We characterize the completeness and frame/basis property of a union of under-sampled windowed exponentials of the form
$$ {\mathcal{F}}(g): =\bigl\{ e^{2\pi i n x}: n\ge 0\bigr\} \cup \bigl\{ g(x)e^{2\pi i nx}: n< 0\bigr\} $$for \(L^{2}[-1/2,1/2]\) by the spectra of the Toeplitz operators with the symbol \(g\). Using this characterization, we classify all real-valued functions \(g\) such that \({\mathcal{F}}(g)\) is complete or forms a frame/basis. Conversely, we use the classical non-harmonic Fourier series theory to determine all \(\xi \) such that the Toeplitz operators with the symbol \(e^{2\pi i \xi x}\) is injective or invertible. These results demonstrate an elegant interaction between frame theory of windowed exponentials and Toeplitz operators. Finally, we use our results to answer some open questions in dynamical sampling, and derivative samplings on Paley-Wiener spaces of bandlimited functions.
相似文献We prove that given any \(\epsilon >0\), a non-zero adelic Hilbert cusp form \({\mathbf {f}}\) of weight \(k=(k_1,k_2,\ldots ,k_n)\in ({\mathbb {Z}}_+)^n\) and square-free level \(\mathfrak {n}\) with Fourier coefficients \(C_{{\mathbf {f}}}(\mathfrak {m})\), there exists a square-free integral ideal \(\mathfrak {m}\) with \(N(\mathfrak {m})\ll k_0^{3n+\epsilon }N(\mathfrak {n})^{\frac{6n^2+1}{2}+\epsilon }\) such that \(C_{{\mathbf {f}}}(\mathfrak {m})\ne 0\). The implied constant depends on \(\epsilon , F\).
相似文献Discrete and continuous frames can be considered as positive operator-valued measures (POVMs) that have integral representations using rank-one operators. However, not every POVM has an integral representation. One goal of this paper is to examine the POVMs that have finite-rank integral representations. More precisely, we present a necessary and sufficient condition under which a positive operator-valued measure \(F: \varOmega \to B(H)\) has an integral representation of the form
$$ F(E) =\sum_{k=1}^{m} \int _{E} G_{k}(\omega )\otimes G_{k}(\omega )\, d \mu (\omega ) $$for some weakly measurable maps \(G_{k}\ (1\leq k\leq m) \) from a measurable space \(\varOmega \) to a Hilbert space ℋ and some positive measure \(\mu \) on \(\varOmega \). Similar characterizations are also obtained for projection-valued measures. As special consequences of our characterization we settle negatively a problem of Ehler and Okoudjou about probability frame representations of probability POVMs, and prove that an integral representable probability POVM can be dilated to a integral representable projection-valued measure if and only if the corresponding measure is purely atomic.
相似文献We extend to the multilinear setting classical inequalities of Marcinkiewicz and Zygmund on \(\ell ^r\)-valued extensions of linear operators. We show that for certain \(1 \le p, q_1, \dots , q_m, r \le \infty \), there is a constant \(C\ge 0\) such that for every bounded multilinear operator \(T:L^{q_1}(\mu _1) \times \cdots \times L^{q_m}(\mu _m) \rightarrow L^p(\nu )\) and functions \(\{f_{k_1}^1\}_{k_1=1}^{n_1} \subset L^{q_1}(\mu _1), \dots , \{f_{k_m}^m\}_{k_m=1}^{n_m} \subset L^{q_m}(\mu _m)\), the following inequality holds
$$\begin{aligned} \left\| \left( \sum _{k_1, \dots , k_m} |T(f_{k_1}^1, \dots , f_{k_m}^m)|^r\right) ^{1/r} \right\| _{L^p(\nu )} \le C \Vert T\Vert \prod _{i=1}^m \left\| \left( \sum _{k_i=1}^{n_i} |f_{k_i}^i|^r\right) ^{1/r} \right\| _{L^{q_i}(\mu _i)}. \end{aligned}$$ (1)In some cases we also calculate the best constant \(C\ge 0\) satisfying the previous inequality. We apply these results to obtain weighted vector-valued inequalities for multilinear Calderón-Zygmund operators.
相似文献The carpet subgroups admitting a Bruhat decomposition and different from Chevalley groups are exhausted by the groups lying between the Chevalley groups of type \( B_{l} \), \( C_{l} \), \( F_{4} \), or \( G_{2} \) over various imperfect fields of exceptional characteristic 2 or 3, the larger of which is an algebraic extension of the smaller field. Moreover, as regards the types \( B_{l} \) and \( C_{l} \), these subgroups are parametrized by the pairs of additive subgroups one of which may fail to be a field and, for the type \( B_{2} \), even both additive subgroups may fail to be fields. In this paper for the carpet subgroups admitting a Bruhat decomposition we present the relations similar to those well known for Chevalley groups over fields.
相似文献Let \(p(\cdot ):\ {{\mathbb {R}}}^n\rightarrow (0,\infty ]\) be a variable exponent function satisfying the globally log-Hölder continuous condition, \(q\in (0,\infty ]\) and A be a general expansive matrix on \({\mathbb {R}}^n\). Let \(H_A^{p(\cdot ),q}({{\mathbb {R}}}^n)\) be the anisotropic variable Hardy–Lorentz space associated with A defined via the radial grand maximal function. In this article, the authors characterize \(H_A^{p(\cdot ),q}({{\mathbb {R}}}^n)\) by means of the Littlewood–Paley g-function or the Littlewood–Paley \(g_\lambda ^*\)-function via first establishing an anisotropic Fefferman–Stein vector-valued inequality on the variable Lorentz space \(L^{p(\cdot ),q}({\mathbb {R}}^n)\). Moreover, the finite atomic characterization of \(H_A^{p(\cdot ),q}({{\mathbb {R}}}^n)\) is also obtained. As applications, the authors then establish a criterion on the boundedness of sublinear operators from \(H^{p(\cdot ),q}_A({\mathbb {R}}^n)\) into a quasi-Banach space. Applying this criterion, the authors show that the maximal operators of the Bochner–Riesz and the Weierstrass means are bounded from \(H^{p(\cdot ),q}_A({\mathbb {R}}^n)\) to \(L^{p(\cdot ),q}({\mathbb {R}}^n)\) and, as consequences, some almost everywhere and norm convergences of these Bochner–Riesz and Weierstrass means are also obtained. These results on the Bochner–Riesz and the Weierstrass means are new even in the isotropic case.
相似文献In this article, we begin a systematic study of the boundedness and the nuclearity properties of multilinear periodic pseudo-differential operators and multilinear discrete pseudo-differential operators on \(L^p\)-spaces. First, we prove analogues of known multilinear Fourier multipliers theorems (proved by Coifman and Meyer, Grafakos, Tomita, Torres, Kenig, Stein, Fujita, Tao, etc.) in the context of periodic and discrete multilinear pseudo-differential operators. For this, we use the periodic analysis of pseudo-differential operators developed by Ruzhansky and Turunen. Later, we investigate the s-nuclearity, \(0<s \le 1,\) of periodic and discrete pseudo-differential operators. To accomplish this, we classify those s-nuclear multilinear integral operators on arbitrary Lebesgue spaces defined on \(\sigma \)-finite measures spaces. We also study similar properties for periodic Fourier integral operators. Finally, we present some applications of our study to deduce the periodic Kato–Ponce inequality and to examine the s-nuclearity of multilinear Bessel potentials as well as the s-nuclearity of periodic Fourier integral operators admitting suitable types of singularities.
相似文献This paper is concerned with a free boundary problem modeling the growth of a spherically symmetric tumor with angiogenesis. The unknown nutrient concentration \(\sigma =\sigma (r,t)\) occupies the unknown tumor region \(r< R(t)\) and satisfies a nonlinear reaction diffusion equation, and the unknown tumor radius \(R=R(t)\) satisfies a nonlinear integro-differential equation. Unlike existing literatures on this topic where Dirichlet boundary condition for \(\sigma \) is imposed, in this paper the model uses the Robin boundary condition for \(\sigma \). We prove existence and uniqueness of a global in-time classical solution (\(\sigma (r,t),R(t)\)) for arbitrary \(c>0\) and establish asymptotic stability of the unique stationary solution (\(\sigma _{s}(r),R_{s}\)) for sufficiently small \(c\), where \(c\) is a positive constant reflecting the ratio between nutrient diffusion scale and the tumor cell-doubling scale.
相似文献In this article, we prove that a general version of Alladi’s formula with Dirichlet convolution holds for arithmetical semigroups satisfying Axiom A or Axiom \(A^{\#}\). As applications, we apply our main results to certain semigroups coming from algebraic number theory, arithmetical geometry, and graph theory.
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