Namangan Institute of Engineering and Technology
nammti.uz
10.25.2023
Pg.583
1
0
1
N
j
j
c
(1´)
Spherical Sobolev spaces are considered as classes of
s
H
integrands. The parameter
s
can
be a fractional number, with
1 / 2
s
.
The error of the cubature formula (1) is the difference
2
1
0
1
0
2
2
0
1
0
1
( , )
( ) ( )
(
)
( )
( )
(
)
( )
( )
N
N
j
j
j
j
j
l
f
p
f
d
c f
p
p
c
f
d
p
where
( )
x
is the Dirac delta function,
2
0
1
( )
(
)
( )
N
j
j
p
l
c
p
is the error functional of the cubature formula.
Task. Among all cubature formulas of the form (1) with the condition (1´), find one whose
error functional has the minimum
s
H
- norm.
If there is such a cubature formula, then we call it
s
H
- optimal.
Theorem. Among all cubature formulas of the form (1) with a given set of nodes
0,1, 2,...,
1
j
j
N
on the unit circle and weights satisfying the condition
1
0
1
N
j
j
c
, there
is exactly one for which the
s
H
-norm of the error functional,
1 / 2
s
, takes the minimum
possible value.
References
1. Sobolev S.L., Vaskevich V.L. Cubature formulas. Novosibirsk: Publishing House of the
Institute of Mathematics, 1996.
2. Vaskevich V.L. Error, conditionality and guaranteed accuracy of multidimensional
spherical cubatures // Sibirsk. mat. jur..2012. T. 53, No. 6.P. 1245-1262.
3. Salikhov G.N. Cubature formulas for multidimensional spheres. Tashkent: FAN, 1985.
