-1/4) ,

|Ai(u)| = O(V1/4 exp (

3/2

Ai(u) =

--u

3

'Ai(u), u ^ 0,

(|Ai(u)|2 + |Bi(u)|2)1/2, u € 0.

Also, note the estimates for Laguerre functions L^(x), which obtained in [1], [13]:

(10) (11) (12)

(13) were

La(x)

O (x 22 v 2) ,

O ( v 1 x 4

O ^v 4 (v1 — |x — v|) 4

lO (e-i) ,

0 € x € 1,

1 < x € 2,

v ^ 2 5

v << x ^ ,

2 < x.

In (14) for n = 0 and — 1 < a < 0 we will assume v = 2.

3. The proof of Theorem 1. First, we obtain an asymptotic representation for Aar+n(x), 0 < x € bv in terms of the Bessel function Ja(x). To this end, we use formula (6) and write

ia_ 1 rrz x , .X »1,1 .1

2a-2 yh" f (x — t)r-V

r'r+nV^ v a -1 (r — 1)iy 12 2

t 2

3

v 2

Aa,r+n(x) = -^ / -.1 , - J«(v^) + JT Ja(v

-Ja(v0)dt+

0

x

2a-1 yha ? (x — t)r-vi

vI-1 (r — 1)i 7 11 2

0

1 ) f (x — t)r-V2

0

Further, we note that ^(r) = t 1/2(1 + O(t)), t = V and = v Vt (1 + O(t )) = vvt + O ( ).

Then

v^ € 1 if 0 <t € 1,

v

> 1 if t > 1.

v

Using the definition of the function Ja(u), we find:

x

r1

«O / J-W)* = o ^ / 1

Vv/ J ) 1 V ' J 2

00

x

O (v^ ^ (v) '+1 dt =

0

a +1 r i

/ a_x2 2x f a_3 r+a+1\ 1

O v2 4 -r = O v2 2 x; + 2 + 2 , 0 <x €-,

( v - x)

x

x

x 1

0( 1)/ ^ = 0(^) +

0

+ of-4) /(x — t)r-|14 dt = of 3 xr+1 A, 1 < x < bv.

^v (v — t) 4 Vv3 (v — x) 4/ v

1/v

So, we have

^ _ 2a-2 ^ x (x — t)r—V1

A"r+n(x) = —T^-nr / i 'i J(v^)dt + K'r(x), (15)

v 2 2 (r — J-)! J 12 («') 2

0

where

a 3 ^ i a | 1

O fv a 2 xr+ a + ^ , 0 < x ^ 1,

<*>=< y, 1 <x<bv. (16)

x_

3 , si

V 2 (v—x) 4

Now we estimate the function Aar+n(x) for 0 < x ^ bv. Let 0 < x ^ 1. Then from (2) and (14) it follows that

A"r+n(x) = O (v2) (x — t)r—1t2dt = O (v2xr+2) . (17)

Consider the case 1 < x ^ bv, 0 < b < 1. From (15)-(17) and (8) we obtain

A"r+n (x) = O - ) + a 1,_V n, X

1N + y/hn

vV + v2—2 (r — 1)!

x (x — t)r—1V1 I" / 2 / an n y 1 y x / -—^—^ J-r cos ---- + O -3

J t1 2 LVnv^ V r 2 4) V(v^) 1A

1/v

/ xr+1 y

+ Ra,r (x) = /1 + 12 + Of -3--, (18)

Vv 2 (v — x) 4 /

dt+

where

x

/2=of 1 )f(x ,/r'dt=of-1 y /(x—t)r—1 dt=

2 J VtVW VvV J 14(v — t)4

1/v 1/v

r-3 x' 4

O -n-Y , (19)

^ v 2 (v — x) 4 /

x

x

x

Il

2av/hO"

y^va (r — 1)!

1/v

(x - t)r—1 cos (v^ - ^ - f) _

v? dt=

20+^L f(x - t)r—3*54t) cos u - 0? - n ) dt.

yfnv2 (r - 1)! y (v - t)

1/v

24

Using the second mean-value theorem [6, p.600], we have:

I1 = °(1)

(1/v )1/4

(x - t)r—V (t) cos (W - an - 9 dt+

(v - 1/v)3/4

1/v

x

1/4

(x - t)r—y(cos (W - 02n - 4) dt

(v - x)3/4

O(1)

(>-M(x - 1/v)r—1/ cos - f - 4)dt+

1/v

+ (x - £)r—W cos (W - an - 4) dt) +

x1/4 , ..„ -, f ..(1\ ( , an n\ ,

__ (x _ {r1 y v/( _) cOS - _ _ _),,

O

^ r- 1

v2x

+

r- 3 x' 4

(v2 - 1)3 / V(v - x)

O

/ xr 4 \ 1

-s , -<y^e^t^x.

\(,/ _ x) 4 / v

(v - x)

From (18) and estimates (19), (20) we deduce

(20)

x'—3

Ar,r+ra(x) = ° ( 7 TT )

V ( v — x ) 4 /

(v - x)

- < x ^ bv, 0 < b < 1. (21)

x

4

Ç

x

y

Ç

Now let v — v1 < x < to. From (17) and (21) we get

1/v V—v 3

A"r+n(x) = (^T^ + J + J j(x — t)r—(t)dt =

! 0 1/v V—v3

x

= Ofi") + O (vr—+ (r — ^ / (x — 1'2e—2M*-

v—v 3

We use formula (7):

, (—1)nn12 5 N N+1

x (x — t)r—1 r 2,. /Ai(—v30)

X

t1 (—0') 2

Ai(—v 3 0) + O

dt = O(vr—1) +

1

v—v 3

r1

(-1)nn2 26 Nn+6 C (x-t) 2

+ (( )n , N (1— i 1 Ai(—v30)dt + Vf'r(x), (22) (r — 1)! n! e^h" i 12 (—0') 2

v—v 3

where

{I) = (— !)-„125N^ J of^i—vMU (23)

v V ' (r — 1)! n! eNVhS Jt 12 (—0')2 V t > v '

v—v 3

For v — v1/3 < x ^ v from (9)-(11) and (13) it follows that

Ai(—v 2/30) = O(v—1/60—1/4) = f y 1/4 o f y (24)

= v—07 = ^v — J OLv«,J, (24)

of Ai(—v2/3 0)y = O(v—1/60—1/4) = 1 f t )1/4Of 1 y (25)

Ol tV—07 ) = tV=07 = tlv — t) O(vv 1/^. (25)

Further, from the Stirling formula n! = nne" "V2nn(1 + O(1/n)) we have

n 2 2 5 NN+1 n!e^v/ha

O(n1/6). (26)

x

From (23), (25) and (26) we get

(x) = O(1)

(x - t)

'—1

i

V—V 3

— / \ 1 14 (v - t) 4

dt =

O

(x - v + v3 )' 1(v1 - (v - x) 3 )

(v - v3 ) 4

Oiv ^

(27)

For v < x < to, from the definition of the function 0 = 0(t), t = - it

follows that _

—0 =

and

-0 = (t - 1)

3 g (-1)fc(1/2)fc (t - 1)k

4 (k + 3/2)k!

2/3

This series is of the Leibniz type, therefore,

-0 ^ (t - 1)

3 (1 - 1(t -1)

a3 5v '

2/3

(t - 1)

— (13 - 3t) 20V ;

2/3

Then from (12) and (13) we obtain

Ai(-v2/30) _ °(v—1/6(-0)—1/4exp(-2v(-0)3/2))

O

t 1/4(-0)1/4

v 1/6 7 (-0)1/4(t - 1)1/4 exp(2v(-0)3/2)

<

< O

t1/4

v1/6/ (t - v)1/4 exp (30(^)3/2(13v - 3t))

(28)

O,Ai(-v2/30)) = O(Ai(-v2/30)) <

Ol tv-0 / ^ tv-07 ^ <

<O

1

v 1/6 7 t3/4(t - v)1/4 exp (30(^)3/2(13v - 3t))

Taking into account estimates (27) and (29), we deduce

x

1

1

C'r (x) = O (v+

? (x - t)r—1 1 7

+ O (1V t5/4(t - v)1/4 exp (30(^)3/2(13v - 3t)) dt ^

^ O(v^ + Ol-J^i-(x - t)r;^ . _ dt

v (t - v)1/4 exp(

(x—v)1/4

O(v1-1Ч + «Ш / (x-УТу2dy. (30)

v S 0 exp

For v ^ x ^ v + v1/3, we have

(x—v)1/4

f (x - v - y4)r—Уdy ^

exp ^ 3vl/2

^ (x - v)r—1 (x - v)1/2 (x - v)1/4 = (x - v)r—4 ^ vi—12 . (31)

When v + v1/3 < x ^ 3V, we get

(x—v)1/4

f (x — v — y4)r—1y2 , ' —dy ^

exp ^ 3vl/2^

(x—v)1/4

^ v3 — li + (x - v - v1/3)r—11 Í y.2dy л ^

Vl/li exn 3^2.

(x-v)3/2 3V1/2 l

^ r--1 , r- 1 11 I ь 3 -t Í, ^ v3 12 + v' 1 vs v12 -re "dt

13

1/3

Б t i

(x-v)3/2 3V1/2

v3 —12 + vr—3 J e—tdt = O fvr—f) . (32) 1/3 t

Let x > "2". In this case, we can write

(x—v)1/4 (x—v)1/4

/ (x — v — yy4)r)1y2 dy = O(vr-4) + / (x — v — yy4)r)1y2 dy €

0 exp( ^J (v/2)1/4 exp(

(i-v)3/2 3V1/2

3) / 3v)r-1 f 1

€ Ofvr-4) + (x — yj J ve-tdt €

V

6V2

3v)r-1

Vv

Therefore, from (27) and (30)-(33) we have

€ O(vr-3) + fx — 3-\r ^e-= O(vr-3). (33) 2v

iO(vV), v — v1/3 < x € v + v1/3, C'r (x)=^ ( ) (34)

\o(vr-2), x > v + v1/3. V '

So, from relation (22) and estimates (34) we obtain the following asymptotic representation for Aar+n(x), v — v 3 < x < to:

r-1

"+"1 ' (r — 1)1 nl eN Vhs Jl t1 (—0') 2 1

v-v 3

In turn, from (24), (26), (28) and (31)-(34) we deduce the estimate A^+Jx) = O (vr-1) , v — v3 < x < TO.

Theorem 1 is proved.

Note another important property of functions Aar+n(x) for a = 0 and r = 1. As noted in the introduction, Fourier-Sobolev series are a convenient tool for solving initial-value problems for ordinary differential equations. In [17], an iterative method for solving the Cauchy problem for ODEs was developed. It was shown that if the system of functions {^1>n(x)}^=0 orthonormal with respect to the inner product (1) for r = 1 satisfies the condition of the form

b

r. OO

= ^^(^1;n(x))2w(x)dx < to,

n=1

then the iterative process converges. When a = 0, r = 1, we get

» _

„ » = I e 2ax ^^ ( A1, 1+n (x )) 2 dx.

A0 2

n=0

The following statement holds:

Theorem 2. Let 1 < a G R. Then the following estimate holds:

/e"2ax£WW*»2«** €

0 n=0

Proof. We use the second mean-value theorem [6, p. 600]:

x § x

J e-t/2Ln(t)dt = J Ln(t)dt + e-x/2 y Ll(t)dt, 0 € £ € x. 0 0 §

Then

§ x

(A?,1+„(x))2 = ((1 — e-x/2) / L^(t)dt + e-x/2 J ^(t)dt)2 =

00 —§ —x

= (1 — L*)*)2 + -(/ L^

00

—§ —x

+2(1—e"'/2 ^^

00

From (3) and (4), we have

y

r y

0

Hence,

L^(t )dT = ^ Li(y).

(A?,1+„(x))2 = (1 — e-x/2)2 ( L^)) 2 + e-x ( L^V

+ 2(1 - e-x/2)e-x/2 -++— L^C)L>(x), 0 ^ C ^ x. (n + 1)2

Further, from the formula (5) (for a = 1) it follows that £x

(n + 1)2~

E T^XT^Ln(C)Ln(x) = eÇ+x7(1,C)r(1,x)

n=0

Ç œ

= eÇ+x y e-tdt = eÇ+x(1 - e-Ç)e-x = (eÇ - 1), 0 ^ C ^ x.

0 x

Then we have

E(A0,i+n(x))2 = (1 - e-X) V - 1) + e-x(ex - 1) +

n=0

+ 2(1 - e-x)e-x (eÇ - 1) ^ ex - 1.

Thus,

œ oo œ

J e-2ax g(A0,1+ra(x))2dx ^ J e-2ax(ex - 1)dX =

□

Acknowledgment. The author thanks the anonymous reviewers for their valuable comments and suggestions. They contributed much to improvement of the manuscript.

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Received September 29, 2020.

In revised form, January 29, 2021.

Accepted January 29, 2021.

Published online February 3, 2021.

Dagestan Federal Research Center of the RAS 45, M.Gadzhieva st., Makhachkala, 367025, Russia E-mail: [email protected]