共查询到20条相似文献，搜索用时 46 毫秒
1.
We study the rate of uniform approximation by Nörlund means of the rectangular partial sums of double Fourier series of continuous functions f( x, y), 2 π-periodic in each variable. The results are given in terms of the modulus of symmetric smoothness defined by $$\begin{gathered} \omega _2 \left( {f,\delta _1 ,\delta _2 } \right) = \mathop {\sup }\limits_{x,y} \mathop {\sup }\limits_{\left| u \right| \leqslant \delta _1 ,\left| v \right| \leqslant \delta _2 } \left| {f\left( {x + u,y + v} \right)} \right. + f\left( {x + u,y - v} \right) + f\left( {x - u,y + v} \right) \hfill \\ + \left. {f\left( {x - u,y - v} \right) + 4f\left( {x,y} \right)} \right| for \delta _1 ,\delta _2 \geqslant 0. \hfill \\ \end{gathered} $$ As a special case we obtain the rate of uniform approximation to functions f( x,y) in Lip({α, β}), the Lipschitz class, and in Z({α, β}), the Zygmund class of orders α and β, 0< α, β ≤ l, as well as the rate of uniform approximation to the conjugate functions \(\tilde f^{(1,0)} (x,y), \tilde f^{(0,1)} (x,y)\) and \(\tilde f^{(1,1)} (x,y)\) . 相似文献
2.
Using the following notation: C is the space of continuous bounded functions f equipped with the norm
, V is the set of functions f such that
, the set E consists of fCV and possesses the following property:
is summable on each finite interval,
we establish some assertions similar to the following theorem: Let
0$$
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, Then for fV the series uniformly converges with respect to
and the following equality holds: This theorem develops some results obtained by Zubov relative to the approximation of probability distributions. Bibliography: 4 titles. 相似文献
3.
We prove that an absolute constant c>0 exists such that |
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4.
The main result proved in the paper is: if f is absolutely continuous in (–, ) and f' is in the real Hardy space Re H
1, then
for every n1, where R
n(f) is the best uniform approximation of f by rational functions of degree n. This estimate together with the corresponding inverse estimate of V. Russak [15] provides a characterization of uniform rational approximation.Communicated by Ronald A. DeVore. 相似文献
5.
For f L
n
( T
d
) and
, the modulus of smoothness is shown to be equivalent to where T
n is the best trigonometric polynomial approximant of degree n to f in L
p and is the Laplacian. The above result is shown to be incorrect for 0 < p
. 相似文献
6.
Let T
n
(x) be trigonometric polynomial of degree n, and
be conjugation of T
n
(x). In this paper we obtain the complete asymptotic expansion for |
相似文献
7.
Let u( x) xR
q
be a symmetric nonnegative definite function which is bounded outside of all neighborhoods of zero but which may have u(0)=. Let p
x, (·) be the density of an R
q
valued canonical normal random variable with mean x and variance and let { G
x, ; ( x, ) R
q
×[0,1 ]} be the mean zero Gaussian process with covariance A finite positive measure on R
q
is said to be in
with respect to u, if
When
, a multiple Wick product chaos
is defined to be the limit in L
2, as 0, of where ,
denotes the Wick product of the m
j
normal random variables
.Consider also the associated decoupled chaos processes
,
defined as the limit in L
2, as 0, of where
are independent copies of G
x,.Define Note that a neighborhood of the diagonals of
in
is excluded, except those points on the diagonal which originate in the same Wick product in (i). Set
One of the main results of this paper is:
Theorem A. If
is continuous on ( R
q
)
r
for all
then
is continuous on
.When u satisfies some regularity conditions simple sufficient conditions are obtained for the continuity of
on ( R
q
)
r
. Also several variants of (i) are considered and related to different types of decoupled processes. These results have applications in the study of intersections of Lévy process and continuous additive functionals of several Lévy processes. 相似文献
8.
In this paper we prove that the maximal commutator of singular integral operator [b, T] * satisfies the inequality: where f is any smooth function with compact support, λ>0 and C is a positive constant independent of f and λ. 相似文献
9.
Let
be the Fejér kernel, C be the space of contiuous 2π-periodic functions f with the norm
, let
be the Jackson polynomials of the function f, and let
be the Fejér sums of f. The paper presents upper bounds for certain quantities like
which are exact in order for every function f ∈ C. Special attention is paid to the constants occurring in the inequalities obtained. Bibliography: 14 titles.
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 357, 2008, pp. 90–114. 相似文献
10.
It is proved that if P( D) is a regular, almost hypoelliptic operator and
|
$
L_{2,\delta } = \left\{ {u:\left\| u \right\|_{2,\delta } = \left[ {\int {\left( {|u(x)|e^{ - \delta |x|} } \right)^2 dx} } \right]^{1/2} < \infty } \right\},\delta > 0,
$
L_{2,\delta } = \left\{ {u:\left\| u \right\|_{2,\delta } = \left[ {\int {\left( {|u(x)|e^{ - \delta |x|} } \right)^2 dx} } \right]^{1/2} < \infty } \right\},\delta > 0,
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11.
The paper deals with the system where
and
are
-matrix functions;
is a boundary control;
is the solution. The singularities of the fundamental solution corresponding to the controls
(
is the Dirac
-function) are under investigation. In the case of
, the singularities of the fundamental solution are described in terms of the standard scale
. In the presence of points
an interesting effect occurs: singularities of intermediate (fractional) orders appear. Bibliography: 1 title. 相似文献
12.
Suppose that X is a Banach space, K denotes the set of real numbers R or the set of nonnegative real numbers R
{+},
is a family of linear operators from X into X such that T
0=I is the identity operator in X,
for all
, and there exists M such that
for all
. The expression
is called the rth order modulus of continuity of an element x with step h in the space X with respect to the family A(K). The properties of
are studied. Bibliography: 3 titles. 相似文献
13.
This is mainly concerned with the following functional equation: |
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14.
In this note we prove the following theorem: Let u be a
harmonic function in the unit ball
and
. Then there is a
constant C =
C( p,
n) such that
. 相似文献
15.
For 2-periodic functions
and arbitrary q [1, ] and p (0, ], we obtain the new exact Kolmogorov-type inequality
which takes into account the number of changes in the sign of the derivatives ( x
(k)) over the period. Here, = ( r – k + 1/ q)/( r + 1/ p),
r
is the Euler perfect spline of degree r,
and
. The inequality indicated turns into the equality for functions of the form x( t) = a
r
( nt + b), a, b R, n N. We also obtain an analog of this inequality in the case where k = 0 and q = and prove new exact Bernstein-type inequalities for trigonometric polynomials and splines. 相似文献
16.
Let f C[0, 1], k = 5, 6, 7. We prove that if f( i/( k – 1)) = 0, i = 0, 1,..., k – 1, then
相似文献
18.
In this paper, we prove that the maximal operatorsatisfiesis homogeneous of degree 0, has vanishing moment up to order M and satisfies Lq-Dini condition for some 相似文献
19.
In what follows, C is the space of
-periodic continuous real-valued functions with uniform norm,
is the first continuity modulus of a function
with step h, H
n is the set of trigonometric polynomials of order at most n,
is the set of linear positive operators
(i.e., of operators such that
for every
),
is the space of square-integrable functions on
, It is proved that
coincides with the smallest eigenvalue of some matrix of order n+1. The main result of the paper states that, for every
does not exceed and, for
, is equal to the minimum of the quadratic functional over the unit sphere of
. Then it is calculated that
Bibliography: 19 titles. 相似文献
20.
Estimates for deviations are established for a large class of linear methods of approximation of periodic functions by linear combinations of moduli of continuity of different orders. These estimates are sharp in the sense of constants in the uniform and integral metrics. In particular, the following assertion concerning approximation by splines is proved: Suppose that
is odd,
. Then moreover, for
it is impossible to decrease the constants on
. Here,
are some explicitly constructed constants,
is the modulus of continuity of order r for the function f, and
are explicitly constructed linear operators with the values in the space of periodic splines of degree
of minimal defect with 2 n equidistant interpolation points. This assertion implies the sharp Jackson-type inequality . Bibliography: 17 titles. 相似文献
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