$ \left\langle p,q\right\rangle _{S}=\int_{0}^{\infty }p(x)q(x)x^{\alpha }e^{-x}dx+Np^{\prime }(a)q^{\prime }(a),\alpha >-1 $ \left\langle p,q\right\rangle _{S}=\int_{0}^{\infty }p(x)q(x)x^{\alpha }e^{-x}dx+Np^{\prime }(a)q^{\prime }(a),\alpha >-1 相似文献
2.
The Ces\aro operator $\mathcal{C}_{\alpha}$ is defined by \begin{equation*} (\mathcal{C}_{\alpha}f)(x) = \int_{0}^{1}t^{-1}f\left( t^{-1}x \right)\alpha (1-t)^{\alpha -1}\,dt~, \end{equation*} where $f$ denotes a function on $\mathbb{R}$. We prove that $\mathcal{C}_{\alpha}$, $\alpha >0$, is a bounded operator in the Hardy space $H^{p}$ for every $0 < p \leqq 1$. 相似文献
3.
Ilia Krasikov 《Constructive Approximation》2008,28(2):113-125
T. Erdelyi, A.P. Magnus and P. Nevai conjectured that for
the orthonormal Jacobi polynomials
satisfy the inequality
4.
Jacek Dziubanski 《Constructive Approximation》2008,27(3):269-287
Let
be the standard Laguerre functions of type a. We denote
. Let
and
be the semigroups associated with the orthonormal systems
and
. We say that a function f belongs to the Hardy space
associated with one of the semigroups if the corresponding maximal function belongs to
. We prove special atomic decompositions of the elements of the Hardy spaces. 相似文献
5.
Arrigo Cellina 《Proceedings of the American Mathematical Society》2002,130(2):413-418
This paper presents a necessary and sufficient condition on the convex function in order that continuous solutions to
satisfy a Strong Maximum Principle on any open connected . 6.
Mark A. Pinsky 《Proceedings of the American Mathematical Society》2002,130(3):753-758
If we can define the Hilbert transform almost everywhere (Lebesgue) and obtain an estimate for where is a suitable finite measure. The classical Kolmogorov inequality for the Lebesgue measure of is obtained by a scaling argument. 7.
John J. Benedetto Alexander M. Powell 《Transactions of the American Mathematical Society》2006,358(6):2489-2505
Let satisfy We construct an orthonormal basis for such that and are both uniformly bounded in . Here . This generalizes a theorem of Bourgain and is closely related to recent results on the Balian-Low theorem.
8.
Weibing Deng Yuxiang Li Chunhong Xie 《Proceedings of the American Mathematical Society》2003,131(5):1573-1582
This paper establishes a new criterion for global existence and nonexistence of positive solutions of the non-local degenerate parabolic system
0, \end{align*}"> with homogeneous Dirichlet boundary conditions, where is a bounded domain with a smooth boundary and are positive constants. For all initial data, it is proved that there exists a global positive solution iff , where is the unique positive solution of the linear elliptic problem 9.
Horst Alzer 《Proceedings of the American Mathematical Society》2007,135(11):3641-3648
Let and be real numbers. The inequality
10.
Weiyang Chen & Xiaoli Chen 《数学研究》2014,47(2):208-220
In this paper, we are concerned with the properties of positive solutions of the following nonlinear integral systems on the Heisenberg group $\mathbb{H}^n$, \begin{equation} \left\{\begin{array}{ll} u(x)=\int_{\mathbb{H}^n}\frac{v^{q}(y)w^{r}(y)}{|x^{-1}y|^\alpha|y|^\beta}\,dy,\\ v(x)=\int_{\mathbb{H}^n}\frac{u^{p}(y)w^{r}(y)}{|x^{-1}y|^\alpha|y|^\beta}\,dy,\\ w(x)=\int_{\mathbb{H}^n}\frac{u^{p}(y)v^{q}(y)}{|x^{-1}y|^\alpha|y|^\beta}\,dy,\\ \end{array}\right.\end{equation} for $x\in \mathbb{H}^n$, where $0<\alpha
1$ satisfying $\frac{1}{p+1} $+ $\frac{1}{q+1} + \frac{1}{r+1} = \frac{Q+α+β}{Q}.$ We show that positive solution triples $(u,v,w)\in L^{p+1}(\mathbb{H}^n)\times L^{q+1}(\mathbb{H}^n)\times L^{r+1}(\mathbb{H}^n)$ are bounded and they converge to zero when $|x|→∞.$ 相似文献 11.
Stamatis Koumandos. 《Mathematics of Computation》2008,77(264):2261-2275
Let , where is Euler's gamma function. We determine conditions for the numbers so that the function is strongly completely monotonic on . Through this result we obtain some inequalities involving the ratio of gamma functions and provide some applications in the context of trigonometric sum estimation. We also give several other examples of strongly completely monotonic functions defined in terms of and functions. Some limiting and particular cases are also considered.
12.
R. G. Novikov 《Selecta Mathematica, New Series》1997,3(2):245-302
We consider the Dirac-ZS-AKNS system (1) where (the space of functions with n derivatives in L
1), (2) We consider for (1) the transition matrix and, in addition, for the case of the Dirac system (i.e. for the selfadjoint case the scattering matrix We can divide main results of the present work into three parts. I. We show that the inverse scattering transform and the inverse Fourier transform give the same solution, up to smooth functions,
of the inverse scattering problem for (1). More preciseley, we show that, under condition (2) with , the following formulas are valid: (3) and, in addition, for the case of the Dirac system (4) where denotes the factor space. II. Using (3), (4), we give the characterization of the transition matrix and the scattering matrix for the case of the Dirac
system under condition (2) with
III. As applications of the results mentioned above, we show that 1) for any real-valued initial data , the Cauchy problem for the sh-Gordon equation has a unique solution such that and for any t > 0, 2) in addition, for , for such a solution the following formula is valid: where
denotes the space of functions locally integrable with n derivatives. We give also a review of preceding results. 相似文献
13.
H. Bavinck 《Proceedings of the American Mathematical Society》1997,125(12):3561-3567
We consider the polynomials orthogonal with respect to the Sobolev type inner product
where and is a nonnegative integer. It is the purpose of this paper to show that these polynomials are eigenfunctions of a class of linear differential operators containing one that is of finite order if is a nonnegative integer and 14.
In this paper, quasi-interpolating splines are used to approximate the Cauchy principal value integral $$J(w_{\alpha \beta } f;\lambda ): = \smallint - _{ - 1}^1 w_{\alpha \beta } (x)\frac{{f(x)}}{{x - \lambda }}dx, \lambda \in ( - 1,1)$$ where $w_{\alpha \beta } (x): = (1 - x)^\alpha (1 + x)^\beta ,\alpha ,\beta > - 1.$ . We prove uniform convergence for the quadrature rules proposed here and give an algorithm for the numerical evaluation of these rules. 相似文献
15.
D.S. Lubinsky 《Constructive Approximation》2007,25(3):303-366
Assume
is not an integer. In papers published in 1913 and 1938,
S.~N.~Bernstein established the limit
16.
Tewodros Amdeberhan Olivier Espinosa Victor H. Moll 《Proceedings of the American Mathematical Society》2008,136(9):3211-3221
The definite integral
17.
Let V be a convex subset of a normed space and let a nondecreasing function α : [0, ∞) → [0, ∞) be given. A function ${f : V \rightarrow \mathbb{R}}$ is called α-midconvex if $$f\left(\frac{x+y}{2} \right)\leq \frac{f(x)+f(y)}{2}+\alpha(\|x-y\|) \quad \,{\rm for}\, x,y\in V.$$ It is known (Tabor in Control Cybern., 38/3:656–669, 2009) that if ${f : V \rightarrow \mathbb{R}}$ is α-midconvex, locally bounded above at every point of V then $$f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)+P_\alpha(\|x-y\|) \quad \,{\rm for}\, x, y \in V,t \in [0,1],$$ where ${P_\alpha(r):=\sum_{k=0}^\infty \frac{1}{2^k} \alpha(2{\rm dist}(2^kr, \mathbb{Z}))}$ for ${r \in \mathbb{R}}$ . We show that under some additional assumptions the above estimation cannot be improved. 相似文献
18.
Steve Hofmann 《Proceedings of the American Mathematical Society》2008,136(12):4223-4233
We consider divergence form elliptic operators , defined in , where the coefficient matrix is , uniformly elliptic, complex and -independent. Using recently obtained results concerning the boundedness and invertibility of layer potentials associated to such operators, we show that if in , then for any vector-valued we have the bilinear estimate
19.
Manuel del Pino Monica Musso Bernhard Ruf 《Calculus of Variations and Partial Differential Equations》2012,44(3-4):543-576
Let Ω be a bounded, smooth domain in ${\mathbb{R}^2}$ . We consider the functional $$I(u) = \int_\Omega e^{u^2}\,dx$$ in the supercritical Trudinger-Moser regime, i.e. for ${\int_\Omega |\nabla u|^2dx > 4\pi}$ . More precisely, we are looking for critical points of I(u) in the class of functions ${u \in H_0^1 (\Omega )}$ such that ${\int_\Omega |\nabla u|^2 \, dx = 4\, \pi \, k\, (1+\alpha)}$ , for small α > 0. In particular, we prove the existence of 1-peak critical points of I(u) with ${\int_\Omega |\nabla u|^2dx = 4\pi(1 + \alpha)}$ for any bounded domain Ω, 2-peak critical points with ${\int_\Omega |\nabla u|^2dx = 8\pi(1 + \alpha)}$ for non-simply connected domains Ω, and k-peak critical points with ${\int_\Omega |\nabla u|^2 dx = 4k \pi(1 + \alpha)}$ if Ω is an annulus. 相似文献
20.
Jeongkeun Lee 《Proceedings of the American Mathematical Society》2000,128(8):2381-2391
We consider orthogonal polynomials in two variables whose derivatives with respect to are orthogonal. We show that they satisfy a system of partial differential equations of the form where , , is a vector of polynomials in and for , and is an eigenvalue matrix of order for . Also we obtain several characterizations for these polynomials. Finally, we point out that our results are able to cover more examples than Bertran's. |