Completeness and minimality of systems of Bessel functions Текст научной статьи по специальности «Математика»

ISSN 2074-1863 Уфимский математический журнал. Том 5. № 2 (2013). С. 132-141.

УДК 517.5

COMPLETENESS AND MINIMALITY OF SYSTEMS OF BESSEL FUNCTIONS

B.V. VYNNYTS’KYI, R.V. KHATS’

Abstract. We find the necessary and sufficient conditions for the completeness and minimality in the space L2(0; 1) of system (^xpk JV(xpk) : k G N) generated by Bessel function of the first kind of index v > -1/2. Moreover, we establish a criterion for the completeness and minimality of system (x-2^xpkJ3/2(xpk) : k G N) in the space L2((0; 1); x2dx).

Keywords: Paley-Wiener theorem, Bessel function, entire function, complete system, minimal system, biorthogonal system, basis.

Mathematics Subject Classification: 33C10, 30B60, 42A65, 42A38, 30D20, 42B10, 44A15, 30E15.

1. Introduction and preliminaries

Let p G [0;+ro), L2((0; 1); xpdx) be the space of functions f : (0; 1) ^ C such that tp/2f (t) G L2(0; 1) with the inner product (f^ f2) = f0 tpf\(t)f2(t) dt and the norm

If II2 := fo tPlf (t)|2 dt. Let JV(x) = Er=o ];r(VX/fe+-+) be Bessel’s function of the first kind of index v. It is known (see [3], [25, p. 345], [32]) that the function Jv is a solution of the equation x2y'' + xy' + (x2 — v2)y = 0, i.e. the equation y" + y'/x + (1 — v2/x2)y = 0, the function y(x) = Jv(xp) is a solution of the equation y" + y'/x — yv2/x2 = — p2y, and the function y(x) = ^xpJV (xp) satisfies the equation

11 v2 — 1/4 2

—y + —2— y = p y. x2

The function Jv for v > —1 has (see [3], [25, p. 350], [32]) an infinite set of zeros, among them positive zeros pk, k G N, and negative zeros p-k := — pk, k G N. All zeros are simple, except perhaps, po = 0.

Theorem A. (see [3], [25, p. 357], [32]) Let v > —1 and (pk : k G N) be a sequence of positive zeros of the function Jv. Then the system (\fxJv(xpk) : k G N) is an orthogonal basis in the space L2(0;1).

The system (\JxJv(xpk) : k G N) is also complete in L2(0; 1) if pkJV(pk) + aJv(pk) = 0, a + v > 0 (see [16, p. 124], [25, pp. 356-357]). From [8] it follows that if v > —1/2 and (pk : k G N) is a sequence of distinct positive numbers such that pk < n(k + v/2) for all sufficiently large k G N, then the system (\fxJV(xpk) : k G N) is complete in the space L2(0; 1).

We say that an entire function G is of formal exponential type a G (0; +ro) if

|G(z)| < c(e)exp((a + e)lzl), z G C, for each e > 0 and some constant c(e).

© Vynnyts’kyi, B.V., Khats’, R.V. 2013. Submitted on 30 january 2012.

Theorem 1. Let v > —1/2 and (pk : k G N) be an arbitrary sequence of distinct nonzero complex numbers. For a system (\ftp~kJV(tpk) : k G N) to be incomplete in the space L2(0; 1) it is necessary and sufficient that a sequence (pk : k G N) is a subsequence of zeros of some even entire function G of formal exponential type a < 1 such that the function f (z) = zV+1/2G(z) belongs to the space L2'

The proof by standard methods (see [20, pp. 131-132], [21]) follows immediately from the following lemmas.

Lemma B. (see [2], [13]) Let v > —1/2. A function f has the representation

1

f (z) = J fztJV(zt)j(t) dt, y G L2(0;1), o

if and only if f G L2(0;+to) and f (z) = zV+1/2G(z), where G is an even entire function of formal exponential type a < 1. Moreover, if f ^ 0 then G is a transcendental entire function.

Lemma C. (see [11, p. 67], [24]) Let v > —1. Then every function f G L2(0;+to) can be represented in the form

f (z) = J fztJV(zt)h(t) dt

o

with some function h G L2(0;+to). Moreover, ||f || = ||h|| and

h(t)= fztJV(zt)f (z) dz.

A system (ek : k G N0) of the Hilbert space is said to be minimal (see [20, p. 131], [21, p. 4258], [22]) if for each n G N0 the element en does not belong to the closure of the linear span of the system (ek : k G N0 \ {n}). A system is minimal if and only if it has a biorthogonal system. A complete system has, at most, one biorthogonal system (see [21], [22]).

Similarly to [20, Lecture 18], [21], from Lemmas B, C and Theorem 1, we obtain the following result.

Theorem 2. Let v > —1/2 and (pk : k G N) be an arbitrary sequence of distinct nonzero complex numbers such that pk = p2m if k = m. The system (^tpk JV(tpk) : k G N) is complete and minimal in the space L2(0; 1) if and only if the sequence (pk : k G Z \ {0}), p-k := — pk, is a sequence of zeros of some even entire function G of formal exponential type a < 1 such that the function zV+1/2G(z) does not belongs to the space L2(0;+to) and the function (z2 — p1 )-1zV+1/2G(z) belongs to L2(0; +ro). Moreover, the biorthogonal system (Yk : k G N) is formed, in particular, by the functions Yk, defined by the equality

—7— 2 f fztJv(zt)zV+1/2G(z)

Yk (t)= v-1/2^,, ' -~^2--dz.

pk ' G'(pk) 0 z — pk

Using methods of [18], [20] and [21], we can obtain a number of other various necessary and sufficient conditions for the completeness and minimality of system (^tpk JV (tpk) : k G N) in the space L2(0; 1). In particular, following the arguments of [20, Lecture 18], [21, §§1.7, 3.3], Theorem 1 yields the next statement.

Theorem 3. Let v > —1/2 and (pk : k G N) be an arbitrary sequence of distinct nonzero complex numbers such that |Spk| > £|pk| for all k G N and some 6 > 0. The system

(Vtpk JV(tpk) : k G N) is complete in the space L2(0; 1) if and only if

-j

1—i = +^. k=1|pk|

At studying of some non-classical boundary-value problems (see [26]—[31]) and generalized eigenvectors of some linear operators [28], [29] we needed to obtain the analogues of Theorems 13 for weighted spaces and establish an approximation properties of the special finite linear combinations of Bessel functions. We don’t understand to the end the nature of expected results for an arbitrary v G R. For advance in the given direction it is important to investigate in details the simplest model cases v = —3/2 and v = 3/2. The case v = —3/2 was considered in [27], [30] (see also [26], [28], [31]). Here we consider the case v = 3/2 more detail. But even in this case we cannot obtain the all necessary facts. In particular, remains an open one for us the problem formulated at the end of this paper. In our view, its solution is very important for the construction of some spectral theory that is based on the notion of a generalized eigenvector (see [28], [29]).

It is well known (see [3], [25, p. 350], [32]) that fzJ3/2(z) = — 2/nz—1(z cos z — sin z). The

fxpJ3/2(xp) 2 2

function --------——----- belongs to the space L2((0; 1); x2dx) for each p = 0. From Theorem A

x2p2

it follows that if (pk : k G N) is a sequence of positive zeros of the function J3/2 then the

system (©k : k G N), 0k(x) := ^ pU 3/^( pU), is complete in the space L2((0; 1); x4dx). But

x2pk

from this statement it does not follows that the system (0k : k G N) is complete in the space L2((0; 1); x2dx). We investigate some approximation properties of the system (0k : k G N) in L2((0; 1); x2dx) with an arbitrary sequence of nonzero complex numbers (pk : k G N). The main result of the paper is contained in Theorem 9 where is found a criterion for the completeness and minimality of system (0k : k G N) in the space L2((0; 1); x2dx).

2. Main results

Denote by PW2 the set of all entire functions of formal exponential type a G (0; +ro) belonging to the space L2(R) on the real axis R in C, and by PW% — we denote the class of odd entire functions from PW2. According to the Paley-Wiener theorem (see [12], [19]—[21]), the class PW2 coincides with the class of functions G admitting the representation

a

G(z) = J eitzg(t) dt, g G L2(—a; a),

—a

and the class PW2 consists of the functions G of the form

a, —

a

G(z) = J sin(tz)g(t) dt, g G L2(0; a). o

Moreover, ||g||L2(o;a) = v/2Tn|G|L2№+TO) and

+<^

g(t) = — sin(tz)G(z) dz.

n J

o

Theorem 4. An entire function Q can be represented in the form

1

Q(z) = zftzJ3/2(tz)h(t) dt, h G L2((0; 1); x2dx), (1)

if and only if Q is an odd entire function, Q(0) = Q'(0) = Q''(0) = 0 and the function Q'(z)/z belongs to the space PW12—. If these conditions hold then

— +^

w . Pi f Q'(z) . . . , h(t) = y— ——sin(tz) dz.

o

Proof. Let the function Q is representable in the form (1). Since

,— T . . /2 tz cos(tz) — sin(tz)

z\ftz J3/2 (tz) = — y ---------y-^t---------^ ,

we have

j— 1

2 tz cos(tz) — sin(tz)

Q(z) = —y_ y ------------------^--------h(t) dt.

o

Therefore,

1 1

'2 f Q'(z) /2

Q'(z) = y — J tz sin(tz)h(t) dt, --------= y — J sin(tz)q(t) dt,

oo where q(t) = th(t). Since h G L2((0; 1); x2dx), we have q G L2(0; 1) and, hence, according to Paley-Wiener theorem, the function Q'(z)/z belongs to the space PW-j2 —. Conversely, if all the

conditions of the theorem hold then the function q(t) = \j2/n sin(tz) dz belongs to

the space L2(0; 1) and Q'(z) = y/2f n J0 z sin(tz)q(t) dt. Using Fubini’s theorem, we get

1 z

Q(z) = Q(z) — Q(0) = y — J q(t) dt J w sin(tw) dw

oo

1 1

2 f sin(tz) — tz cos(tz) q(t) , f ,—

= Jn —y~^~t---------dt = zVTz.J3/2(tz)h(t) dt,

oo where h(t) = q(t)/t. Since q G L2(0; 1), one has that h G L2((0; 1); x2dx), and the proof of the

theorem is completed. □

Let E2, — be the class of the entire functions Q that can be represented in the form (1), and

let E2 — be the class of nonzero odd entire functions Q such that Q(0) = Q'(0) = Q''(0) = 0 and

the function Q'(z)/z belongs to the space PWf —.

Corollary 1. E2 — = E2 —.

Corollary 2. The class E2,— coincides with the set of the entire functions Q that can be

represented in the form

_ 1

2 f sin(tz) — tz cos(tz) u t2

Q(z) = y nj--------------12--------q(t) dt, qG L(0;1). (2)

o

Theorem 5. Let (pk : k G N) be an arbitrary sequence of distinct nonzero complex numbers such that pk = pn if k = n. For a system (0k : k G N) to be incomplete in the space L2((0; 1); x2dx) it is necessary and sufficient that a sequence (pk : k G Z \ {0}), p—k := — pk, is a subsequence of zeros of some nonzero even entire function G such that the function Q(z) = z3G(z) belongs to the space E2,—.

Proof. Incompleteness of a system (0k : k G N) is equivalent to the incompleteness of the system (p30k : k G N). According to the well-known completeness criterion, the last system is incomplete in the space L2((0; 1); x2dx) if and only if there exists a nonzero function h G L2((0; 1); x2dx) such that

1

J pk fxpk J3/2 (xpk )h(x) dx = 0 (3)

o

for all k G N. If the system (0k : k G N) is incomplete, then the function (1) has zeros at points pk, belongs to the space E2 — and Q(z) ^ 0. Hence, the function G(z) = z—3Q(z) is required. Conversely, if the sequence (pk : k G Z \ {0}), p—k := — pk, is a subsequence of zeros of some

even nonzero entire function G such that the function Q(z) = z3G(z) belongs to E2 — then,

using (1), we obtain (3). The theorem is proved. □

Lemma 1. Let an entire function Q G E2,— be defined by the formula (2). Then (here and so on by C1, C2, ... we denote arbitrary positive constants) for all z G C, we have

e|»z| / ^3^1 \ 1/2

|Q(z)| < C‘(1 + |z|)+ |32| + C2|z| + 1 + |3;|

Proof. Indeed, let

1/2

. . 2 f sin(tz) — tz cos(tz) . . ,

Il(z) = v( ; t2——q(‘)dt■

o

l~ 1 l~ 1

h(z) = — y n J cos(tz) dt, I:i(z) = y 2 J q2isin(tz) dt.

1/2 1/2

Then Q(z) = I1(z) + zI2(z) + I3(z). According to the Paley-Wiener theorem, the functions I2(z) and I3(z) belong to the space PW-j2,

1— 1 1- —1/2

I2(z) = — yf Je‘Z f jeitZ^ dt,

1/2 —1

and applying Schwartz’s inequality, we get

el3zl

^2 (z)|< "WPi ■ z G C'

Similarly,

el3zl

|I3(z)| < 64^===, z G C.

z

x/1 +

Finally, since | sin(tz) — tz cos(tz)|2 = (sin(tx) — tx cos(tx))2 + (sinh(ty) — ty cosh(ty))2+ +t2(x2 sinh2(ty) — y2 sin2(tx)) for any t G R and z = x + iy G C, we obtain

1/2 1/2

r | sin(tz) — tz cos(tz)|2 r (sin(tx) — tx cos(tx))2

J t4 t J t4 t

oo

1/2 1/2

+ j (sinh(ty) — ty cosh(ty))2 ^ + j x2 sinh2(ty) — y2 sin2(tx) ^

oo

x/2 y/2

3 f (sin t — t cos t)2 3 f (sinh t — t cosh t)2 ,

x3 I (--------------14------- dt + y3 (------------14--------- dt

y/2 x/2

2 sinh2 t 2 sin2 t

+xy ——— dt — yx dt.

Therefore, for z G C

/ e|y| \1/ 2 / e|3z| \1/ 2

|I1(z)| < C^|x|3 + |z|2+ y2|x^ = Ca|z^|»z| + .

This completes the proof of the lemma. □

Theorem 6. Let (pk : k G N) be a sequence of distinct nonzero complex numbers such that pk = 9m if k = m, and let a sequence (pk : k G Z \ {0}), p—k := — pk, be a sequence of zeros of the some even entire function G of finite formal exponential type, for which on the rays {z : argz = pj}, j G {1; 2; 3; 4}, p1 G [0; n/2), p2 G [n/2; n), p3 G (n;3n/2], p4 G (3n/2;2n),

we have

|G(z)| > Ca(1 + |z|)—2exp(|3z|).

Then the system (0k : k G N) is complete in the space L2((0; 1); x2dx).

Proof. Assume the converse. Then, according to Theorem 5, there exists an entire function Q G E2,— for which the sequence (pk : k G Z \ {0}) is a subsequence of zeros. Let V(z) = Q(z)/(z3G(z)). Then V is an even entire function of finite exponential type, for which (see Lemma 1)

|V(z)| < C7 , 1 =, arg z = pj, j G {1;2;3;4}.

y 1 + |^z |

Hence, according to the Phragmen-Lindelof theorem (see [20], [21]), V(z) = 0. Therefore, Q(z) = 0. This contradiction proves the theorem. □

Corollary 3. Let (pk : k G Z), p—k := — pk, be a sequence of zeros of the function J3/2. Then the system (0k : k G N) is complete in the space L2((0; 1); x2dx).

Proof. Indeed, the sequence (pk : k G Z \ {0}) is a sequence of zeros of the entire function G(z) = z—3(z cos z — sin z), and this function satisfies the conditions of Theorem 6. Therefore, the system (0k : k G N) is complete in the space L2((0; 1); x2dx). □

Theorem 7. Let (pk : k G N) be a sequence of distinct nonzero complex numbers such that pk = pm if k = m, and let a sequence (pk : k G Z \ {0}), p—k := — pk, be a sequence of zeros of the some even entire function G of finite formal exponential type such that the function z3G(z) does not belongs to the space E2,— and for which on the rays {z : arg z = pj}, j G {1; 2; 3; 4}, p1 G [0; n/2), p2 G [n/2; n), p3 G (n;3n/2], p4 G (3n/2;2n), the inequality

|G(z)| > Cg(1 + |z|)—2—aexp(|Sz|)

holds, where a < 5/2 is a some constant. Then the system (0k : k G N) is complete in the space L2((0; 1); x2dx).

Proof. Assume the converse. Then, according to Theorem 5, there exists an entire function Q G E2 — for which the sequence (pk : k G Z \ {0}) is a subsequence of zeros. Let V(z) = Q(z)/(z3G(z)). Then V is an even entire function of finite formal exponential type, for which (see Lemma 1)

|V(z)| < C9(1 + Mr—1/2, argz = pj, j g{1;2;3;4}.

Since a — 1/2 < 2 and V is an even entire function, then, according to the Phragmen-Lindelof theorem, the function V is a constant. Hence, Q(z) = C10z3G(z). Therefore, Q G E2 —. Thus, we have a contradiction and the proof of the theorem is completed. □

Lemma 2. If an odd entire function L belongs to the space E2 — and has a root at a point p = 0, then the function L(z) = L(z)/(z2 — p2) also belongs to E2,— .

Proof. Indeed, the function L is an odd entire function of formal exponential type a < 1,

L' (z)

and L(0) = L'(0) = L''(0) = 0. Besides,

L'(z )(z2 — p2) — 2zL(z)

(z2 — p2)2

L'(z) L'(z)

2L(z)

z(z2 — p2) (z2 — p2)2 '

1+^p

L'(x)

x(x2 — p2)

dx < C11

1+^p

L'(x)

x

dx < +oo,

and according to Lemma 1

1 + ^p

L(x)

(x2 — p2)2

dx < C12

1+^p

(1 + |x|)3

(x2 — p2)4

dx < +to.

□

Hence, the function L'(z)/z belongs to L2(R). This concludes the proof of the lemma.

Lemma 3. If an odd entire function L has zeros at points pk = 0, k G N, and the function L(z)/(z2 — pi) belongs to the space E2,—, then the functions Lk(z) = L(z)/(z2 — pk) also belong to E2 — for every k G N \ {1}.

Proof. In fact, let Qk(z) = (p)2 — p1)-

L(z)

Then Qk(z) = (pk — p1)

2 L1

L1(z)

z2 — pk

and

(z2 — pk )(z2 — p?)

Lk = Qk + L1. Therefore, taking into account the previous lemma, we obtain the required proposition. □

Theorem 8. Let (pk : k G N) be an arbitrary sequence of distinct complex numbers such that p\ = pm if k = m. If the sequence (pk : k G N) is a subsequence of zeros of some even entire function G which has simple roots at all points pk and the function z3(z2 — p2)—1G(z) belongs to E2_, then the system (0k : k G N) has a biorthogonal system (Yk : k G N) in the space L2((0; 1); x2dx). The biorthogonal system (Yk : k G N) is formed, in particular, by the functions Yk, defined by the equality

+ ^

'2 f Vk'(z) . 2pkz3G(z)

Yk(t) = W-

sin(tz) dz, Vk(z) :=

G' (pk )(z2 — pk)

(4)

Proof. In fact, according to Lemma 3, the functions Vk belong to the space E2 —. Therefore, there exist nonzero elements Yk of the space L2((0; 1); x2dx) such that

y1

Vk(z)= zVTzJ3/2(tz)Yk(t) dt,

and by Theorem 4 the functions Yk can be found by (4). Moreover,

Vk(pn) _ f 1, n = k,

0, n = k,

p3

n

z

2

2

2

and we obtain the required proposition. □

Theorem 9. Let (pk : k G N) be an arbitrary sequence of nonzero complex numbers such that p2k = pm as k = m. The system (0k : k G N) is complete and minimal in the space

L2((0; 1); x2dx) if and only if the sequence (pk : k G Z \{0}), p—k := — pk, is a sequence of zeros

of some even entire function G such that the function z3(z2 — p1)—1G(z) belongs to the space E2,— and the function z3G(z) does not belongs to this space.

Proof. If the considered system is minimal then there exists a nonzero function

Y1 G L2((0; 1); x2dx) such that

1

f ( 1 k=1

pk^ftpk J3/2(tpk)Y1(t) dt = 1 0, k = 1

o

Let T(z) = J0 z\[-zJ3/2(-z)y1 (t) dt. The function G(z) = z—3(z2 — p‘^)T(z) is the required,

because the function T(z) = z3(z2 — p2) —1G(z) belongs to the space E2 — and has zeros at all points pk, all its zeros are simple and it has no other zeros. Indeed, if p is another root of the function G, then the function V(z) = G(z)/(z2 — p2) which has roots at all points pk, would belongs to the space E2,— that, according to Theorem 5, contradicts the completeness of the considered system. Besides, the function z3G(z) does not belongs to E2 —, because otherwise the system would be incomplete. Conversely, if all the conditions of the theorem hold then, basing on Theorem 8, we obtain the required proposition. The proof of theorem is thus completed. □

Corollary 4. Let (pk : k G Z), p—k := — pk, be a sequence of zeros of the function J3/2. Then the system (0k : k G N) has in the space L2((0; 1); x2dx) a biorthogonal system (Yk : k G N) which formed by the functions Yk, defined by the formula

Yk (t) = п(1 + pk }v/tpfc Js/2(tpk).

This corollary can be proved by standard methods of the theory of Bessel functions (see [3], [25, p. 347], [32]). However, it can be proved by Theorem 8. In fact, the sequence (pk : k Є Z \ {0}), p-k := — pk, is a sequence of zeros of even entire function G(z) = z-3(z cos z — sin z). Further, the function z3G(z) dose not belongs to the space E2-and the function z3(z2 — p2)-1G(z) belongs to this space. Furthermore, according to Theorem 8, the system (©k : k Є N) has in the space L2((0; 1); x2dx) a biorthogonal system (Yk : k Є N) which formed by the functions Yk, defined by the equality (4), where

2pk (z cos z — sin z) , _ sin pk

Vk (z):= G'(Pk )(z2 — p2 ) . G(Pk ) = --p-.

Therefore,

--

—7—T- 2 fr. .tz cos(tz) — sin(tz) ,

Yk (t) = —\l - J Vk (z)-----------(—^—---(-) dz

0

__

2 2pk f (z cos z — sin z)(tz cos(tz) — sin(tz)) ^

ПІЩРй) J z2(z2 — pk) z.

0

Let n(z; t) = tz2ei(1+t)z +tz2ei(1-t)z+izel(1+t')z—izei(1-t')z+itzel(1+t')z+itzei(1-t)z—el(1+t)z+ei(1-t)z. Then (zcosz — sinz)(tzcos(tz) — sin(tz)) = 4(n(z;t) + n(—z;t)). Hence,

1 pk f n(z; t)

Yk(t) = —^+гK \ 2( 2 —^ dz

\J2ntG<(pk)J z2(z2 — pk)

— OO

-(tpk cos(tpk) - sin(tpfc))(pfc sin Pk + cos pk) = n(1 + pk)Vtpk J3/2(tpk).

t sin pk

Problem. Let (pk : k G Z), p—k := —pk, be a sequence of zeros of the function J3/2. Since (see [32], [25, p. 352]) pk ~ nk as k ^ to and

Pk Pk

|0k fil'ft ||2 = j2<1 p+ pk )2 J" |2VtJз/2^2)|2 dt J dt

0 0

n(1 + p|)2 , , ( /“ ^(t)2 I

= ----3p3--------(1 + o(1)) \J------1?-----dt + o(1) I —' +^ k ,

then the system (0k : k G N) is not uniformly minimal (see [21, p. 4258], [22, p. 62]) in the space L2((0; 1); x2dx) and therefore is not a basis in this space (see [21, p. 4258], [22, p. 62]). However, it is easy to show that the biorthogonal system (Yk : k G N) is complete in L2((0; 1); x2dx). Therefore, the numbers dk = t2 f (t)Yk(t) dt determine the function f G L2((0; 1); x2dx)

uniquely. But the series ^=1 dk0k(x) not for each function f G L2((0; 1); x2dx) converges in

L2((0; 1); x2dx) to the function f. We do not know the methods of restoration of the function f G L2((0; 1); x2dx) by numbers dk and, in particular, whether the given series converges in L2((0; 1); x2dx) to f in the sense of Cesaro.

Similar problems are studied in [1], [4]—[7], [9], [10], [14], [15], [23], [32, Ch. XVIII], [33] and for exponential systems in [17], [18], [21], but we cannot use these results.

СПИСОК ЛИТЕРАТУРЫ

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Institute of Physics, Mathematics and Informatics,

Ivan Franko Drohobych State Pedagogical University,

3 Stryiska Str.,

82100 Drohobych, Ukraine E-mail: [email protected]

Ruslan V. Khats’,

Institute of Physics, Mathematics and Informatics,

Ivan Franko Drohobych State Pedagogical University,

3 Stryiska Str.,

82100 Drohobych, Ukraine E-mail: [email protected]