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Abstract: Abstract We prove that if G is a discrete group and \((A,G,\alpha )\) is a C*-dynamical system such that the reduced crossed product \(A\rtimes _{r,\alpha } G\) possesses property (SOAP) then every completely compact Herz–Schur \((A,G,\alpha )\) -multiplier can be approximated in the completely bounded norm by Herz–Schur \((A,G,\alpha )\) -multipliers of finite rank. As a consequence, if G has the approximation property (AP) then the completely compact Herz–Schur multipliers of A(G) coincide with the closure of A(G) in the completely bounded multiplier norm. We study the class of invariant completely compact Herz–Schur multipliers of \(A\rtimes _{r,\alpha } G\) and provide a description of this class in the case of the irrational rotation algebra. PubDate: 2022-05-13
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Abstract: Abstract In this paper we discuss and prove an analogy of the Carleson–Hunt theorem with respect to Vilenkin systems. In particular, we use the theory of martingales and give a new and shorter proof of the almost everywhere convergence of Vilenkin–Fourier series of \(f\in L_p(G_m)\) for \(p>1\) in case the Vilenkin system is bounded. Moreover, we also prove sharpness by stating an analogy of the Kolmogorov theorem for \(p=1\) and construct a function \(f\in L_1(G_m)\) such that the partial sums with respect to Vilenkin systems diverge everywhere. PubDate: 2022-05-13
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Abstract: Abstract Under some restrictions on weight functions we obtain sufficient conditions for the boundedness of the Hilbert transform from weighted Sobolev space of the first order on the semi-axis to weighted Lebesgue space. PubDate: 2022-05-09
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Abstract: Abstract For Fatou’s interpolation theorem of 1906 we suggest a new elementary proof. PubDate: 2022-05-05
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Abstract: Abstract We study the restriction estimates in a class of conical singular space \(X=C(Y)=(0,\infty )_r\times Y\) with the metric \(g=\mathrm {d}r^2+r^2h\) , where the cross section Y is a compact \((n-1)\) -dimensional closed Riemannian manifold (Y, h). Let \(\Delta _g\) be the Friedrichs extension positive Laplacian on X, and consider the operator \(\mathcal {L}_V=\Delta _g+V\) with \(V=V_0r^{-2}\) , where \(V_0(\theta )\in \mathcal {C}^\infty (Y)\) is a real function such that the operator \(\Delta _h+V_0+(n-2)^2/4\) is positive. In the present paper, we prove a type of modified restriction estimates for the solutions of wave equation associated with \(\mathcal {L}_V\) . The smallest positive eigenvalue of the operator \(\Delta _h+V_0+(n-2)^2/4\) plays an important role in the result. As an application, for independent of interests, we prove local energy estimates and Keel–Smith–Sogge estimates for the wave equation in this setting. PubDate: 2022-05-03
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Abstract: Abstract We obtain a characterization of the weighted inequalities for the Riesz transforms on weighted local Morrey spaces. The condition is sufficient for the boundedness on the same spaces of all Calderón–Zygmund operators suitably defined on the functions of the space. In the case of the fractional maximal operator and the fractional integral we obtain a characterization valid for exponents satisfying the Sobolev relation. For power weights we get sharp results for these operators in the usual versions of weighted Morrey spaces, neither restricted to the Sobolev relation of the exponents nor to the one-weighted setting. PubDate: 2022-04-28
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Abstract: Abstract Let \(A^q_s({{\mathbf {T}}})\) denote the space of all Lebesgue integrable functions f on the torus \({\mathbf {T}}\) such that \(\sum _{m \in {{\mathbf {Z}}}} \widehat{f}(m) ^q \langle m \rangle ^{sq} < \infty \) , where \(\{ \widehat{f}(m) \}_{m \in {{\mathbf {Z}}}}\) denote the Fourier coefficients of f. We consider necessary and sufficient conditions for all functions \(F \in A^1_\beta ({{\mathbf {T}}})\) to operate on all real-valued functions in \(A^q_s({{\mathbf {T}}})\) . PubDate: 2022-04-28
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Abstract: Abstract The paper introduces new sufficient conditions of strict positive definiteness for kernels on d-dimensional spheres which are not radially symmetric but possess specific coefficient structures. The results use the series expansion of the kernel in spherical harmonics. The kernels either have a convolutional form or are axially symmetric with respect to one axis. The given results on convolutional kernels generalise the result derived by Chen et al. (Proc Am Math Soc 131:2733–2740, 2003) for radial kernels. PubDate: 2022-04-18
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Abstract: Abstract We prove that there exists no window function \(g \in {L^2(\mathbb {R})}\) and no lattice \({\mathcal {L}} \subset \mathbb {R}^2\) such that every \(f \in {L^2(\mathbb {R})}\) is determined up to a global phase by spectrogram samples \( V_gf({\mathcal {L}}) \) where \(V_gf\) denotes the short-time Fourier transform of f with respect to g. Consequently, the forward operator $$\begin{aligned} f \mapsto V_gf({\mathcal {L}}) \end{aligned}$$ mapping a square-integrable function to its spectrogram samples on a lattice is never injective on the quotient space with \(f \sim h\) identifying two functions which agree up to a multiplicative constant of modulus one. We will further elaborate this result and point out that under mild conditions on the lattice \({\mathcal {L}}\) , functions which produce identical spectrogram samples but do not agree up to a unimodular constant can be chosen to be real-valued. The derived results highlight that in the discretization of the STFT phase retrieval problem from lattice measurements, a prior restriction of the underlying signal space to a proper subspace of \({L^2(\mathbb {R})}\) is inevitable. PubDate: 2022-04-04
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Abstract: Abstract In this paper, we study dot-product sets and k-simplices in \(\mathbb {Z}_n^d\) for odd n, where \(\mathbb {Z}_n\) is the ring of residues modulo n. We show that if E is sufficiently large then the dot-product set of E covers the whole ring. In higher dimensional cases, if E is sufficiently large then the set of simplices and the set of dot-product simplices determined by E, up to congurence, have positive densities. PubDate: 2022-03-30
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Abstract: Abstract Let \(X(\Gamma )\) be the space of all finite Borel measure \(\mu \) in \(\mathbb R^2\) which is supported on the smooth curve \(\Gamma \) and absolutely continuous with respect to the arc length on \(\Gamma \) . For \(\Lambda \subset \mathbb R^2,\) the pair \(\left( \Gamma , \Lambda \right) \) is called a Heisenberg uniqueness pair for \(X(\Gamma )\) if any \(\mu \in X(\Gamma )\) satisfies \(\hat{\mu }\vert _\Lambda =0,\) implies \(\mu =0.\) We prove a characterization of the Heisenberg uniqueness pairs corresponding to finitely many parallel lines with an irregular gap. We observe that the size of the determining sets \(\Lambda \) for \(X(\Gamma )\) depends on the number of lines and their irregular distribution that further relates to a phenomenon of the interlacing of certain trigonometric polynomials. PubDate: 2022-03-30
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Abstract: Abstract The paper is mainly concerned with the interconnection of the boundary behaviour of the solutions of the exterior Dirichlet and Neumann problems of harmonic analysis for the unit disk in \({\mathbb {R}}^2\) and the unit ball in \({\mathbb {R}}^3\) with the corresponding behaviour of the associated ergodic inverse problems for the entire space. The basis is the theory of semigroups of linear operators mapping a Banach space X into itself. The classical one-parameter theory for semigroups applies in the present particular applications, actually for \(X= L^2_{2\pi }\) in case of the unit disk, and \(X=L^2(S)\) in the three dimensional setting, S being the unit sphere in \({\mathbb {R}}^3\) . Another tool is a Drazin-like inverse operator B for the infinitesimal generator A of a semigroup that arises naturally in ergodic theory. This operator B is a closed, not necessarily bounded, operator. It was introduced in a paper with Butzer and Westphal (Indiana Univ Math J 20:1163–1174, 1970/1971) and extended to a generalized setting with Butzer and Koliha (J Oper Theory 62:297–326, 2009). PubDate: 2022-03-29
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Abstract: Abstract In compressed sensing, the sensing matrices of minimal sample complexity are constructed with the help of randomness. Over 13 years ago, Tao (Open question: deterministic UUP matrices. https://terrytao.wordpress.com/2007/07/02/open-question-deterministic-uup-matrices/) posed the notoriously difficult problem of derandomizing these sensing matrices. While most work in this vein has been in the setting of explicit deterministic matrices with uniform guarantees, the present paper focuses on explicit random matrices of low entropy with non-uniform guarantees. Specifically, we extend the techniques of Hügel et al. (Found Comput Math 14:115–150, 2014) to show that for every \(\delta \in (0,1]\) , there exists an explicit random \(m\times N\) partial Fourier matrix A with \(m\le C_1(\delta )s\log ^{4/\delta }(N/\epsilon )\) and entropy at most \(C_2(\delta )s^\delta \log ^5(N/\epsilon )\) such that for every s-sparse signal \(x\in {\mathbb {C}}^N\) , there exists an event of probability at least \(1-\epsilon \) over which x is the unique minimizer of \(\Vert z\Vert _1\) subject to \(Az=Ax\) . The bulk of our analysis uses tools from decoupling to estimate the extreme singular values of the submatrix of A whose columns correspond to the support of x. PubDate: 2022-03-29
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Abstract: Abstract We prove that if P(D) is some constant coefficient partial differential operator and f is a scalar field such that P(D)f vanishes in a given open set, then the integrals of f over all lines intersecting that open set determine the scalar field uniquely everywhere. This is done by proving a unique continuation property of fractional Laplacians which implies uniqueness for the partial data problem. We also apply our results to partial data problems of vector fields. PubDate: 2022-03-26
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Abstract: Abstract We consider the commutators [b, T] and \([b,I_{\rho }]\) , where T is a Calderón–Zygmund operator, \(I_{\rho }\) is a generalized fractional integral operator and b is a function in the closure of \(C^{\infty }_{\mathrm {comp}}(\mathbb {R}^n)\) with respect to generalized Campanato spaces. We give a necessary and sufficient condition for the compactness of [b, T] and \([b,I_{\rho }]\) on Orlicz-Morrey spaces. The Orlicz-Morrey spaces unify Orlicz and Morrey spaces, and the Campanato spaces unify \(\mathrm {BMO}\) and Lipschitz spaces. Therefore, our results contain many previous results as corollaries. PubDate: 2022-03-23
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Abstract: Abstract In a previous work we proved a spectral multiplier theorem of Mihlin–Hörmander type for two-dimensional Grushin operators \(-\partial _x^2 - V(x) \partial _y^2\) , where V is a doubling single-well potential, yielding the surprising result that the optimal smoothness requirement on the multiplier is independent of V. Here we refine this result, by replacing the \(L^\infty \) -Sobolev condition on the multiplier with a sharper \(L^2\) -Sobolev condition. As a consequence, we obtain the sharp range of \(L^1\) -boundedness for the associated Bochner–Riesz means. The key new ingredient of the proof is a precise pointwise estimate in the transition region for eigenfunctions of one-dimensional Schrödinger operators with doubling single-well potentials. PubDate: 2022-03-22
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Abstract: Abstract The boundedness of the Hardy–Littlewood maximal operator, and the weighted extrapolation in grand variable exponent Lebesgue spaces are established provided that Hardy–Littlewood maximal operator is bounded in appropriate variable exponent Lebesgue space. Moreover, we give some bounds of the norm of the Hardy–Littlewood maximal operator in these spaces. As corollaries, we have appropriate norm inequalities and the boundedness of operators of Harmonic Analysis such as maximal and sharp maximal functions; Calderón–Zygmund singular integrals, commutators of singular integrals in grand variable exponent Lebesgue spaces. Finally, applying the boundedness results of integral operators of Harmonic Analysis, we have the direct and inverse theorems on the approximation of \(2\pi \) -periodic functions by trigonometric polynomials in the framework of grand variable exponent Lebesgue spaces. PubDate: 2022-03-21
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Abstract: Abstract Phase retrieval is concerned with recovering a function f from the absolute value of its Fourier transform \( {\widehat{f}} \) . We study the stability properties of this problem in Lebesgue spaces. Our main results shows that $$\begin{aligned} \Vert f-g\Vert _{L^2({\mathbb {R}}^n)} \le 2\cdot \Vert {\widehat{f}} - {\widehat{g}} \Vert _{L^2({\mathbb {R}}^n)} + h_f\left( \Vert f-g\Vert ^{}_{L^p({\mathbb {R}}^n)}\right) + J({\widehat{f}}, {\widehat{g}}), \end{aligned}$$ where \(1 \le p < 2\) , \(h_f\) is an explicit nonlinear function depending on the smoothness of f and J is an explicit term capturing the invariance under translations. A noteworthy aspect is that the stability is phrased in terms of \(L^p\) for \(1 \le p < 2\) : in this region \(L^p\) cannot be used to control \(L^2\) , our stability estimate has the flavor of an inverse Hölder inequality. It seems conceivable that the estimate is optimal up to constants. PubDate: 2022-03-21
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Abstract: Abstract We investigate analytic properties of the double Fourier sphere (DFS) method, which transforms a function defined on the two-dimensional sphere to a function defined on the two-dimensional torus. Then the resulting function can be written as a Fourier series yielding an approximation of the original function. We show that the DFS method preserves smoothness: it continuously maps spherical Hölder spaces into the respective spaces on the torus, but it does not preserve spherical Sobolev spaces in the same manner. Furthermore, we prove sufficient conditions for the absolute convergence of the resulting series expansion on the sphere as well as results on the speed of convergence. PubDate: 2022-03-21
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