CC BY
1652
Probl. Anal. Issues Anal. Vol. 5(23), No. 2, 2016, pp. 52-68
DOI: 10.15393/j3.art.2016.3390
UDC 517.548
E. L. Shishkina
ON WEIGHTED GENERALIZED FUNCTIONS ASSOCIATED WITH QUADRATIC FORMS
Abstract. In this article we consider certain types of weighted generalized functions associated with nondegenerate quadratic forms. Such functions and their derivatives are used for constructing fundamental solutions of iterated ultra-hyperbolic equations with the Bessel operator and for constructing negative real powers of ultra-hyperbolic operators with the Bessel operator.
Key words: weighted generalized function, quadratic form, ultra-hyperbolic operator, Bessel operator
2010 Mathematical Subject Classification: 46T12, 46F05
1. Introduction and main definitions. The weighted generalized functions associated with nondegenerate indefinite quadratic forms considered in this article are necessary for construction of the ultra-hyperbolic Riezs potential with the Bessel operator. Riezs potential with the Bessel operator and other operators with the Bessel differential operator are very interesting subjects with many applications (see, for example, [1]-[9]). We deal with the part of the Euclidean space
R+={x=(xi, ...,xn) e Rn, xi >0,.. .,xn >0}.
Let Q be finite or infinite open set in Rn, symmetric with respect to each hyperplane Xj =0, i = 1,..., n, Q+ = Q If R + and Q+ = Q If R + where
n
={x=(x1,..., xn) e Rn, x1>0,..., xn>0}.
We have C R + and C R +.
We consider the class C) consisting of infinitely differentiable on functions. We denote the subset of functions from Csuch that all derivatives of these functions with respect to x for any i = 1,..., n are
©Petrozavodsk State University, 2016
continuous up to xi =0 by C^(0+). A function f E C^(0+) will be called
d2k+1f
even with respect to xi5 i = 1, ...,n if 2k+1 = 0 for all nonnegative
Xi_ x=0
integer k (see [10], p. 21). Class C^(0+) consists of functions from
__o _
C^(0+), even with respect to each variable xi, i = 1,..., n. Let C ^(0+) be the space of all functions f E C^(0+) with a compact support. We will
o __o _
call elements of C ^ (0+) test functions and use the notation C ^ (0+) = = D+(0+).
We define K as an arbitrary compact in Rn symmetric with respect to each hyperplane xi=0, i = 1, ...,n, K+ = K if R + . A distribution u on 0+ is a linear form on D+(0+) such that for all compacts K+ C 0+, constants C and k exist and
|u(f )|< C £ sup |Daf |, f E C e°S(K+),
|a| <k
where Da = D^1 ...D^, a = (ai,...,an), ai, ...,an are integer nonnegative numbers, DXj = i-JXr, i is imaginary unit, j = 1, ...,n. The set of all
distributions on the set 0+ is denoted by D+ (0+) (see [10], p. 11 and [11], p. 34).
Multiindex y=(y1 ,...,Yn) consists of positive fixed reals Yi > 0, i=1,..., n and |y|=Yi+.. .+Yn. Let (0+), 1 < p < to be the space of all measurable in 0+ functions even with respect to each variable xi, i = 1 , ... , n such that
rY dx < m xY =1 I x7i
J |f(x)|pxYdx< to, xY = JJ
0+ i=1
For a real number p > 1, the (0+)-norm of f is defined by
LP (0+)
( \
J |f (x)|pxYdx
V°+ J
1/p
Weighted measure of 0+ is denoted by mes7 (0) and is defined by formula
mes7 (0+) = J xYdx. o+
For every measurable function f (x) defined on R+ we consider
(f, t) = mesY{x G R+ : |f(x)| > t} = I xYdx
+
{x: |f(x)|>t}
where {x : |f (x)| > t}+={x G R+ : |f (x)| > t}. We will call the function = (f, t) a weighted distribution function |f (x)|. A space L^ (^+) is defined as a set of measurable on and even with respect to each variable functions f (x) such as
L2o(n+) = esssuP7|f (x)| = ^mf (f,a) = 0} < to.
xgO+
aGÜ+
For 1 < p < to the Lp loc (0+) is the set of functions u(x) defined almost
o _
everywhere in such that uf G LY (0+) for any f G C e (0+). Each function u(x) G LYloc(0+) will be identified with the functional u G G D+ (0+) acting according to the formula
/n
u(x) f (x) XY dx, XY = n x7l, f G C £ (R +). (1)
+ i=1
Functionals u G D+ (0+) acting by the formula (1) will be called regular weighted functionals. All other functionals u G D+ (0+) will be called singular weighted functionals.
2.Weighted generalized functions concentrated on the part of the cone. In this section we consider weighted generalized functions SY (P) concentrated on the part of the cone and give formulas for their derivatives.
Generalized function 5Y is defined by equality (by analogy with [12] p. 247)
(¿7, ^)7 = ^(0), <^(x) G K +. For convenience we will write
(¿7, ^)7 = SY(x)^(x)x7dx = ^(0).
Let p, qGN, n=p+q and
p _ i/12 _ I //12 _ 2 | | 2 _ 2 _ _ 2
p |x 1 |x 1 xi + ... + xp xp+i ... xp+q,
where x=(x1,..., xn) = (x/, x")eR+, x/=(x1,...,xp), x//=(xp+1,..., xp+q).
Definition 1. Let pED+(R+) vanishes at the origin. For such p we define generalized function 57 (P) concentrated on the part of the cone P=0 belonging to R+ by the formula
(57(P),p)7 = / 57(|x'|2 -|x"|2)p(x)xYdx. (2)
If the function pED+(R+) does not vanish at the origin then (57 (P), p)7 is defined by regularizing the integral.
Lemma 1. Let pED+ (R +) vanishes at the origin, p>1 and q>1. For 5y (P) the representation
CC
(57(P),p)7 = 1// ^ ¥>(sw)sn+|Y|-3wYdSpdSqds (3)
o S+ S+
holds true. In (3) w=(w/,w//), w/=(w1,...,wp)eR+, =(wp+1 , ...,wp+q)e
n
E R+, n=p+q, |w/| = |w//| = 1, = n w7i, dSp and dSq are elements of
i=i
surface area on the part of the unit sphere
Sp+ = {w/ E R+ : |w/|=1} S+=K E R+ : |w//| = 1}, respectively. For the k-th derivative (kEN) of 57 (P) we have
(5« (P ),p)7 = where
C r k
(¿1) "'"2
rp+|7'|-1dr, (4)
^(r, s) = ^ I I p(rw/,sw//)w7dSpdSq. (5)
Proof. Let us transform (2) to bipolar coordinates defined by
xi = rwi, ...,xp = rWp, xp+1 = SWp+i, ...,xp+q = SWp+q, (6)
where
r = A/ x2 + ... + xp, S = A/xp+i + ... + xp+q,
s=r
W = V + ... + ^p = 1, W I = J "p+1 + ... + ^p+q = 1.
We obtain
(¿7 (P ),p(x))7 =
57(r2 - s2)^(rw',sw//)rp+|Y'|-1sq+|Y"|-1wYdSpdSqdrds.
0 0
Now let us choose the coordinates to be r2 = u, s2 = v. In these coordinates we have
(57 (P ),^)7 = 4
¿7(u — vMvW, v/v^//)uP+27 ' 1x
00
q+' Y// ' 1 1
xv 2 wY dSpdSq dudv = -
n+ ' Y ' 2
4 I I I rw—/ - W'UOp dSq d
0 S+ S+
^__n+ ' y ' i-j
^(V vw)v ~ wY dSp dSq dv.
Returning to variable s by the formula v=s2, we obtain (3).
Now we prove the formula (4). After the change of variables by and r2=u, s2=v in (¿(k)(P),^)7 we get
(p ),^)7=4
d k
dvk
[¿7(v — u)]^^v/uw/, v/vw//) x
00
u
p+'y/' _ 1 q+'i"' —1
2 wY dSp dSq dudv =
(—1)k dk
x
4 dvk
/ i— / r~ n\ q+'i ' 1 ^(Vuu ^vw )v 2
dk
SY (v — u) x
0 0 S+ S+
(—1)k
p+ ' y ' _1
u 2 wY dSpdSq dudv =
p+'i
u 2 w
-1 uY
q+'i'
dvk
v/vu//)v 2
1
dSpdSq du.
0 S+ S+
Returning to variables r, s and using notation (5) we obtain (4). This completes the proof of Lemma 1. □
4
Remark 1. Similarly, we can get the formula
(¿(k)(P =(-1)k
2rdr J rK J
sq+lY"|-1ds. (7)
Remark 2. Noticing that when k=0 formulas (4) and (7) are equivalent to the formula (3) we will examine intergals (4) and (7) at kGN U {0}.
Let (R+). Assuming that the function ^ vanishes at the origin
we have that integrals (4) and (7) converge for all k G NU{0}. If the function ^ does not vanish at the origin then integrals (4) and (7) converge only for k < p+q+|Y|-2 . In this case for k > p+q+JY|-2 we will consider the regularization of (4) and (7) denoting them ¿i^i (P) and ¿Yk2 (P), re-
spectively. So using the expression have
for p > 1, q > 1 and kGN U {0} we
1 (p ),^)7 =
-^(r,s)sq+|Y"|-2 2s3sJ rv ' 7
rp+|7,|-1dr, (8)
(¿Yfc2 (p = (-i)k
i-^(r,s)rP+|Y/|-2 \2rdrJ rv ' 7
S^'V1 ¿s.
(9)
The integrals (8) and (9) converge and coincide for k < p+g+2'7' 2 and for k > p+q+27|-2 these integrals must be understood in the sense of their regularizations.
2.Weighted generalized function P^ +. Let n=p+q, p>1, q>1 and P(x) = xi + ... + xp — xp+1 — ... — xp+q. Here and further let p E D+ (R +). We define the weighted generalized function P7;+ by
(P7A+,^)7 =
PA(x)p(x)x7dx,
(10)
{P (x)>0} +
where {P(x) > 0}+ = {x G R+ : P(x) > 0}, A G C.
Weighted generalized function P7,+ and its derivatives are used for constructing fundamental solutions of iterated B-ultra-hyperbolic equations of the form L^u = f (x), k G N, x G Rn, x^ > 0, i = 1,..., n,
r=s
s=r
r=s
where LB is B-ultra-hyperbolic operator (see [9] and [13]-[15])
LB = + ...+BXp —Bxp+i —...— BXn,
BXi = -¡X + xtaX" is the Bessel operator, Yi>0, i=1,...,n.
It should also be noted that negative real powers of an operator LB called generalized B-hyperbolic potentials (see [16]) are constructed using function PA,+. Let us find singularities of (PA,+, ^)7. For this purpose
we transform (10) to bipolar coordinates (6) and using notation (5) for
integral (10) we obtain
(P7A + ,^)7 = y J (r2 — s2)V(r,s)rp+|Y'|-1sq+||-1drds. (11) 00
We now make change of variables u=r2, v=s2 in (IT):
e u
/t->x \ 1 i f/ \\ i / \ p+'y'' 1 q+'i''' 1 , ,
(P^, = - / (u — v)A^1 (u,v)u 2 1 s 2 1dudv, 00
where ^1(u, v)=^(r, s) when u=r2, v=s2. If we write v=ut then we obtain
(PA+,^)7= / uA+-1 $(A,u)du, (12)
where
$(A,u) = 1 /"(1 — t)A t -1 ^(u,tu)dt. (13)
The formula (12) shows that PA,+ has two sets of poles. The first consists of poles of $(A, u). Namely for t=1 function $(A,u) has singularity when
A = —1, —2,..., — k,... (14)
in which $(A,u) has simple poles with residues
^ X 1 (— 1)k-1 dk-1 r q +'y''' 2 l
res $(A,u) = ^, —^r t 2 ^(u,tu) . (15)
a=_k v ' y 4 (k — 1)! dtk
t=1
oo r
1
Moreover integral (12) has poles at the points
A = _ n + M _ n + |Y1 - 1 - n + M - k
2
2
2
where n = p + q, 7 = (7', 7"). Wherein
A= —
res
n+l Yl
-k
1 d* ( n + |y| ,
(P7V = m ITT M--^ _ k,u
k! duk
2
(16)
(17)
u=0
We have three cases. The first case is when a singular point A belongs
to the first set (14), but not to the second (16). The second case is when
singular point A belongs to the second (16), but A=_k, kGN. And the third case is when A belongs both to the first set (14) and the second
set (16). Let us now study each case separately in the following three theorems.
Theorem 1. If A=—k, k E N and n + |y|eR\N or n + |y|eN and n + |y|=2k — 1, kEN and also if n+1y| is even and k< n+,'7' the weighted generalized function PA,+ has simple pole with residue
A = (_1)k 1 ¿(k-1) (P) Y,+ = n- 1M °Y,1 (P).
res PA , =
A=_V y,+ " (k _ 1)!
(18)
Proof. Let us write $(A,u) in the neighborhood of A = —k in the form
$(A,u) = ^^ + $1 (A,u), $0(u) = res $(A,u), A + k A=-k
where function $1 (A, u) is regular at A=—k. We obtain
A+k
u
A+
n+ | y |
^-1$o(u)du+ / uA+^-1$1 (A, u)du. (19)
n+1 y |
The integrals in ([191) are regular functions of A at A=_k. Therefore
(PA +, p)7 has a simple pole at such a point and using (15) we have
res (PA,+,p) =
(_1)
k1
n±M_ k_ 1 d
k1
= k
4(k_1)!
u 2
dtk-1
q+lY'
t" 2 ' (u, tu)
du.
t=1
1
If in (20) we get tu = v then we may write
CO
Ar=e-sk(P"+= Ick-iY.J dV^
0
■ 9+Iy''I V 2
-1
(u, v)
p+1 y' 1
u 2
-1
du, (21)
where the integral is to be understood in the sense of its regularization for k > n. We now make the change of variables u = r2 and v = s2 in (8) and have
1 2
d
k- 1
q+l y'' i
1
dvk
1
^l(u,v)
u
p+1 y' 1
1
du, (22)
where
^i(u,v) = ^
^^x/uw', \fvw")w7dSpdSq
Formulas (21) and (22) imply (17). For k > n integral in (22) is to be
understood in the sense of its regularization. In the case when n+|Y|gR\N
or n + |y|gN and n + |y|=2k — 1, kGN regularization of the integral in (22) is defined by analytic continuation. This proves the desired result. □
Now we study the case when the singular point A is in the second set (16), but not in the first (14). If A=— — k, k=0,1, 2,..., and n + |y|gR\N or n + |y|gN and n + |yi =2k — 1, kGN, then function $(A, u) is regular in the neighborhood of A=— n+271 —k. Therefore function (PA,+, will have a simple pole with residue given by (17).
Before proceeding to the expression of the residue res
A=— -k
(P^)
through derivatives of function ^(x) at the origin we will obtain one useful formula. Consider the B-ultra-hyperbolic differential operator
Lb = B7i + ... + — B,
TP'+i
-B.
Yp+q,
„ d2 Yi d By = — + h
dx2 x, dx,
Applying an operator LB to quadratic form
/ \ *) OO *)
P(x)=x1 + ...+xp—xp+1 —...—xp+q, n = p + q, p> 1, q> 1
we obtain
LbPA+1 (x)=4(A+1) (A+ ^^ PA(x).
(23)
Theorem 2. Let n + |y| be not integer or n + |y|eN and n + |y|=2k — 1, kEN. When p + |y/| is not integer or p+|y/|eN, p+|y/|=2m—1, mEN and q + |y"| is even weighted functional PA,+ has simple poles at A=— n±'7' —k, kENU{0} with residues
n
^ nr m
(_1) q '2' ' 11 " v 2 res PA,+ = (0n+2kM x LB5y(x).
A=-n+Y -k 7'+ 2n+2kk! r fn+|7|
+k
If p + |y/| is even then weighted functional PA,+ is regular at A=— n+2'7' — k, kENU{0}. '
Proof. We first consider A=— n+,'7'. Using formula (fTT) we can write
1
res (PA,+,p)y=$( ,0)=^^^- /(1— t)-"^t^dt=
n+' 7 '
A = - 2
n+|Y | ^1(0, 0W n+' 7 ' q+' 7" '
V 2 ' 4
1 r(q±^) r (—n±M + 1)
=4 ^(0,0) v r (:p±M+1- . (24)
From the last formula we can see that if p+1y/ | is even then
A= —
res+ (P7A+,p)=0.
n+ ' 7 ' "
Now assume that p + |y/| is not integer or p + |y/|eN andp + |y/|=2k — 1, kEN and q+1y"| is even. We have
(0, 0)= ^(0, 0) = p(0) / [ ^7dSpdSq = p(0)|S± (p)|y |S± (q) |y,
(25)
Sp Sq
where
U r№) n rf^
i=1 + i=1
= ¿-wi+EEn, |S+^ = ¡q-iF7q±E^ (26)
2 / -1- \ 2
(see [1], p. 20, formula (1.2.5)). After some simple calculations, we obtain
n
S+WH n r №)
res+ (P7a + , y>)7 = ( 1)2w 2 ±, ,, m(0).
a=_ n+|7| " 1 ' 2n r( n+jrl J
Also we have
^ n r №)
res PA+ = (—)-i=1/ , ,N Ay (x). (27)
A=_n+|7| Y,± 2n r(
_(—1)M- v 2 2 r I 2 J
Using Green's theorem and formula (23) we derive
J (^(x)[LsPA±1(x)j — PA±1 (x)[Lb^(x)]) xYdx = 0,
{P (x)>0} +
therefore
(P-±,^)Y = 2(A + 1)(2A + n +H) (Py±±1,£B^• (28)
Then k-fold iteration of (28) leads to
(pA m) =_v Y,± ' B7_ (2Q)
( Y,± )Y 22k(A +1)...(A + k) (A+n+bl) ...(A+ n+bi+k—1) • ( ) Consequently
ne+s (P7a,± ,m)7 = re+s (P7A+k ,LB m)7 x
A=_ n+pl _k " A=_ n+M _k "
A= — n+M-k
22k(A+1)...(A+k) (A+ n±!Yl) ... (A+^+k—l) and
re+s (P7A+k, LBm)7 = res (P7A,±, LBm)7.
A=— n+hl-k A= — n^-L
1
Therefore if p + |y/| is even this residue vanishes. If p + |y/ | is not integer
or p + |y'|gN and p + |y'|=2k _ 1, kGN then (27) gives
res
A=— k
(pa±,p)y =
(_1)
n
^ nr №)
¿=1
2n+2kk!
r
n+|Y| 2
+k
(LB ¿y (x),p)
Y"
This completes the proof of Theorem 2. □
Theorem 3. If n+|Y| is even and p + |y/| and q + | y" I are also even, kENU{0}, then function PA,+ has a simple pole in A=— n+,'7' —k with residue
res PA±
, n+1 y l ■ Y,+ A=--2-
1
k Y,+ + k)
(_i) ^+k-15Y^+k-1)(P )+
/ . \ q+1 y'' l n
+<_У- n r
22k k!
¿=1
^ ) LB ¿Y (x)
If p+|y' | and q+1 y'' | are not integer or p+|y' |, q+1 y'' |gN and p+|y' | = = 2m _ 1, q+1 y'' | =2k _ 1, m, kGN then function PA,+ a pole of order two
at A=_ n±,|Y| _k. Coefficients c-2 and c-1 of expansion of function PA,±
in Laurent series at A = _ n~t|Y| _ k are expressed by formulas
c(0) = c-1 =
r
i±M
+k
n+ l y l i
(_1) ±'-¿<7" ±k-1)(p H^KF1
x
x II r (Y^ mf^^ ^ i^) ) LB¿Y(x)
¿=1
2
2
2
ik) - ±1
sin n r ( Yif1 )
¿=1
c(k = (_1)-
where ^(x) = r^X)
/ ±1 I±M "LB¿Y(x),
2n±2kk!nW n±|Y|±k N
1
Proof. Let n+'Y| be even and A=— n±Y—k, kENU{0}. We express this (PA,+, p)Y in the form
(P7a + ,p)7=^ J uA+ ^- 1 $o(u)du^ uA+ ^- 1 $ i(A, u)du, (30)
00
where $0(u)= res $(A, u) and $1(A, u) is a regular at A=— n±2'7' —k
A= — k
function. By virtue of the proposal each integral in (30) may have at
A=—n±bl—k a simple pole therefore function (PA,+, p)Y may have a pole
of order two at A=—n±2Y—k. In the neighborhood of such a point we may expand PA,+ in the Laurent series
c(k) c(k) P A = _-2__|__-1__L
(A + n±±H + k)2 + A + + k
Let us find c(k), . We have
(c(k^, p)7 = res / uA+^-1$o(u)du = ■1$0k) (0). a=-™±m-kJ k!
20
If k = 0 then cL0) = $0(0). According to (13)
$0(0) = ^^i(0, 0) res /(1 — t)Atq+'72'' 2 dt =
4 A= — n+' 7 '
2 0
r(q±P) r(A +1)
= ^1(0, 0) res —^-J--).
a=-^ 4r (A + + 1]
Considering that ^1(0, 0)=p(0)|S± (p)|y |S± (q)|y where |S± (p)|y and |S±(q)|7» were determined in (26) we obtain
(c-02,p)7 =
( 1)B (p+lVI q+li"I) . . ... _ (—1) 2 , . n(p + 'y/I)
4n 2
si^ n IS± (p) 17, | S± (q) 17» p (0)
l
When p + |y' | is even (in this case q + |y'' I is also even) we have c^ = 0 i.e. function (PY,+, ^)7 has a simple pole at A = — n+,171. If p + |y' | is not integer or p + |y'|gN and p + |y'I =2k — 1, kGN then
sin ft r ()
c-2 = (—1) 2 +1-(^-¿7 (x).
„(0) _ +1_i=L
±1
2 J
As well as in Theorem 2 we obtain that if p + |y' | and q + |y'' | are even then function PY,± has a simple pole at A = — n±2lYl — k. If p+|y'| and q +|y'' I are not integer or p+|y'I, q+1y''|gN andp+|y'|=2m—1, q+1Y''|=2k—1, m, kGN then
sin n(p±Ml ft r (^)
c(k2 = (—1) ^+1-71 ^ LB ¿7 (x).
-2 ( ) 2n±2kn±M±k) 7( )
Let's find c(k1. We have
(¿k1,^)= / u-k-1 $o(u)du+
+ res /uA± ^-1 — ^ — k,u)du A=-n+M -J \ 2 '
20
Since $0(u) = res $(A, u) then using the formulas (15) and (22) we
A=-k
obtain
/u-k-1*0(u)du = (~±),"+^±k-1 x f±k-1)(P), ^
0 r(=±w+k—1
Thus
r„e+s7l fc /uA±^—— k,u)du =
2 0
1 dk $1 (— ^ — k,u)
k! duk
= (<^)7
u=0
. . n+ ' 7 ' +7 _i
and c- = 7—+) ,2 + - ) +k-1)(P) + af.
r(+ k — 1J 7' 7
For k = 0 we obtain
(a<0),p)7 = $ i (— ^,e).
In order to find $ 1 (— n±ili, 0) we consider $(A, 0). Using ((24), (SB) and
(26) we obtain
n
r(A +1) n r (^)
$(A, 0) = p(0) i=1
2n^p±M\ r (A + + 1
Taking into account the formula r(1 — x)r(x) = ^J1" we can write
sin n7A + q+p) r (—A — ^) n r (^)
$(A,0) =-^--/ , i=1-p(0).
(,) sin nA r( p±Mj r(—A) p()
If p+Iy/1 and q+1y"I are even then
r sin n (A+ ^ )
lim ------ = (—1) 2 ,
sin nA
hence function $(A, 0) is regular at A=—n±J^ and
n + 'y' ( n + 'y'
--^, — 2 whence
n ( ) + „. n r ()
«> ,p)Y = (—1) *HM ^ p(0).
Ifp+|Y/| and q+1y"I are not integer or p+'y/I, q+1Y"|EN andp+|Y/|=2m—1, q+1y"|=2k—1, m, kEN then $(A, 0) has a pole at A = — n±ri. In this case
«Y") ,P)y = —^, <>) = (—1) ^-1 n r' Yi + 1 2 i=1
x ■
sin ( n
p+Iy ; I 2
- ^
i+IyI
r
n+|Y| 2
¥>(0),
where ^(x) = r^X) • We obtain
c(0) = c-1 =
r
î+Jy!
(-1) ^-142
+ iyi
-1)
(P ) + 057 (x)
with a value
»=(-1) ^n r
i=1
Yi + 1
if p+|Y/| and 1y"I are even. If p+|yand q+|Y"I are not integer or p+iy'i, q+IVI^N and p+IY|=2m-1, q+|Y"|=2k-1, m, keN then
» = (-1) ^r( Y^ smfP+Mn , x
i=1
2
V 2
x
^(p+jyt) vn + m
2
2
Finally, in order to obtain c^l for arbitrary k, we again use the formula
(29). This proves the desired result. □
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Received October 12, 2016.
In revised form, December 06, 2016. Accepted December 06, 2016.
Voronezh State University
1, Universitetskaya pl., 394006 Voronezh, Russia
E-mail: [email protected]
