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Necessary conditions of summability of spectral expansion on eigenfuction of the operator laplace.

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Necessary conditions of summability of spectral expansion on eigenfuction of the operator laplace
The proving of theorem 3. Let 2λ1 ≥ 1. We Make sure
that the multi-valued mapping D (t ) , 0 ≤ t ≤ T is
0
weakly invariant. For all z (⋅) ∈ D ( 0 ) we choose the control
u (t , x ) = 0, t ∈ [0,T ] , x ∈∂Ω, i. e.
∫ u (t , s )ϕk ( s )ds = 0, k = 1, 2,... . Then,
∂Ω
T
z (⋅, ⋅) = ∫ z (t , ⋅) dt =
2
2
0
1 − e −2 λ T b 2
≤
≤ b2.
2λ1
2λ1
k =1 0
Consequently, z (⋅, ⋅) ∈ D (t ) for all 0 ≤ t ≤ T , i. e.
D (t ) , 0 ≤ t ≤ T is weakly invariant concerning the problem
(1)–(3). Theorem 3 is proved.
∞ T
= ∑ ∫ ( z k0e − λ t ) dt ≤ b 2
k
2
1
References:
1. Egorov A. I. The optimal control of the heat and diffusion processes. Nauka Moscov, 1978 (in Russian).
2. Feuer A., Heymann M. Ω — invariance in control systems with bounded controls. J. Math.Anal.And Appl.53, no.2 (1976),
266–276.
3. Tukhtasinov M., Kh.Ya. Mustapokulov. Invariant sets in systems with distributed parameters. Theses of the reports of
the International Russian-Balkarsky symposium. Nalchik city. 2010. 232–233. (in Russian).
Pirmatov Shamshod Turgunboevich,
Head of Higher mathematics department,
candidate of physico-mathematical sciences,
an associate professor of Tashkent state technical
university named after Abu Raikhon Beruni
E‑mail: shamshod@rambler.ru
Necessary conditions of summability of spectral
expansion on eigenfuction of the operator laplace
Abstract: The spectral decomposition connected with self-conjugate expansion are considered the operator
Laplace in N of dimensional area. It is proved that if spectral decomposition of any function in some point is summarized by Riesz’s means, its average value about α in the specified point possesses the generalized continuity.
Keywords: eigenfuction, eigenvalues, spectral expansion, local bounded variation, Riesz’s means, generalized
continuity, summability, operator Laplace.
Introduction. The question on sufficient conditions at
which performance it is possible to approximate function
by its spectral decomposition connected with self-interfaced
expansion of the elliptic operator, by present time is well studied and in detail shined in the mathematical literature as in our
country, and abroad. Recently interest to these problems has
noticeably increased, and the delicate questions connected
with spectral decomposition of rough functions have undergone to research more.
From the mathematical literature well-known the examples showing, that spectral decomposition can converge
even in those points where decomposed function has break so
the usual requirement of smoothness is not necessary. However in all these examples functions in some the generalized
sense nevertheless is continuous in a considered point. This
fact for the spectral decomposition ad equating to self-interfaced expansion of operator Laplace in any multivariate area
for the first time has been established in E. C. Titchmarsha’s
work [1, 255–258]. Corresponding results have been received
by them for Riesz means of spectral decomposition in case of
when the order of averages is an integer, and also in the assumption, that dimension of considered area is not so great.
Conseder the following orthonormal system of eigenfuctions of Laplace operator:
− ∆uk (x ) = λkuk (x ),
x ∈Ω ⊂ RN
Define the spectral expansions:
E λ (x , f ) = ∑ f kuk (x ) .
λk <λ
We introduce Riesz means of order s ≥ 0 :
s
s
λ
 λk 
 t 
s
E λ f (x ) = ∑  1 −  f kuk (x ) = ∫  1 −  dEt f .
λ
λ
λ <λ 
0 
Denote
k
α
2
−N

y 
R
S f (x ) =
 f (x + y )dy ,
1 −
ω(α , N ) y ∫<R  R 2 
where
α
R
ω(α , N ) =
∫
y ≤1
(1 − y ) dy .
2 α
Theorem1. Let f ∈ L2 (Ω) and s ≥ 0 . Suppose that for some
x ∈Ω the following ratio is carried out
lim
E s f (x ) = A.
λ →∞ λ
Then for any α > s − (N − 2) 4 the following equality
lim
S α f (x ) = A
R →0 R
is valid.
29
Section 6. Mathematics
First we prove some Propositions.
ν 1
Proposition 1. Let −1 < s < − . Then
2 4
∞
−
s
s +1 − s −1
∫ z (z − 1) Jν (a z )dz = Γ(s + 1)2 a Jν −s −1(a).
ν
λ
g s (λ ) = ∫ (1 − λt )s dg (t ).
0
If l > β + s + 1 2 then the following equality
2
∞
∫ (R
1
For the proof we can refer, for example, on [3, 717–718].
Set
(l +s +1)
−
A
s
 ∫ (λ − t ) λ 2 J l +s +1 (R λ )d λ , 0 ≤ t ≤ A ,
Φ A (t , R) =  t

0,
t > A.
Proposition 2. Let l > s −1 2 and s > −1 . Then the following equality
limΦ
A
A →∞
(t , R) = t
( s −l +1)
2
t
Φ A (t , R) = ∫ t st
0
=2
Then
(u − 1)su
A
=t
t
∫ (u − 1) u
s
− 2l − 12 − 2s
J l +s +1 (R tu )du.
1
Φ (t ,u) t
A
(l −s −1)
−
2
−(l +12+s )
J
[(R t ) u ]du ≤
= ∫ (u − 1)s u
l +1+s
1
A
t
s l 1
∞ − −
(R ut ) du ≤
≤ ∫ u2 2 2 J
l +1+s
1
s l 3
− −
∞
−
1
4
2
≤ C (R)t
∫ u 2 4 cos(R tu )du =
1
= C (R)t
1∞
−
4
∫v
s −l −
3
2
cos(R t v )dv ≤
1
1
1
≤ C (R)t ∞ v s −l − 2 cos(R t v )dv =C (R)t − 4 .
∫
1
Q. E.D.
Proposition 4. Let g (t ) be the function of local bounded variation on the half-line t ≥ 0 and for some β > 0 the following inequality
−
1
4
∞
∫t
0
is valid. Set
30
−β
dg (t ) < ∞
∫
l +1+s
A
0
J l +s +1 (R tu )tdu =
We set ν = l + 1 + s .Then since l > s −1 2 and s > −1 we
have
ν − 1 4 = l 2 + s 2 + 1 4 = (l − s + 1 2) 2 + s > s ,
or
−1 < s < ν 2 − 1 4 .
In this case the required equality follows from Proposition1.
Q. E.D.
Proposition 3. Let l > s +1 2 . The following inequality
Φ A (t , R) ≤ Φ(t , R),
t ≥1,
where
Φ(t , R) = C (R)t −(l −s −1 2) 2
is valid.
Proof. We have
λ
I A = ∫ ( λ )−l −1+s J l +s +1 (R λ )g s (λ )d λ =
 λ  t s

= ∫λ
J l +s +1 (R λ )  ∫  1 −  dg (t ) d λ =
0
0  λ 

l 1 s
λ
A
− − +


= ∫ λ 2 2 2 J l +s +1 (R λ )  ∫ (λ − t )s λ − sdg (t ) d λ =
0
0

A
− 2l − 12 − 2s
.
0
− 2l − 12 − 2s
R s +1−l ∞
− l −1+s
s
∫ ( λ ) J l +1+s (R λ )E fd λ
Γ(s + 1) 0
A
t
s l 1
− +
2 2 2
− s −1
is valid.
Proof. Consider the following integral
A
I = ( λ )−l −1+s J (R λ )g s (λ )d λ .
Γ(s + 1)2s +1 (R t )− s −1 J l (R t )
is valid.
Proof. We have
A
t )−l J l (R t )dg (t ) =
l 1 s
− − +
2 2 2
A
A
0
t
= ∫ dg (t )∫ (λ − t )s λ
l 1 s
− − −
2 2 2
J l +s +1 (R λ )d λ =
A
= ∫ Φ A (t , R)dg (t ) .
Hence,
0
∞
I A = ∫ Φ A (t , R)dg (t ).
0
Remaind l > s −1 2. In this case, according to Proposition 2,
Φ A (t , R) = Γ(s + 1)2s +1 (R t )− s −1 J l (R t )
lim
A →∞
Further, since l > s +1 2 according to Proposition 3,
∞
∞
1
1
∫ Φ(t ,u) dg (t ) ≤ C (h)∫ t
−(l −1−1 2 ) 2
∞
dg (t ) ≤ C ∫ t − β dg (t ) < ∞
1
Hens, we may apply theorem of Lebesgue:
∞
I = lim
I = lim
Φ A (t , R)dg (t ) =
A →∞ A
A →∞ ∫
0
∞
l
= Γ(s + 1)2s +1 R − s −1 ∫ t 2 J l (R t )dg (t ).
0
Q. E.D.
From this moment we set l = N 2 + α .
Proposition 5. Let l = N 2 + α , where α > −1 2 . Then
∞
2l Γ(l + 1) (R λ )−l J (R λ )dE f (x ) = S α f (x ) ,
l
λ
R
∫
0
and integral converges absolutely and uniformly.
For the proof we can refer on [1, 255–258].
Proposition 6. Set g (λ ) = E λ f (x ) . Let β > N 4 . Then
∞
λ − β dg (t ) ≤ C f .
∫
0
Proposition 7. Let l = N 2 + α , where α > s − (N − 2) 4 .
Then
Γ(l + 1) s +1−l ∞
SRα f (x ) = 2l −s −1
R ∫ ( λ )−l −1+s J l +1+s (R λ )E λs fd λ .
Γ(s + 1)
0
Necessary conditions of summability of spectral expansion on eigenfuction of the operator laplace
Proof follows from Proposition 2 and 3.
Corollary. Let l = N 2 + α , where α > s − (N − 2) 4 .
Then
∞
α
l −s −1 Γ(l + 1)
g (λ R −2 )( t )−l −1+s J l +1+s ( t )dt .
SR f (x ) = 2
∫
Γ(s + 1) 0
Proposition 8. Let l = N 2 + α , where α > s − (N − 2) 4 .
Then
∞
∫t
−(l +1−s ) 2
J l +1+s ( t )dt < ∞ .
0
Proof of the Theorem1 follows from Propositions 7, 8.
Q. E.D.
References:
1.
2.
3.
4.
Titchmarsh E. C. Eigenfunction Expansions Associated with Second Order Differential Equations, part II, Oxford, 1958.
Bochner S. Summation of multiple Fourier series by spherical means. Trans. Amer. Math. Soc., Volume 40, 1936.
Gradshteyn, Ryzhik, Tables of Integrals and Sums. M.1963.
Alimov Sh. A., Ilyin V. A. Condition exactes de convergence uniforme des developpements spectraux et de leurs moyennes
de Riesz pour une extension autoadjoint arbitraire de l’operateur de Laplace. Comptes Rendus Acad. Scien. Paris, Serie
A, t. 237, 1970.
5. Sogge C. D. Eigenfunction and Bochner-Riesz estimates on manifolds with boundary, Mathematical Research Letter, Volume 9, 2002.
6. Alimov Sh. A. Sets of uniform convergence of Fourier expansions of piecewise smooth functions, J. of Fourier Analysis and
Applications, Volume 10 (6), 2004.
31
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