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1.
стАтьь ьВльЕтсь пРОД ОлжЕНИЕМ пРЕДыДУЩЕИ ОДНОИМЕННОИ РАБОты АВтОРА, гДЕ ИжУ ЧАлсь пОРьДОк ВЕлИЧИН пРИ УслОВИьх, ЧтО α>-1/2, Рα >- 1 И ЧтО МАтРИцА t nk УДОВлЕтВОРьЕт НЕкОт ОРОМУ УслОВИУ РЕгУльРНОстИ. жДЕсь ДОкАжыВАЕтсь, Ч тО ЕслИ f∈ H Ω, тО ВыпОлНь Етсь ОцЕНкА $$\left\{ {\frac{1}{{\lambda _n }}\mathop \Sigma \limits_{k = n - \lambda _n + 1}^n \left| {\sigma _k^\alpha \left( x \right) - f\left( x \right)} \right|^p } \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} = O\left( {\left\{ {\frac{1}{{\lambda _n }}\mathop \Sigma \limits_{k = n - \lambda _n + 1}^n \left( {\frac{1}{k}\mathop \smallint \limits_{{1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-\nulldelimiterspace} k}}^{2\pi } \frac{{\omega \left( t \right)}}{{t^2 }}dt} \right)^p } \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} + \left( {\frac{{\lambda _n }}{n}} \right)^\alpha \omega \left( {\frac{1}{n}} \right)} \right)$$ (λ 1=1, λ n+1-λ n≦1), А тАкжЕ ЧтО Ёт А ОцЕНкА ОкОНЧАтЕльН А В сВОИх тЕРМИНАх; пОДОБ НыИ РЕжУль-тАт спРАВЕДлИВ тАкжЕ И Дль сОпРьжЕННОИ ФУНкцИИ . ДОкАжыВАЕтсь, ЧтО Усл ОВИь α>?1/2 И pα>?1, кОтОРыЕ Б ылИ НАлОжЕНы В УпОМьНУтО И ВышЕ ЧАстИ I, сУЩЕстВЕН Ны. 相似文献
2.
We introduce and study new separation axioms in generalized topological spaces, namely,
m- T\frac14\mu\mbox{-}T_{\frac{1}{4}},
m- T\frac38\mu \mbox{-}T_{\frac{3}{8}} and
m- T\frac12\mu\mbox{-}T_{\frac{1}{2}}.
m- T\frac14\mu\mbox{-}T_{\frac{1}{4}} spaces are strictly placed between μ- T
0 spaces and
m- T\frac38\mu\mbox{-}T_{\frac{3}{8}},
m- T\frac38\mu\mbox{-}T_{\frac{3}{8}} spaces are strictly placed between
m- T\frac14\mu\mbox{-}T_{\frac{1}{4}} spaces and
m- T\frac12\mu \mbox{-}T_{\frac{1}{2}} spaces, and
m- T\frac12\mu\mbox{-}T_{\frac{1}{2}} spaces are strictly placed between
m- T\frac38\mu\mbox{-}T_{\frac{3}{8}} spaces and μ- T
1 spaces. 相似文献
3.
Recently, Girstmair and Schoissengeier studied the asymptotic behavior of the arithmetic mean of Dedekind sums
\frac1j( N) ? 0 £ m < Ngcd(m,N)=1 | S( m, N)|\frac{1}{\varphi(N)} \sum_{\mathop{\mathop{ 0 \le m< N}}\limits_{\gcd(m,N)=1}} \vert S(m,N)\vert
, as N → ∞. In this paper we consider the arithmetic mean of weighted differences of Dedekind sums in the form
Ah( Q)=\frac1? \fracaq ? FQh(\frac aq) ×? \fracaq ? FQh(\frac aq) | s( a¢, q¢)- s( a, q)|A_{h}(Q)=\frac{1}{\sum_{\frac{a}{q} \in {\cal F}_{Q}}h\left(\frac{a}{q}\right)} \times \sum_{\frac{a}{q} \in {\cal F}_{\!Q}}h\left(\frac{a}{q}\right) \vert s(a^{\prime},q^{\prime})-s(a,q)\vert
, where
h:[0,1] ? \Bbb Ch:[0,1] \rightarrow {\Bbb C}
is a continuous function with
ò 01 h( t) d t 1 0\int_0^1 h(t) \, {\rm d} t \ne 0
,
\frac aq{\frac{a}{q}}
runs over
FQ{\cal F}_{\!Q}
, the set of Farey fractions of order Q in the unit interval [0,1] and
\frac aq < \frac a¢q¢{\frac{a}{q}}<\frac{a^{\prime}}{q^{\prime}}
are consecutive elements of
FQ{\cal F}_{\!Q}
. We show that the limit lim
Q→∞
A
h
( Q) exists and is independent of h. 相似文献
4.
In this work, we consider the Jacobi-Dunkl operator Λ
α,β
,
a 3 b 3 \frac-12\alpha\geq\beta\geq\frac{-1}{2}
,
a 1 \frac-12\alpha\neq\frac{-1}{2}
, on ℝ. The eigenfunction
Y la,b\Psi_{\lambda}^{\alpha,\beta}
of this operator permits to define the Jacobi-Dunkl transform. The main idea in this paper is to introduce and study the Jacobi-Dunkl
transform and the Jacobi-Dunkl convolution product on new spaces of distributions 相似文献
5.
For x = ( x 1, x 2, ..., x n ) ∈ ℝ + n , the symmetric function ψ n ( x, r) is defined by $\psi _n (x,r) = \psi _n \left( {x_1 ,x_2 , \cdots ,x_n ;r} \right) = \sum\limits_{1 \leqslant i_1 < i_2 \cdots < i_r \leqslant n} {\prod\limits_{j = 1}^r {\frac{{1 + x_{i_j } }}
{{x_{i_j } }}} } ,$\psi _n (x,r) = \psi _n \left( {x_1 ,x_2 , \cdots ,x_n ;r} \right) = \sum\limits_{1 \leqslant i_1 < i_2 \cdots < i_r \leqslant n} {\prod\limits_{j = 1}^r {\frac{{1 + x_{i_j } }}
{{x_{i_j } }}} } , 相似文献
6.
We prove that max |p′(x)|, where p runs over the set of all algebraic polynomials of degree not higher than n ≥ 3 bounded in modulus by 1 on [−1, 1], is not lower than
( n - 1 ) \mathord | / |
\vphantom ( n - 1 ) ?{1 - x2} ?{1 - x2} {{\left( {n - 1} \right)} \mathord{\left/{\vphantom {{\left( {n - 1} \right)} {\sqrt {1 - {x^2}} }}} \right.} {\sqrt {1 - {x^2}} }} for all x ∈ (−1, 1) such that
| x | ? èk = 0[ n \mathord | / |
\vphantom n 2 2 ] [ cos\frac2k + 12( n - 1 )p, cos\frac2k + 12np ] \left| x \right| \in \bigcup\nolimits_{k = 0}^{\left[ {{n \mathord{\left/{\vphantom {n 2}} \right.} 2}} \right]} {\left[ {\cos \frac{{2k + 1}}{{2\left( {n - 1} \right)}}\pi, \cos \frac{{2k + 1}}{{2n}}\pi } \right]} . 相似文献
7.
It is proven that the set of eigenvectors and generalized eigenvectors associated to the non-zero eigenvalues of the Hilbert-Schmidt
(non nuclear, non normal) integral operator on L2(0, 1)
[Ar (a)f](q) = ò01 r( \fracaq x )f(x)dx [A_{\rho } (\alpha )f](\theta ) = {\int_0^1 {\rho {\left( {\frac{{\alpha \theta }} {x}} \right)}f(x)dx} } 相似文献
8.
We consider a new Sobolev type function space called the space with multiweighted derivatives $
W_{p,\bar \alpha }^n
$
W_{p,\bar \alpha }^n
, where $
\bar \alpha
$
\bar \alpha
= ( α
0, α
1,…, α
n
), α
i
∈ ℝ, i = 0, 1,…, n, and $
\left\| f \right\|W_{p,\bar \alpha }^n = \left\| {D_{\bar \alpha }^n f} \right\|_p + \sum\limits_{i = 0}^{n - 1} {\left| {D_{\bar \alpha }^i f(1)} \right|}
$
\left\| f \right\|W_{p,\bar \alpha }^n = \left\| {D_{\bar \alpha }^n f} \right\|_p + \sum\limits_{i = 0}^{n - 1} {\left| {D_{\bar \alpha }^i f(1)} \right|}
,
$
D_{\bar \alpha }^0 f(t) = t^{\alpha _0 } f(t),D_{\bar \alpha }^i f(t) = t^{\alpha _i } \frac{d}
{{dt}}D_{\bar \alpha }^{i - 1} f(t),i = 1,2,...,n
$
D_{\bar \alpha }^0 f(t) = t^{\alpha _0 } f(t),D_{\bar \alpha }^i f(t) = t^{\alpha _i } \frac{d}
{{dt}}D_{\bar \alpha }^{i - 1} f(t),i = 1,2,...,n
相似文献
9.
Subject to the abc-conjecture, we improve the standard Weyl estimate for cubic exponential sums in which the argument is a
quadratic irrational. Specifically. we show that
?n \leqslant N e( an3 ) << e, aN\tfrac57 + e \sum\limits_{n \leqslant N} {e\left( {\alpha {n^3}} \right){ \ll_{\varepsilon, \alpha }}{N^{\tfrac{5}{7} + \varepsilon }}} 相似文献
10.
Let Hk\mathcal{H}_{k} denote the set { n∣2| n,
n\not o 1 (mod p)n\not\equiv 1\ (\mathrm{mod}\ p) ∀ p>2 with p−1| k}. We prove that when
X\frac1120(1-\frac12k) +e\leqq H\leqq XX^{\frac{11}{20}\left(1-\frac{1}{2k}\right) +\varepsilon}\leqq H\leqq X, almost all integers
n ? \allowbreak Hk ?( X, X+ H]n\in\allowbreak {\mathcal{H}_{k} \cap (X, X+H]} can be represented as the sum of a prime and a k-th power of prime for k≧3. Moreover, when
X\frac1120(1-\frac1k) +e\leqq H\leqq XX^{\frac{11}{20}\left(1-\frac{1}{k}\right) +\varepsilon}\leqq H\leqq X, almost all integers n∈( X, X+ H] can be represented as the sum of a prime and a k-th power of integer for k≧3. 相似文献
11.
We take up in this paper the existence of positive continuous solutions for some nonlinear boundary value problems with fractional
differential equation based on the fractional Laplacian
(-D |D) \fraca2{(-\Delta _{|D})^{\frac{\alpha }{2}}} associated to the subordinate killed Brownian motion process ZaD{Z_{\alpha }^{D}} in a bounded C
1,1 domain D. Our arguments are based on potential theory tools on ZaD{Z_{\alpha }^{D}} and properties of an appropriate Kato class of functions K
α
( D). 相似文献
12.
By means of a method of analytic number theory the following theorem is proved. Let p be a quasi-homogeneous linear partial differential operator with degree m, m > 0, w.r.t a dilation
given by ( a 1, …, a n). Assume that either a 1, …, a n are positive rational numbers or
for some
Then the dimension of the space of polynomial solutions of the equation p[u] = 0 on ℝ n must be infinite 相似文献
13.
Let a n≥0 and F(u)∈C [0,1], Sikkema constructed polynomials:
, if α
n
≡0, then B n (0, F, x) are Bernstein polynomials.
Let
, we constructe new polynomials in this paper:
Q
n
(k)
( α
n
, f( t))= d
k
/ dx
k
B
n+k
( α
n
, F
k
( u), x), which are called Sikkema-Kantorovic polynomials of order k. If α
n
≡0, k=1, then Q n
(1) (0, f(t), x) are Kantorovič polynomials P n(f). If α
n
=0, k=2, then Q n
(2), (0, f(t), x) are Kantorovič polynomials of second order (see Nagel). The main result is:
Theorem 2. Let 1≤p≤∞, in order that for every f∈L P [0, 1],
, it is sufficient and necessary that
,
§ 1. Let f(t) de a continuous function on [a, b], i. e., f∈C [a, b], we define [1–2],[8–10]:
.
As usual, for the space L p [a,b](1≤p<∞), we have
and L[a, b]=l 1[a, b].
Let α
n
⩾0 and F(u)∈C[0,1], Sikkema-Bernstein polynomials
[3] [4].
The author expresses his thanks to Professor M. W. Müller of Dortmund University at West Germany for his supports. 相似文献
14.
We generalize a Hilbert space result by Auscher, McIntosh and Nahmod to arbitrary Banach spaces X and to not densely defined injective sectorial operators A. A convenient tool proves to be a certain universal extrapolation space associated with A. We characterize the real interpolation space
( X, D( Aa ) ? R( Aa ) ) q,p{\left( {X,\mathcal{D}{\left( {A^{\alpha } } \right)} \cap \mathcal{R}{\left( {A^{\alpha } } \right)}} \right)}_{{\theta ,p}}
as
{ x ? X|t - q\textRea y1 ( tA )x, t - q\textRea y2 ( tA )x ? L*p ( ( 0,¥ );X ) } {\left\{ {x\, \in \,X|t^{{ - \theta {\text{Re}}\alpha }} \psi _{1} {\left( {tA} \right)}x,\,t^{{ - \theta {\text{Re}}\alpha }} \psi _{2} {\left( {tA} \right)}x \in L_{*}^{p} {\left( {{\left( {0,\infty } \right)};X} \right)}} \right\}} 相似文献
15.
Let X1, X2, ... be i.i.d. random variables with EX1 = 0 and positive, finite variance σ2, and set Sn = X1 + ... + Xn. For any α > −1, β > −1/2 and for κn(ε) a function of ε and n such that κn(ε) log log n → λ as n ↑ ∞ and
, we prove that
*Supported by the Natural Science Foundation of Department of Education of Zhejiang Province (Grant No. 20060237 and 20050494). 相似文献
16.
A k-dimensional box is a Cartesian product R
1 × · · · × R
k
where each R
i
is a closed interval on the real line. The boxicity of a graph G, denoted as box( G), is the minimum integer k such that G can be represented as the intersection graph of a collection of k-dimensional boxes. That is, two vertices are adjacent if and only if their corresponding boxes intersect. A circular arc
graph is a graph that can be represented as the intersection graph of arcs on a circle. We show that if G is a circular arc graph which admits a circular arc representation in which no arc has length at least
p(\fraca-1a){\pi(\frac{\alpha-1}{\alpha})} for some
a ? \mathbb N 3 2{\alpha\in\mathbb{N}_{\geq 2}}, then box( G) ≤ α (Here the arcs are considered with respect to a unit circle). From this result we show that if G has maximum degree
D < ?\frac n(a-1)2a?{\Delta < \lfloor{\frac{n(\alpha-1)}{2\alpha}}\rfloor} for some
a ? \mathbb N 3 2{\alpha \in \mathbb{N}_{\geq 2}}, then box( G) ≤ α. We also demonstrate a graph having box( G) > α but with
D = n\frac(a-1)2a+ \frac n2a(a+1)+(a+2){\Delta=n\frac{(\alpha-1)}{2\alpha}+ \frac{n}{2\alpha(\alpha+1)}+(\alpha+2)}. For a proper circular arc graph G, we show that if
D < ?\frac n(a-1)a?{\Delta < \lfloor{\frac{n(\alpha-1)}{\alpha}}\rfloor} for some
a ? \mathbb N 3 2{\alpha\in \mathbb{N}_{\geq 2}}, then box( G) ≤ α. Let r be the cardinality of the minimum overlap set, i.e. the minimum number of arcs passing through any point on the circle, with
respect to some circular arc representation of G. We show that for any circular arc graph G, box( G) ≤ r + 1 and this bound is tight. We show that if G admits a circular arc representation in which no family of k ≤ 3 arcs covers the circle, then box( G) ≤ 3 and if G admits a circular arc representation in which no family of k ≤ 4 arcs covers the circle, then box( G) ≤ 2. We also show that both these bounds are tight. 相似文献
17.
Let X, X1, X2,... be i.i.d, random variables with mean zero and positive, finite variance σ^2, and set Sn = X1 +... + Xn, n≥1. The author proves that, if EX^2I{|X|≥t} = 0((log log t)^-1) as t→∞, then for any a〉-1 and b〉 -1,lim ε↑1/√1+a(1/√1+a-ε)b+1 ∑n=1^∞(logn)^a(loglogn)^b/nP{max κ≤n|Sκ|≤√σ^2π^2n/8loglogn(ε+an)}=4/π(1/2(1+a)^3/2)^b+1 Г(b+1),whenever an = o(1/log log n). The author obtains the sufficient and necessary conditions for this kind of results to hold. 相似文献
18.
In studying local harmonic analysis on the sphere S n, R.S. Strichartz introduced certain zonal functions ϕ 2( d( x, y)) which satisfy the equation
, where Δ z is the Laplace operator and δ −y the Dirac measure. The explicit expression of the constant a (λ) is given by R.S. Strichartz in the case that n is odd. Appyling
the Apéry identity, we show in this paper that
for n even, where w n-1 is the surface area of S n-1,
.
The author's research was supported by a grant from NSFC. 相似文献
19.
In the case where a 2π-periodic function f is twice continuously differentiable on the real axis ℝ and changes its monotonicity at different fixed points y
i
∈ [− π, π), i = 1,…, 2 s, s ∈ ℕ (i.e., on ℝ, there exists a set Y := { y
i
}
i∈ℤ of points y
i
= y
i+2s
+ 2π such that the function f does not decrease on [ y
i
, y
i−1] if i is odd and does not increase if i is even), for any natural k and n, n ≥ N( Y, k) = const, we construct a trigonometric polynomial T
n
of order ≤ n that changes its monotonicity at the same points y
i
∈ Y as f and is such that
*20c || f - Tn || £ \fracc( k,s )n2\upomega k( f",1 \mathord\vphantom 1 n n ) ( || f - Tn || £ \fracc( r + k,s )nr\upomega k( f(r),1 \mathord | / |
\vphantom 1 n n ), f ? C(r), r 3 2 ), \begin{array}{*{20}{c}} {\left\| {f - {T_n}} \right\| \leq \frac{{c\left( {k,s} \right)}}{{{n^2}}}{{{\upomega }}_k}\left( {f',{1 \mathord{\left/{\vphantom {1 n}} \right.} n}} \right)} \\ {\left( {\left\| {f - {T_n}} \right\| \leq \frac{{c\left( {r + k,s} \right)}}{{{n^r}}}{{{\upomega }}_k}\left( {{f^{(r)}},{1 \mathord{\left/{\vphantom {1 n}} \right.} n}} \right),\quad f \in {C^{(r)}},\quad r \geq 2} \right),} \\ \end{array} 相似文献
20.
Let S n = X 1 + · · · + X
n
, n ≥ 1, and S
0 = 0, where X
1, X
2, . . . are independent, identically distributed random variables such that the distribution of S
n/ B
n converges weakly to a nondeoenerate distribution F
α
as n → ∞ for some positive B
n
. We study asymptotic behavior of sums of the form
where
a function d( t) is continuous at [0,1] and has power decay at zero,
Bibliography: 13 titles.
Translated from Zapiski Nauchnylch Serninarov POMI, Vol. 361, 2008, pp. 109–122. 相似文献
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