Recently, an approach to graph signal processing based on graphons was proposed. Here we show how such a graphon-driven approach to the Fourier transform can be used on graphs sampled from a stochastic block model (SBM). In particular, we show how a Fourier basis can be easily calculated from the block sizes and the block probability matrix. Using perturbation theory, we derive bounds on the sensitivity of the basis with respect to variations in the block sizes. We then consider SBMs constructed from weighted Cayley graphs. When block sizes are equal, a nice Fourier basis can be derived from the representation theory of the underlying group. When block sizes are nearly uniform, we demonstrate that this Fourier basis closely approximates the SBM Fourier basis. For highly non-uniform block sizes, the group-based Fourier basis is no longer applicable, though, as we show, the underlying group still provides partial information about the SBM Fourier basis.
This paper introduces a special affine shearlet transform, as a generalization of the nD-shearlet transform. For this purpose, we first study an nD-special affine Fourier transform and examine its fundamental properties for both complex and quaternion valued functions. Subsequently, we extend an existing convolution associated with the special affine Fourier transform to the multidimensional setup and establish the corresponding convolution theorem. Building on this framework, we introduce an nD-special affine shearlet transform (nDSAST) via the convolution and investigate its key properties, including Parseval’s relation and an inversion formula. Furthermore, we derive a range characterization theorem and establish a Heisenberg-type uncertainty principle. Finally, we extend the nDSAST to the setting of quaternion valued functions in a consistent manner.
In this paper we propose a dual version of the Furstenberg set problem and obtain partial results via L^p estimates of orthogonal projections. Examples are also discussed. Moreover, compared with general sets, we find that special structure like Cartesian product has better L^p -behavior. This leads to an improvement on some discretized sum-product estimates.
Let {φ _j(x,y)}_j =1^∞ be a uniformly bounded orthonormal system on [0,1]^2 , {A_n}_n=1^∞ be sequence of bounded subsets of ℤ_+={1, 2, …} and {N_n} denote the number of elements in A_n . Let also L_A_n(x,y) and S_A_n(f; x,y) denote, respectively, the Lebesgue function and the partial sum of the Fourier series of a function f with respect to the system {φ _j(x,y)} corresponding to the indices from A_n . It is proved that if the limit of the sequence {L_A_n(x,y)/log N_n} is ∞ almost everywhere on [0,1]^2 , then there exists an integrable on [0,1]^2 function h and a strictly increasing sequence {n_k} of positive integers, such that lim _k→∞| S_A_n_k(h;x,y)| =∞ almost everywhere.
Pairs of (a, b)-Gabor dual frames for L^2(ℝ) have been extensively studied in the classical “painless-expansion” region with ab≤1/2 . Beyond these regions, this paper considers ab∈ (1/2,2/3] with the normalized translation parameter a=1 and modulation parameter b∈ (1/2,2/3] . For any given primal window g supported inside [-1,1] , we characterize all pairs (g, h) of compactly supported Gabor dual frames for L^2(ℝ) with the dual windows h having explicit parametric expressions. We further characterize these dual windows having continuity, high smoothness, and symmetry. Based on our characterizations, we present several examples of pairs (g, h) of Gabor dual frames such that the dual windows h have small supports and several desired properties such as high smoothness, symmetry, and small condition numbers, which are nearly optimal with respect to the primal window g and exhibit excellent stability and strong robustness. In particular, by employing smoothing techniques, we construct a family of pairs of smooth Gabor dual frames with the near smallest optimal condition number such that both primal and dual Gabor frames are very close to Gabor tight frames for L^2(ℝ) .
We identify all classical Markov processes obtained by restricting Biane’s quantum Ornstein–Uhlenbeck semigroups to commutative C^* -algebras related to Gelfand pairs built on Heisenberg groups. It turns out that they form a family of (mostly) multivariate continuous-time pure birth and pure death chains, with birth and death rates defined in terms of the generalized binomial coefficients for multiplicity free actions. The state spaces for some of those processes are some sets of partitions (equivalently, Young diagrams).
In this paper, we establish a normed fractional Faà di Bruno inequality within the framework of Lebesgue spaces. This extends the classical fractional chain rule from the range 0< s < 1 to an arbitrarily large value of s .
The space CMO is a proper subspace of BMO typically employed in harmonic analysis to prove compactness results for commutators. It has been shown, however, that there exists another space XMO, which is also a proper subspace of BMO but larger than CMO, so that the commutators of pointwise multiplications by its members with certain pseudodifferential operators of order zero (associated with singular integral operator of Calderón-Zygmund type) still produce compact operators on Lebesgue spaces. The purpose of this article is to proved the analogous result for smoothing pseudodifferential operators of finite order (associated with fractional integral operators).
A closed subspace X of L^p[0,1] (1≤ p <∞ ) is called a Λ (p) -space if there exists a number 10 , we establish a representation of Λ (p) -spaces in terms of such operators. We prove that if for 1≤p
The so-called DCD-matrices consist of products of two diagonal matrices with a circulant matrix, merging diagonal and circulant matrices with rank-one matrices. By introducing a notion of double orthogonality for rectangular matrices, an iterative double orthogonalization process is devised for approximating a given M∈ ℂ^n × n with DCD-matrices. This simultaneous orthogonality of columns and rows leads to several approximation schemes of which the best rank-one approximation is just a special case. Expanding through summing yields, like principal component analysis (PCA), a novel optimal technique for reducing the dimensionality of datasets. Being notably general and flexible, this approach provides a natural way to merge fast Fourier methods with low rank matrix approximations. Least squares solution methods play a significant role in algorithms.
We establish an analogue of Pitt’s inequality for Fourier series on 𝕋^d , extending the range of parameters by employing averages of Fourier coefficients. We show that this inequality is closely connected to sharp estimates for the L_p norms of the Dirichlet kernels over various subsets of ℤ^d .
This paper aims to establish global well-posedness results for nonlinear wave equations (NLWs) in a broader class of weak-Besov spaces. We consider nonlinearities of both single- and double-power types, and carry out the analysis in higher dimensions, n≥ 3 . To achieve these results, we develop suitable composition-type estimates within our functional framework. These estimates are of independent interest and provide a detailed understanding of how the nonlinearity influences the behavior of solutions in such spaces. In addition, we derive certain time-weighted dispersive estimates for the wave group, which naturally arise in the course of the well-posedness analysis.
We study the pointwise convergence of Landau type Schrödinger operators on Bessel potential spaces L_s^p (ℝ) . Our results extend those established by Bailey (Rev. Mat. Iberoam., 29 (2): 531–546, 2013) and Yuan, Zhao and Zheng (Nonlinear Anal., 208: Paper No. 112312, 28, 2021). Furthermore, we also analyze the convergence rate of Landau type Schrödinger operators along curves and derive a sharp result for the case of convergence along vertical lines.
We propose a survey on composition operators acting in Sobolev spaces including their continuity and the Faà di Bruno’s formula, together with similar properties for composition operators in homogeneous Adams-Frazier spaces Ẇ^m_p∩Ẇ^1_mp(ℝ^n) .
In this note, we investigate orthogonal expansions adapted to exponential weights and function spaces endowed with Hermite-frequency decompositions. Our objective is to formulate the Hausdorff–Young inequality within the framework of Hermite based Triebel–Lizorkin type spaces. The results rest on two underlying principles. First, since the Hermite functions {ℋ_m(x)} , along with their generalized counterparts, are particular instances of generalized Freud functions associated with generalized Freud weights, we rely on the L^∞ and L^q , q<∞ , estimates of Kasuga and Sakai which extend Hille’s estimate |ℋ_m(x)|≲ m^-1/12 , m=1,2,… Second, we apply interpolation techniques involving the Orlicz–Lorentz spaces Λ (φ _X, C) , where C is a Young (or concave) function and φ _X denotes the fundamental function of a rearrangement-invariant space X. This approach enables us to derive Hausdorff–Young inequalities in L^p , Lorentz, Orlicz, and Λ (φ _X, C) based Hermite Triebel–Lizorkin spaces. Finally, in the Coda we present a general Hausdorff–Young inequality for the generalized Freud coefficients of functions f∈ L^p_0(ℝ)+L^p_1(ℝ) , where 1≤ p_0
The purpose of this study is to develop a discrete frame decomposition of the Besov and Triebel-Lizorkin spaces on the product X_1× X_2 of doubling metric measure spaces X_1 , X_2 associated with non-negative self-adjoint operators L_1 , L_2 , whose heat kernels have Gaussian localization. To achieve this, we first establish a pair of frames with sub-exponential spatial localization and compact spectral support. Some advances of independent interest in the theory related to product spaces, including the lower bounds of the L^p norms of kernels, of cut-off functions are also obtained.
Consider weighted Bergman spaces A^p_α (Ω ) , where Ω is a bounded strongly pseudo-convex domain with smooth boundary in ℂ^n . Our first main result demonstrates that if δ ^(n+1+α )μ be a Carleson measure for A^p_α (Ω )(1
Let X be a translation-invariant Banach function space on the unit circle 𝕋 with the associate space X' , let w be a weight such that w∈ X and 1/w∈ X' , let X(w) consist of measurable functions f:𝕋→ℂ such that fw∈ X , and let H[X] and H[X(w)] denote the abstract Hardy spaces built upon X and X(w), respectively. Extending Rudin’s arguments (1962), we show that if 𝒫 is a bounded projection from X(w) onto H[X(w)], then the Riesz projection P is bounded from X onto H[X] and ‖ aI+bP‖ _ℬ(X)≤‖ aI+b𝒫‖ _ℬ(X(w)) for all a,b∈ℂ . Further, for m∈ℕ , let T(e_-m) be the Toeplitz operator with symbol e_-m(t)=t^-m . We prove that ‖ T(e_-m)‖ _ℬ(H[X])≤‖ T(e_-m)‖ _ℬ(H[X(w)]) for all m∈ℕ .
