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1.
We investigate the relationship between the constants K(R) and K(T), where
is the exact constant in the Kolmogorov inequality, R is the real axis, T is a unit circle,
is the set of functions x L
p(G) such that x
(r) L
s(G), q, p, s [1, ], k, r N, k < r, We prove that if
thenK(R) = K(T),but if
thenK(R) K(T); moreover, the last inequality can be an equality as well as a strict inequality. As a corollary, we obtain new exact Kolmogorov-type inequalities on the real axis. 相似文献
2.
Let X be a Banach space of differentiable functions and A: XX be a superposition operator. We consider for A the conditions
相似文献
3.
We consider a finite XXZ spin chain with periodic boundary conditions and an odd number of sites. It appears that for the special value of the asymmetry parameter = –1/2, the ground state of this system described by the Hamiltonian
has the energy E
0 = –3N/2. Although the ground state is antiferromagnetic, we can find the corresponding solution of the Bethe equations. Specifically, we can explicitly construct a trigonometric polynomial Q(u) of degree n = (N–1)/2, whose zeros are the parameters of the Bethe wave function for the ground state of the system. As is known, this polynomial satisfies the Baxter T–Q equation. This equation also has a second independent solution corresponding to the same eigenvalue of the transfer matrix T. We use this solution to find the derivative of the ground-state energy of the XXZ chain with respect to the crossing parameter . This derivative is directly related to one of the spin–spin correlators, which appears to be
. In turn, this correlator gives the average number of spin strings for the ground state of the chain,
. All these simple formulas fail if the number N of chain sites is even. 相似文献
4.
V. A. Yudin 《Proceedings of the Steklov Institute of Mathematics》2011,273(1):188-189
It is established that H. Bohr’s inequality \(\sum\nolimits_{k = 0}^\infty {\left| {{{f^{\left( k \right)} \left( 0 \right)} \mathord{\left/ {\vphantom {{f^{\left( k \right)} \left( 0 \right)} {\left( {2^{{k \mathord{\left/ {\vphantom {k 2}} \right. \kern-\nulldelimiterspace} 2}} k!} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {2^{{k \mathord{\left/ {\vphantom {k 2}} \right. \kern-\nulldelimiterspace} 2}} k!} \right)}}} \right| \leqslant \sqrt 2 \left\| f \right\|_\infty }\) is sharp on the class H ∞. 相似文献
5.
Let = (1,...,d) be a vector with positive components and let D be the corresponding mixed derivative (of order j with respect to the jth variable). In the case where d > 1 and 0 < k < r are arbitrary, we prove that
6.
A celebrated result of Johnson, Maurey, König and Retherford from 1977 states that for
every complex
matrix
satisfies the following eigenvalue estimate:
7.
K. F. Cheng 《Periodica Mathematica Hungarica》1983,14(2):177-187
The nonparametric regression problem has the objective of estimating conditional expectation. Consider the model $$Y = R(X) + Z$$ , where the random variableZ has mean zero and is independent ofX. The regression functionR(x) is the conditional expectation ofY givenX = x. For an estimator of the form $$R_n (x) = \sum\limits_{i = 1}^n {Y_i K{{\left[ {{{\left( {x - X_i } \right)} \mathord{\left/ {\vphantom {{\left( {x - X_i } \right)} {c_n }}} \right. \kern-\nulldelimiterspace} {c_n }}} \right]} \mathord{\left/ {\vphantom {{\left[ {{{\left( {x - X_i } \right)} \mathord{\left/ {\vphantom {{\left( {x - X_i } \right)} {c_n }}} \right. \kern-\nulldelimiterspace} {c_n }}} \right]} {\sum\limits_{i = 1}^n {K\left[ {{{\left( {x - X_i } \right)} \mathord{\left/ {\vphantom {{\left( {x - X_i } \right)} {c_n }}} \right. \kern-\nulldelimiterspace} {c_n }}} \right]} }}} \right. \kern-\nulldelimiterspace} {\sum\limits_{i = 1}^n {K\left[ {{{\left( {x - X_i } \right)} \mathord{\left/ {\vphantom {{\left( {x - X_i } \right)} {c_n }}} \right. \kern-\nulldelimiterspace} {c_n }}} \right]} }}} $$ , we obtain the rate of strong uniform convergence $$\mathop {\sup }\limits_{x\varepsilon C} \left| {R_n (x) - R(x)} \right|\mathop {w \cdot p \cdot 1}\limits_ = o({{n^{{1 \mathord{\left/ {\vphantom {1 {(2 + d)}}} \right. \kern-\nulldelimiterspace} {(2 + d)}}} } \mathord{\left/ {\vphantom {{n^{{1 \mathord{\left/ {\vphantom {1 {(2 + d)}}} \right. \kern-\nulldelimiterspace} {(2 + d)}}} } {\beta _n \log n}}} \right. \kern-\nulldelimiterspace} {\beta _n \log n}}),\beta _n \to \infty $$ . HereX is ad-dimensional variable andC is a suitable subset ofR d . 相似文献
8.
We obtain new unimprovable Kolmogorov-type inequalities for differentiable periodic functions. In particular, we prove that, for r = 2, k = 1 or r = 3, k = 1, 2 and arbitrary q, p [1, ], the following unimprovable inequality holds for functions
:
9.
Jeremy T. Tyson 《Potential Analysis》2006,24(4):357-384
We obtain sharp weighted Moser–Trudinger inequalities for first-layer symmetric functions on groups of Heisenberg type, and for -symmetric functions on the Grushin plane. To this end, we establish weighted Young's inequalities in the form , for first-layer radial weights on a general Carnot group and functions with first-layer symmetric. The proofs use some sharp estimates for hypergeometric functions.Research supported by NSF grant DMS-0228807. 相似文献
10.
Fairly general conditions on the coefficients
of even and odd trigonometric Fourier series under which L-convergence (boundedness) of partial sums of the series is equivalent to the relation
are given. 相似文献
11.
12.
G. Kuba 《Acta Mathematica Hungarica》2001,91(4):325-332
For a large real parameter t and 0 a b
we consider sums
where is the rounding error function, i.e. (z) = z - [z] - 1/2. We generalize Huxley's well known estimate
by showing that
holds uniformly in 0 a b
. Fruther, we investigate an analogous question related to the divisor problem and show that the inequality
, which (due to Huxley) holds uniformly in 0 a b
, and which is in general not true for 1 a b t, is true uniformly in 0 a b
. 相似文献
13.
Let Θ be a bounded open set in ℝ
n
, n ⩾ 2. In a well-known paper Indiana Univ. Math. J., 20, 1077–1092 (1971) Moser found the smallest value of K such that
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