Journal of Fourier Analysis and Applications最新文献

The study of the infinite Hankel matrix acting on analytic function spaces dates back to the influential work of Nehari and Widom on the Hardy space (H^2). Since then, it has been extensively generalized to other settings such as weighted Bergman spaces, Dirichlet type spaces, and Möbius invariant function spaces. Nevertheless, several fundamental operator-theoretic questions, including the boundedness and compactness, remain unresolved in the context of the Dirichlet space. Motivated by this, via Carleson measures, the Widom type condition, and the reproducing kernel thesis, we obtain:

1. (i)

   necessary and sufficient conditions for bounded and compact operators induced by Hankel matrices on the Dirichlet space, thereby answering a folk question in this field (Galanopoulos et al. in Result Math 78(3) Paper No. 106, 2023);

2. (ii)

   necessary and sufficient conditions for bounded and compact operators induced by Cesàro type matrices on the Dirichlet space.

As a beneficial product, we find an intrinsic function-theoretic characterization of functions with positive decreasing Taylor coefficients in the function space ({mathcal {X}}) throughly studied by Arcozzi et al. (Lond Math Soc II Ser 83(1):1–18, 2011). In addition, we also show that a random Dirichlet function almost surely induces a compact Hankel type operator on the Dirichlet space.

A projective representation of a locally compact group does phase retrieval if it admits a maximal spanning frame vector. In this paper, we provide a characterization of maximal spanning vectors for type I and square integrable irreducible projective representations of separable locally compact abelian groups. This generalizes the well-known criterion for the time–frequency case and unifies previous criteria for finite groups case and locally compact Gabor case. As an application, we show that irreducible projective representations of compact abelian groups do phase retrieval.

In this paper, let (Omega ) be homogeneous of degree zero which has vanishing moment of order one, A be a function on (mathbb {R}^d) such that (nabla Ain textrm{BMO}(mathbb {R}^d)), we consider a class of nonstandard singular integral operators, (T_{Omega ,,A}), with rough kernel being of the form ( frac{Omega (x-y)}{vert x-yvert ^{d+1}}big (A(x)-A(y)-nabla A(y)(x-y)big ) ). This operator is closely related to the Calderón commutator. We prove that, under the Grafakos-Stefanov minimum size condition (GS_{beta }(S^{d-1})) with (2<beta <infty ) for (Omega ), (T_{Omega ,,A}) is bounded on (L^p(mathbb {R}^d)) for p with (1+1/(beta -1)< p < beta ).

We discuss the Hausdorff–Young inequality in the context of maximal integral estimates, including the case of Hermite and Laguerre expansions. We establish a maximal inequality for integral operators with bounded kernel on ({mathbb {R}}), which in particular allows for the pointwise evaluation of these operators, including the Fourier transform, for functions in appropriate Lorentz and Orlicz spaces. In the case of the Hermite expansions we prove a refined Hausdorff–Young inequality, further sharpened by considering the maximal Hermite coefficients in place of the Hermite coefficients when estimating the appropriate Lorentz and Orlicz norms. We also consider the refined companion Hausdorff–Young inequality and Hardy–Littlewood type inequalities for the Hermite expansions. Similar results are proved for the Laguerre expansions.

In the present paper we discuss the rate of the approximation by the matrix transform of special partial sums of some two-dimensional rectangle (decreasing diagonal) Walsh-Fourier series in (L^p(G^{2})) space ((1le p <infty )) and in (C(G^{2})). It implies in some special case

norm convergence. We also show an application of our results for Lipschitz functions. At the end of the paper we show the most important result, the almost everywhere convergence theorem. We note that T summation is a common generalization of the following known summation methods Cesàro, Weierstrass, Riesz and Picar and Bessel methods.

In this paper we focus our attention on matrix or operator-valued spherical functions associated to finite groups (G, K), where K is a subgroup of G. We introduce the notion of matrix-valued spherical functions on G associated to any K-type (delta in hat{K}) by means of solutions of certain associated integral equations. The main properties of spherical functions are established from their characterization as eigenfunctions of right convolution multiplication by functions in (A[G]^K), the algebra of K-central functions in the group algebra A[G]. The irreducible representations of (A[G]^K) are closely related to the irreducible spherical functions on G. This allows us to study and compute spherical functions via the representations of this algebra.

In this paper, we first prove a density theorem for operator-valued frames via square-integrable representations restricted to closed subgroups of locally compact groups, which is a natural extension of the density theorem in classical Gabor analysis. More precisely, it is proved that for such an operator-valued frame, the index subgroup is co-compact if and only if the generator is a Hilbert–Schmidt operator. Then we present some applications of this density theorem, and in particular establish necessary and sufficient conditions for the existence of such operator-valued frames with Hilbert–Schmidt generators. We also introduce the concept of wavelet transform for Hilbert–Schmidt operators, and use it to prove that if the representation space is infinite-dimensional, then the system indexed by the entire group is Bessel system but not a frame for the space of all Hilbert–Schmidt operators on the representation space.

We revisit Fourier’s approach to solve the heat equation on the circle in the context of (twisted) reduced group C*-algebras, convergence of Fourier series and semigroups associated to negative definite functions. We introduce some heat properties for countably infinite groups and investigate when they are satisfied. Kazhdan’s property (T) is an obstruction to the weakest property, and our findings leave open the possibility that this might be the only one. On the other hand, many groups with the Haagerup property satisfy the strongest version. We show that this heat property implies that the associated heat problem has a unique solution regardless of the choice of the initial datum.

We introduce a class of FBI transforms using weight functions (which includes the subclass of Sjöstrand’s FBI transforms used by Christ in (Commun Partial Differ Equ 22(3–4):359–379, 1997)) that is well suited when dealing with ultradifferentiable functions (see Definition 2.3) and ultradistributions (see Definition 2.15) defined by weight functions in the sense of Braun, Meise and Taylor (BMT). We show how to characterize local regularity of BMT ultradistributions using this wider class of FBI transform and, as an application, we characterize the BMT vectors (see Definition 1.2) and prove a relation between BMT local regularity and BMT vectors.

We prove the existence of functions that extremize the endpoint (L^2) to (L^4) adjoint Fourier restriction inequality on the one-sheeted hyperboloid in Euclidean space (mathbb {R}^4) and that, taking symmetries into consideration, any extremizing sequence has a subsequence that converges to an extremizer.