共查询到20条相似文献,搜索用时 31 毫秒
1.
Erik Talvila 《Journal of Fourier Analysis and Applications》2012,18(1):27-44
Fourier series are considered on the one-dimensional torus for the space of periodic distributions that are the distributional
derivative of a continuous function. This space of distributions is denoted
Ac(\mathbbT){\mathcal{A}}_{c}(\mathbb{T}) and is a Banach space under the Alexiewicz norm,
||f||\mathbbT=sup|I| £ 2p|òI f|\|f\|_{\mathbb{T}}=\sup_{|I|\leq2\pi}|\int_{I} f|, the supremum being taken over intervals of length not exceeding 2π. It contains the periodic functions integrable in the sense of Lebesgue and Henstock–Kurzweil. Many of the properties of
L
1 Fourier series continue to hold for this larger space, with the L
1 norm replaced by the Alexiewicz norm. The Riemann–Lebesgue lemma takes the form [^(f)](n)=o(n)\hat{f}(n)=o(n) as |n|→∞. The convolution is defined for
f ? Ac(\mathbbT)f\in{\mathcal{A}}_{c}(\mathbb{T}) and g a periodic function of bounded variation. The convolution commutes with translations and is commutative and associative.
There is the estimate
||f*g||¥ £ ||f||\mathbbT ||g||BV\|f\ast g\|_{\infty}\leq\|f\|_{\mathbb{T}} \|g\|_{\mathcal{BV}}. For
g ? L1(\mathbbT)g\in L^{1}(\mathbb{T}),
||f*g||\mathbbT £ ||f||\mathbb T ||g||1\|f\ast g\|_{\mathbb{T}}\leq\|f\|_{\mathbb {T}} \|g\|_{1}. As well, [^(f*g)](n)=[^(f)](n) [^(g)](n)\widehat{f\ast g}(n)=\hat{f}(n) \hat{g}(n). There are versions of the Salem–Zygmund–Rudin–Cohen factorization theorem, Fejér’s lemma and the Parseval equality. The
trigonometric polynomials are dense in
Ac(\mathbbT){\mathcal{A}}_{c}(\mathbb{T}). The convolution of f with a sequence of summability kernels converges to f in the Alexiewicz norm. Let D
n
be the Dirichlet kernel and let
f ? L1(\mathbbT)f\in L^{1}(\mathbb{T}). Then
||Dn*f-f||\mathbbT?0\|D_{n}\ast f-f\|_{\mathbb{T}}\to0 as n→∞. Fourier coefficients of functions of bounded variation are characterized. The Appendix contains a type of Fubini theorem. 相似文献
2.
Liviu I. Ignat Julio D. Rossi 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2008,170(1):918-925
In this paper we study the asymptotic behaviour as t → ∞ of solutions to a nonlocal diffusion problem on a lattice, namely,
u¢n(t) = ?j ? \mathbbZd Jn-juj(t)-un(t)u^{\prime}_{n}(t) = \sum_{{j\in}{{{\mathbb{Z}}}^{d}}} J_{n-j}u_{j}(t)-u_{n}(t) with t ≥ 0 and
n ? \mathbbZdn \in {\mathbb{Z}}^{d}. We assume that J is nonnegative and verifies
?n ? \mathbbZdJn = 1\sum_{{n \in {\mathbb{Z}}}^{d}}J_{n}= 1. We find that solutions decay to zero as t → ∞ and prove an optimal decay rate using, as our main tool, the discrete Fourier transform. 相似文献
3.
We study hypersurfaces in the Lorentz-Minkowski space
\mathbbLn+1{\mathbb{L}^{n+1}} whose position vector ψ satisfies the condition L
k
ψ = Aψ + b, where L
k
is the linearized operator of the (k + 1)th mean curvature of the hypersurface for a fixed k = 0, . . . , n − 1,
A ? \mathbbR(n+1)×(n+1){A\in\mathbb{R}^{(n+1)\times(n+1)}} is a constant matrix and
b ? \mathbbLn+1{b\in\mathbb{L}^{n+1}} is a constant vector. For every k, we prove that the only hypersurfaces satisfying that condition are hypersurfaces with zero (k + 1)th mean curvature, open pieces of totally umbilical hypersurfaces
\mathbbSn1(r){\mathbb{S}^n_1(r)} or
\mathbbHn(-r){\mathbb{H}^n(-r)}, and open pieces of generalized cylinders
\mathbbSm1(r)×\mathbbRn-m{\mathbb{S}^m_1(r)\times\mathbb{R}^{n-m}},
\mathbbHm(-r)×\mathbbRn-m{\mathbb{H}^m(-r)\times\mathbb{R}^{n-m}}, with k + 1 ≤ m ≤ n − 1, or
\mathbbLm×\mathbbSn-m(r){\mathbb{L}^m\times\mathbb{S}^{n-m}(r)}, with k + 1 ≤ n − m ≤ n − 1. This completely extends to the Lorentz-Minkowski space a previous classification for hypersurfaces in
\mathbbRn+1{\mathbb{R}^{n+1}} given by Alías and Gürbüz (Geom. Dedicata 121:113–127, 2006). 相似文献
4.
For automorphic L-functions L(s, π) and L( s,p¢){L( s,\pi^{\prime })} attached to automorphic irreducible cuspidal representations π and π′ of
GLm( \mathbbQA){GL_{m}( \mathbb{Q}_{A})} and
GLm¢(\mathbbQA) {GL_{m^{\prime }}(\mathbb{Q}_{A}) }, we prove the Selberg orthogonality unconditionally for m ≤ 4 and m′ ≤ 4, and under hypothesis H of Rudnik and Sarnak if m > 4 or m′ > 4, without the additional requirement that at least one of these representations be self-contragradient. 相似文献
5.
Françoise Lust-Piquard 《Potential Analysis》2006,24(1):47-62
Let L=?Δ+|ξ|2 be the harmonic oscillator on $\mathbb{R}^{n}Let L=−Δ+|ξ|2 be the harmonic oscillator on
\mathbbRn\mathbb{R}^{n}
, with the associated Riesz transforms R2j−1=(∂/∂ξj)L−1/2,R2j=ξjL−1/2. We give a shorter proof of a recent result of Harboure, de Rosa, Segovia, Torrea: For 1<p<∞ and a dimension free constant Cp,
||(?k=12n|Rk(f)|2)1/2||Lp(\mathbbRn,dx)\leqslant Cp||f||Lp(\mathbbRn,dx).\bigg\Vert \bigg(\sum_{k=1}^{2n}\vert R_{k}(f)\vert ^{2}\bigg)^{{1}/{2}}\bigg\Vert _{L^{p}(\mathbb{R}^{n},\mathrm{d}\xi )}\leqslant C_{p}\Vert f\Vert _{L^{p}(\mathbb{R}^{n},\mathrm{d}\xi )}. 相似文献
6.
In this paper we obtain a new regularity criterion for weak solutions to the 3D MHD equations. It is proved that if
div( \fracu|u|) \mathrm{div}( \frac{u}{|u|}) belongs to
L\frac21-r( 0,T;[(X)\dot]r( \mathbbR3) ) L^{\frac{2}{1-r}}( 0,T;\dot{X}_{r}( \mathbb{R}^{3}) ) with 0≤r≤1, then the weak solution actually is regular and unique. 相似文献
7.
We consider the spectral decomposition of A, the generator of a polynomially bounded n-times integrated group whose spectrum set $\sigma(A)=\{i\lambda_{k};k\in\mathbb{\mathbb{Z}}^{*}\}
8.
Nikolaos D. Atreas 《Advances in Computational Mathematics》2012,36(1):21-38
Let ϕ be a function in the Wiener amalgam space W¥(L1)\emph{W}_{\infty}(L_1) with a non-vanishing property in a neighborhood of the origin for its Fourier transform [^(f)]\widehat{\phi},
t={tn}n ? \mathbb Z{\bf \tau}=\{\tau_n\}_{n\in {{\mathbb Z}}} be a sampling set on ℝ and VftV_\phi^{\bf \tau} be a closed subspace of
L2(\mathbbR)L_2(\hbox{\ensuremath{\mathbb{R}}}) containing all linear combinations of τ-translates of ϕ. In this paper we prove that every function f ? Vftf\in V_\phi^{\bf \tau} is uniquely determined by and stably reconstructed from the sample set
Lft(f)={ò\mathbbR f(t)[`(f(t-tn))] dt}n ? \mathbb ZL_\phi^{\bf \tau}(f)=\Big\{\int_{\hbox{\ensuremath{\mathbb{R}}}} f(t) \overline{\phi(t-\tau_n)} dt\Big\}_{n\in {{\mathbb Z}}}. As our reconstruction formula involves evaluating the inverse of an infinite matrix we consider a partial reconstruction
formula suitable for numerical implementation. Under an additional assumption on the decay rate of ϕ we provide an estimate to the corresponding error. 相似文献
9.
Petros Galanopoulos Daniel Girela Rodrigo Hernández 《Journal of Geometric Analysis》2011,21(3):665-682
This paper is concerned mainly with the logarithmic Bloch space ℬlog which consists of those functions f which are analytic in the unit disc
\mathbbD{\mathbb{D}} and satisfy
sup|z| < 1(1-|z|)log\frac11-|z||f¢(z)| < ¥\sup_{\vert z\vert <1}(1-\vert z\vert )\log\frac{1}{1-\vert z\vert}\vert f^{\prime}(z)\vert <\infty , and the analytic Besov spaces B
p
, 1≤p<∞. They are all subspaces of the space VMOA. We study the relation between these spaces, paying special attention to the membership of univalent functions in them. We
give explicit examples of:
|