CC BY
7932
Probl. Anal. Issues Anal. Vol. 8(26), No1, 2019, pp. 32-46
DOI: 10.15393/j3.art.2019.5150
UDC 517.521
R. M. GADZHIMIRZAEY
SOBOLEV-ORTHONORMAL SYSTEM OF FUNCTIONS GENERATED BY THE SYSTEM OF LAGUERRE FUNCTIONS
Abstract. We consider the system of functions A0,n(x) (r G N, n = 0,1, 2,...), orthonormal with respect to the Sobolev-type inner product (f,g) = f(v)(0)g(v}(0) + f(r)(x)g(r)(x)dx and generated by the orthonormal Laguerre functions. The Fourier series in the system {Aa,n(x)}£=0 is shown to uniformly converge to the function f G W[p for 4 < p < 4, a ^ 0, x G [0,A], 0 ^ A < œ. Recurrence relations are obtained for the system of functions Aa,n(x). Moreover, we study the asymptotic properties of the functions A®n(x) as n ^ œ for 0 ^ x ^ w, where w is a fixed positive real number.
Keywords: Laguerre polynomials, Laguerre functions, inner product of Sobolev type, Sobolev-orthonormal functions, recurrence relations, Fourier series, asymptotic formula 2010 Mathematical Subject Classification: 42C10, 65Q30
1. Introduction.
Let Lp be the space of measurable functions f defined on the semiaxis [0, œ), such that
W£p be the space of r — 1 times continuously differentiable functions f for which f(r-1) is absolutely continuous on an arbitrary segment [a,b] c [0, ro) and f(r) e Lp. By A£(x) (n = 0,1,...) we denote the Laguerre function defined by the formula
A^x) = ^/p(x)l^{x),
(1)
©Petrozavodsk State University, 2019
where p(x) = e-xxa, /n(x) is the orthonormal Laguerre polynomial (13). It is well known that for a > —1 the system of functions (A^ (x)}(=0 is orthonormal with respect to the inner product
<Am Aa> = J Am(x)Aa(x)dx 0
The system of Laguerre functions (Aa(x)}(=0 generates on [0, to) a system of functions A?n(x) (r E N, n = 0,1,...) orthonormal for a > —1 with respect to the Sobolev type inner product
r— 1 X
<f,g> = E f(v) (0)g(v)(0) + / f (r)(x)g(r) (x)dx. (2)
v=0 0
The functions Aan(x) are defined by means of equalities (15) and (16). In this paper, we show that the Fourier series in the system (Aan(x)}(=0 converges uniformly to the function f E W£P for a ^ 0, | < p < 4, x E [0,A], 0 ^ A < to. Recurrence relations are obtained for the system of functions A?n(x) and can be used for calculating the values of Aan(x) for any x and n. Moreover, we study the asymptotic properties of the functions Aan(x) as n ^ to for 0 ^ x ^ w, where w is a fixed positive real number. Using these asymptotic properties, we obtained estimates for the functions A?n(x) on the interval [0,w].
2. Some information on the Laguerre polynomials and Laguerre functions.
To study Sobolev-orthonormal functions generated by Laguerre functions, we need some properties of the Laguerre polynomials and Laguerre functions that are given in this section.
Let a be an arbitrary real number. Then for the Laguerre polynomials we have [12]:
• The Rodrigues formula
La(x) = 1 x-aex {xn+ae-x)(n). n J n! L J
• The orthogonality relations
J La(x)Lm(x)p(x)dx = bnmK (a > —1), (3)
0
where p(x) = e xxa, 6n,m is the Kronecker symbol, h°a = ("+") r(a + 1).
• The equalities
d
dXLa(x) = -L£i(*). (4)
(x) n[fc] Ln-k(x),
where k is a positive integer number and 1 ^ k ^ n, n[0] = 1, n[k] = n(n — 1) ■ ■ ■ (n — k + 1).
xLa+1(x) = (n + a + 1)La(x) — (n + 1)L0+i(x); (5)
• The recurrence formula
Lo(x) = 1, L*(x) = — x + a + 1,
nL^(x) = (—x + 2n + a — 1)La-1 (x) — (n + a — 1)La-2(x), n = 2, 3,...
(6)
• Theorem. [12, p.199, Theorem 8.22.4] For a > —1, we have
x a r a . . a r(n + a + 1) T / / N A ^ / a 3\ /„n e-2x2 La(x) = N- 2 ^---L Ja (2(Nx)2 J + O (n2-4 J , (7)
a + 1
N = n +---—, x > 0,
the bound holding uniformly in 0 < x ^ u (u is a fixed positive number). More precisely, the following bounds are valid:
5 / 3 \ -x4O ^n2 4J , — ^ x ^ u,
c
x2 +2O (na), 0 < x ^ -
n
In (7), Ja(x) is the Bessel function of the first kind; for it the following asymptotic formula holds [12, p.15, formula 1.71.7]:
1
2 \2 / an n
3
Ja(x) = ^—J cos \x-----—J + O (x 2 J , x ^ (9)
The weight estimate [1,4]
e-x|La(x)| ^ c(a)A£(x), a> —1. (10)
Here and henceforth, c and c(a) are positive real numbers depending only on the indicated parameters,
Aa(x)
ea,
en
&n en + |x - era|
e 4,
0 ^ x ^ 1,
"n
è <x ^ ,
"n < x ^ ^,
< x
2 •X' ^
:n)
where = 0n(a) = 4n + 2a + 2.
• The differentiation formula [2, p.191, formula 27]
[xaLa(x)](m) = (n - m + a + 1)mxa-mLa-m(x)
:12)
where (n)0 = 1, (n)m = n(n +1) ■ ■ ■ (n + m — 1), m ^ 1.
It follows from (3) that the corresponding orthonormal system of the Laguerre polynomials has the form:
in(x) = (ha)-2 ¿a (x)
n
0,1,...
:13)
so
ll(x)im(x)p(x)dx = in,m (a > —1).
From (6) and (13), we immediately obtain a recurrence formula for /^(x)
ia(x) =
1
Vr(a + 1)!
, ia(x) =
—x + a + 1 /T(a + 2) ,
ln(x) — (an bnx)ln_ l(x) cnln_ 2(x) n — 2, 3,..._
where
an = an (a) =
2n + a — 1 [n(n + a)] 2
bn = bn(a)
[n(n + a)] 2
cn cn(a)
(n — 1)(n + a — 1) n(n + a)
A----
4 x 2 4
1
A similar recurrence formula holds for the functions A^(x):
Ag(x) = , A?(x) = + " + 1)
Vr(a + 1)' Vr(a + 2) ' \ . (14)
Aa(x) = (an - brax)Aa-i(x) - c„Aa-2(x), n = 2,3,.. .
In the sequel, we need the following property of the functions A^(x). Theorem A. [1, Theorem 1] Let f G Lp, 4 < p < 4, a ^ 0. Define
OO n
= / Aa(x)f (x)dx and set Sn(x) = akAa(x). Then ||Sn — f ||LP ^ 0 0 k=0 as n ^ w.
3. On the Sobolev orthonormal functions generated by the Laguerre functions.
Definition 1. For a given r G N, define the functions Aan(x), n = 0,1,..., by
A^+Jx) = - 1)! I (x - t)r-1Aa(i)dt, n = 0,1,.... (15)
0
xn
Aa„(x)^—, n = 0,1,...,r - 1. (16)
' n!
Consider the problem of computing the functions Aar+n(x) for any n
x
and x. Note that A^Jx) = A£(x), Aa 0(x) = 1, Aa 1(x) = / A£(t)dt by
0
definition.
Theorem 1. Let a> — 1. Then the following recurrence relations hold:
x
A"(x) = -A?„-i (x), 1 ^ n ^ r - 1; (17)
rA?+1 ,r+1(x) = (x - 2r - a)Aa>r(x) + 2xA0-1 ,r-1(x), r ^ 1; (18)
J(n +1)(n + a + 1)Aa;ra+2(x) = 2xAa(x) - A?^ (x)+
+ Jn(n + a)A?>n(x), n ^ 1; (19)
rAa+1,r+n(x) = Jn(n+a) Aar+n(x) +(x - 2n - a+1) Aar+n-1 (x)+ + J(n - 1)(n + a - 1)Aa>r+n-2(x), r ^ 1, n = 2, 3,... (20)
x
Proof. The equality (17) is obvious. Let us prove the relation (18). From the definition of the functions Aar+n(x) and integrating by parts, we have:
x
I (x - t)r-
(r — 1)!
x
Aar (x) = 1 /(x — t)r-1Aa(t)dt
0
x
1 1 ^(x — t)r-1e-2t2 dt =
x
\/r(a + 1) (r — 1)! 7
_2__L_ I
(a + 2)/r(a + 1) (r — 1)! J
x
(x — t)r-1e-2 d(t 2+1)
2 1 (x — t)r-2(x — t — x)A£(t)dt—
a + 2 (r — 2)! J 0
x
1 1 (x — t)r-1 (x — t — x)A£(t)dt
a + 2 (r — 1)!
0
2(r — 1) a 2 a r a
Aar(x) + ^ , 0 xAr-1,r-1(x) — " rr; Ar+1,r+1(x) +
a + 2 r,rv ' ■ a + 2 r-1>r-1^ ' a + 2
+ T"^" xAr,r(x).
a+2
Hence, we obtain (18). We now establish the equality (19):
x x
Ai,n+1 (x) = / Aa(t)dt = f e-2£(t)d(t2+1) = xAa(x)+
' + J a + 2 J a + 2
00
x x
+ i e-212+1at)dt — .A- / e-212+1(/a(t))'dt. (21)
a + 2 a + 2
00
Consider separately the second and the third terms of the right-hand side of the last equality. From (14) we have:
x
e-212+1/a(t)dt = J tAa(t)dt = J y/(n + 1)(n + a + 1)A£+1(t)+ 0 0 0
+ (2n + a + 1)Aa(t) — v7n(n + a)A^_ 1 (t)l dt =
x
x
= —/ (n +1)(n+a + 1) AOn+2(x) + (2n+a+1)AOn+1(x) —v7 n(n+a) Aan(x).
(22)
Further, from the equalities (4), (5), and (13) it follows that
(/act))'=—v<m c+i(t) = /a-1(t) — vn/a(t).
Then
x x
J e-212+1(/n(t))'dt = —V^y e-212tCi(t)dt =
00
x
= —V^y e-212 [Vn + a/ a-1(t) — Vn/a(t)] dt =
0
= — ^n(n + a) A?,n(x) + nAa, n+1(x). (23)
From (22), (23) and (21) we obtain (19).
Let us proceed to the proof of (20). By definition,
x
Ar,r+n(x) = (r — 1)! j(x — t) Aa(t)dt. 0
Replace the function Aa(t) by the right-hand side of the equality (14):
x
Aar+n (x) = (r 11)! J (x — t)r-1 [(an — bnt)Aa-1(t) — CnAa-2(t)] dt = 0
x
= anAar+n-1(x) — (r — 1)! j(x — t) tAa-1(t)dt — CnAar+n-2(x) =
0
x
anAar +n — 1(x) + 7 i I (x t) (x t x)Aa — 1(t)dt cnAa r+n— 2(x)
' + (r — 1)! J ' +
0
= anAr>r+n-1 (x) + bn rAa+1,r+n(x) — bnxAa,r+n-1 (x) — Cn Ar,r+n-2(x). (24)
Now divide both sides of (24) by bn and obtain the relation (20). □
Remark 1. Formula (19) is also valid for n = 0.
Note that the systems defined by means of formulae (15), (16) in the general case, when an arbitrary orthonormal system (x) (k = 0,1,...) is used as the generating system, were considered in the works [5-10]. In particular, in the paper [5] the following theorem was proved.
Theorem B. Assume that the functions (x) (k = 0,1,...) form a complete in ¿2(a, b) orthonormal system with respect to the weight p(x) on the interval [a, b]. Then the system |^r,k(x)}£=0, generated by the (x)}£=0 by means of
x
If,
^r,r+fc(x) = 7-7TT (x - t)r ^fck = 0,1,
(r - 1)! J
Vr,k(x) = (x .,a) , k = 0,1,... ,r - 1, k!
is complete in W£2(af) and orthonormal with respect to the inner product
r-1 f
(/,g) = £ f(v)(a)g(v)(a) + /(r)(t)g(r)(t)p(i)di.
v=0 ^
a
Note that Theorem B holds for infinite intervals too. The following statement is immediately deduced from Theorem B.
Corollary 1. If a > -1, then the system of functions Aan(x), generated by the Laguerre functions A^(x) by means of equalities (15) and (16), is complete in W£2 and orthonormal with respect to the inner product (2).
Further, from (15), (16), and the integrand differentiation formula [3, sec. 509, p. 667] for almost all x G [0,to) we have
(au (x))(v) =
'AO—v,k—v(x), 0 ^ V ^ r - 1, r ^ k, A^_r(x), v = r ^ k,
25)
(
r—v, fc—v V
k < v < r,
A"fc-v (x) V ^ k<r
where Aan(x) = A^ (x) by convention.
0
It is easily seen from (2), (15)-(25) that the Fourier series of the
function f E WL in the system (Aak (x)}(=0
X
f (x) - E ca,k (f )Aa,k (x) k=0
has the following form:
r-1 k x
f (x) - E f(k)(0)f[ + Eca,k(f)A?,k(x), (26)
k=0 k=r
where
X
ca,k (f) = / f(r)(t)Aa-r (t)dt, k = r,r + 1,... (27)
0
Note that the Fourier series (26) can be defined for any function f E W£P, p ^ 1. To this end, we show the existence of the coefficients cak(f) defined by the equality (27). Using the Holder inequality, we have
X X
Kk (f )i ^ (/if (r)(t)ipdt)p (/|Aa-r (t)iq dt)1 ^ 00
^ M|f(r)|| LP , k = r, r + 1,...,
where M is a positive real number and 1/p + 1/q = 1. Consider the problem of uniform convergence of the Fourier series (26) to the function f E WLp. To prove the following theorem, we use the same technique as in [11].
Theorem 2. Let a ^ 0, 0 ^ A < to, 4 < p < 4, f E W£P. Then the series (26) converges uniformly on [0, A] to the function f.
Proof. Since f E W£p, then, first, f(r) E Lp, and, therefore, in the metric of the space Lp we have (see Theorem A)
X
f (r)(x) = e cak (f (r))Aa (x), (28)
k=0
X
ca,k (f (r)) = / f (r)(t)Aa(t)dt, k = 0,1,... 0
Second, we can write the Taylor formula for the function /, with the remainder in the integral form:
1 x
r-1 k
/(x) = £ /(k)(0)^ + T^T / (x - t)r-1/(r)(t)dt. k=0 ! ( )! 0
Further, denote by $r*n(/, x) and Sa(/(r), x) the partial sums of the series (26) and (28), respectively:
r-1
fc=Q
(f,x) = J] f(k)(0)k + J] c^(f)A?,fc(x),
fc=r
Then
sa(f (r),x) = ]>]
ca (f (r))\«(x) °r,fc(f )Afc (x)-
fc=Q
|f (x) San+r (f, x)|
n+r
(r - 1)!
(x - t)r—1 f(r)(t)dt -J] ca,k(f)A£fc(x)
fc=r
(r - 1)!
n+r
(x - t)r—if (r)(t)dt - ^ ca,fc (f w (x - t)r—iAa—r (t)dt
fc=r
(r - 1)!
(x - t)r—i(f (r)(t) - sa(f (r),t))dt
(r - 1)!
(x - t)r—iif (r)(t) - sa (f (r),t)idt ^
(r - 1)!
x x
(/(x - t)q(r—i)dt)1/q (/if (r)(t) - sa(f (r),t)ipdt)
i/p
1 / Aq(r—i)+i \i/q
(r - 1)! Vq(r - 1) + 1
il f(r) - sa(f(r)) Hu • (29)
x
1
x
x
1
x
1
x
1
1
From equality (28) it follows that || f(r) - ££(/(r)) ||LP^ 0 as n ^ to. From this relation and (29) uniform convergence of the series (26) on [0, A] to the function f follows. □
4. Asymptotic properties of the functions Aa1+n(x).
Let us study the behavior of the functions Aa1+n(x) on the segment [0,u], where u is a fixed positive real number.
Theorem 3. Suppose a > —1 and x E [0,u]. Then the following asymptotic formula holds:
r(n +1) xa/2+1e-x r(n + a + 1) n + a + 1
x
(30)
where the remainder
r(n + a + 1) 4(n + a + 1)(n + a + 2)
r(n + 1)
1
x
x
0
satisfies the estimate:
In the case a = 0, the last estimate becomes
Proof. From (15), (1) and (13) it follows that
x
x
0
0
I- x
m+h I ^2LS(t)dt.
' n
Further, integrating by parts and using the equality (12), we obtain:
Al,1+ra(x) =
u
_t
e 2
W2 '
du = taLa(t)dt, u
du = —
1
_t
e 2
(t + a)
2ta/2+1
n + a + 1
ta+1 (t)
r(n +1) / xa/2+1e" f
r(n + a + 1) ^ n + a + 1
+
-La+1(x)+
2(n + a + 1)
ta/2(t + a)e-2 ¿a+1(i)di
u
e 2 (t + a)
t«/2+1 ' du = ta+1La+1(t)dt,
du =
e-2 (t2 + 2at + a2 + 2a)
1
u =
n + a + 2
2t«/2+2 t«+2La+2(t)
r(n +1) Xa/2+1e"f ^La+1(x)+ x + a a+2/
r(n + a + 1) n + a + 1
2(n + a + 2)
¿a+2(x) + Ra (x)
Therefore, (30) holds.
Let us proceed to the estimate of the remainder (x) for 0 ^ x ^ u. To this end, consider the following two cases:
1) Let 0 ^ x ^ nn; then, from estimates (10) and (11), it follows that
x
R(x)| ^ /ta/2(t2 + 2|a|t + a2 + 2|a|)e-2+2(t)|dt ^
_xa/2+3 + 2M xa/2+2 + ^ + 2M xa/2+1
a/2 + 3X + a/2 + 2X + a/2 + 1 X
O(-).
n
If a = 0, R(x)| = O ()•
x
1
2) Let i ^ x ^ w; then, from the formulas (7)-(9), we have:
1/n
|Ra(x)| = 0(^)1 f ta/2(t2 + 2at + a2 + 2a)e-2L^+2(t)dt+
+ ta/2(t2 + 2at + a2 + 2a)e-2 La+2(t)dt| = +
1/n
+0
n 2
+2
t2 + 2at + a2 + 2« r(n + a + 3)
N 2 1
n!
-Ja+2(2VNt)di
1/n
+
+0
1
a/2+2
n
t2 + 2at + a2 + 2a t
t5/40(na/2+1/4)dt
1/n
«0(D + 0(n7,) + 0(n
t2 + 2at + a2 + 2a t
Ja+2(2VNt)dt
1/n
0(i) + 0(i
nn
t2 + 2at + a2 + 2a
t
X
cos -
1/n
(2a + 5)n 4
+ 0(
-x
dt
+0
n5/4
t2 + 2at + a2 + 2a
t5/4
cos ^2 VNi -
(Nt)3/4
(2a + 5)n
^ 0(- ) +
n
4
dt
1/n
< 0(n) + 0(n5/4)
-/NX
/N/ra
y4 + 2aNy2 + (a2 + 2a)N
N 7/4y3/2
= 0( n).
If a = 0, then (x)| = 0^^. □
Further, from Theorem 3 and estimates (10), (11), the following assertion is immediately deduced:
x
x
1
t
x
x
x
1
x
1
2
Corollary 1. The following estimates hold:
, 1, 0 ^ x ^ i, |A?,n(x)| ^ c\ ^ _L ,
l3/4 , ön < x ^ ^
Acknowledgment. The author thanks the anonymous reviewers for their valuable comments and suggestions. They contributed much to improvement of the manuscript.
This work was written with the support of the Russian Foundation for Basic Research (grant 18-31-00477 mol_a)
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Received November 02, 2018.
In revised form, February 03, 2019.
Accepted February 04, 2019.
Published online February 11, 2019.
Dagestan Scientific Center of RAS
45, M.Gadzhieva st., Makhachkala, 367025, Russia
E-mail: [email protected]
