Fast algorithm of adaptive Fourier series

Adaptive Fourier decomposition (AFD, precisely 1‐D AFD or Core‐AFD) was originated for the goal of positive frequency representations of signals. It achieved the goal and at the same time offered fast decompositions of signals. There then arose several types of AFDs. The AFD merged with the greedy algorithm idea, and in particular, motivated the so‐called pre‐orthogonal greedy algorithm (pre‐OGA) that was proven to be the most efficient greedy algorithm. The cost of the advantages of the AFD‐type decompositions is, however, the high computational complexity due to the involvement of maximal selections of the dictionary parameters. The present paper constructs one novel method to perform the 1‐D AFD algorithm. We make use of the FFT algorithm to reduce the algorithm complexity, from the original to , where N denotes the number of the discretization points on the unit circle and M denotes the number of points in [0,1). This greatly enhances the applicability of AFD. Experiments are performed to show the high efficiency of the proposed algorithm.     

Orthogonal polynomials with respect to the sum of an arbitrary measure and a Bernstein–Szegö measure

In the present paper we study the orthogonal polynomials with respect to a measure which is the sum of a finite positive Borel measure on [0,2π] and a Bernstein–Szegö measure. We prove that the measure sum belongs to the Szegö class and we obtain several properties about the norms of the orthogonal polynomials, as well as, about the coefficients of the expression which relates the new orthogonal polynomials with the Bernstein–Szegö polynomials. When the Bernstein–Szegö measure corresponds to a polynomial of degree one, we give a nice explicit algebraic expression for the new orthogonal polynomials.     

A representing system generated by the Szegö kernel for the Hardy space

In this paper we give an explicit construction of a representing system generated by the Szegö kernel for the Hardy space. Thus we answer an open question posed by Fricain, Khoi and Lefèvre. We use frame theory to prove the main result.     

Comparison of the convergence rate of pure greedy and orthogonal greedy algorithms

The following two types of greedy algorithms are considered: the pure greedy algorithm (PGA) and the orthogonal greedy algorithm (OGA). From the standpoint of estimating the rate of convergence on the entire class A ₁(D), the orthogonal greedy algorithm is optimal and significantly exceeds the pure greedy algorithm. The main result in the present paper is the assertion that the situation can also be opposite for separate elements of the class A ₁(D) (and even of the class A ₀(D)): the rate of convergence of the orthogonal greedy algorithm can be significantly lower than the rate of convergence of the pure greedy algorithm.     

Aveiro method in reproducing kernel Hilbert spaces under complete dictionary

Aveiro method is a sparse representation method in reproducing kernel Hilbert spaces, which gives orthogonal projections in linear combinations of reproducing kernels over uniqueness sets. It, however, suffers from determination of uniqueness sets in the underlying reproducing kernel Hilbert space. In fact, in general spaces, uniqueness sets are not easy to be identified, let alone the convergence speed aspect with Aveiro method. To avoid those difficulties, we propose an new Aveiro method based on a dictionary and the matching pursuit idea. What we do, in fact, are more: The new Aveiro method will be in relation to the recently proposed, the so‐called pre‐orthogonal greedy algorithm involving completion of a given dictionary. The new method is called Aveiro method under complete dictionary. The complete dictionary consists of all directional derivatives of the underlying reproducing kernels. We show that, under the boundary vanishing condition bring available for the classical Hardy and Paley‐Wiener spaces, the complete dictionary enables an efficient expansion of any given element in the Hilbert space. The proposed method reveals new and advanced aspects in both the Aveiro method and the greedy algorithm.     

A Comparison Between the Bergman and Szegö Kernels of the Non-smooth Worm Domain $$D'_{\upbeta }$$

In this work we provide an asymptotic expansion for the Szegö kernel associated to a suitably defined Hardy space on the non-smooth worm domain \(D'_{\upbeta }\). After describing the singularities of the kernel, we compare it with an asymptotic expansion of the Bergman kernel. In particular, we show that the Bergman kernel has the same singularities of the first derivative of the Szegö kernel with respect to any of the variables. On the side, we prove the boundedness of the Bergman projection operator on Sobolev spaces of integer order.     

Approximation with Bernstein–Szegö polynomials

We present approximation kernels for orthogonal expansions with respect to Bernstein–Szegö polynomials. Theconstruction is derived from known results for Chebyshev polynomials of the first kind and does not pose any restrictions on the Bernstein–Szegö polynomials.     

Convex symmetrization and Pólya–Szegö inequality

We prove a Pólya–Szegö inequality involving a convex symmetrization of functions and we investigate the equality case.     

Approximation of monogenic functions by higher order Szeg kernels on the unit ball and half space

We study the adaptive decomposition of functions in the monogenic Hardy spaces H2by higher order Szeg kernels under the framework of Clifford algebra and Clifford analysis,in the context of unit ball and half space.This is a sequel and a higher-dimensional generalization of our recent study on the complex Hardy spaces.     

Efficient approximation algorithm for the Schrödinger–Possion system

In this article, we study an efficient approximation algorithm for the Schrödinger–Possion system arising in the resonant tunneling diode (RTD) structure. By following the classical Gummel iterative procedure, we first decouple this nonlinear system and prove the convergence of the iteration method. Then via introducing a novel spatial discrete method, we solve efficiently the decoupled Schrödinger and Possion equations with discontinuous coefficients on no‐uniform meshes at each iterative step, respectively. Compared with the traditional ones, the algorithm considered here not only has a less restriction on the discrete mesh, but also is more accurate. Finally, some numerical experiments are shown to confirm the efficiency of the proposed algorithm.     

Szegö Limit Theorems on the Sierpiński Gasket

We use the existence of localized eigenfunctions of the Laplacian on the Sierpiński gasket (SG) to formulate and prove analogues of the strong Szegö limit theorem in this fractal setting. Furthermore, we recast some of our results in terms of equally distributed sequences.     

Comparing the Bergman and Szegö projections on domains with subelliptic boundary Laplacian

We prove that the difference between the Bergman and Szegö projections on a bounded, pseudoconvex domain (with C ^∞ boundary) is smoothing whenever the boundary Laplacian is subelliptic. An equivalent statement is that the Bergman projection can be represented as a composition of the Szegö and harmonic Bergman projections (along with the restriction and Poisson extension operators) modulo an error that is smoothing. We give several applications to the study of optimal mapping properties for these projections and their difference.     

Hardy–Carleson Measures and Their Dual Poisson–Szegö Transforms

Recently Choe et al. have introduced the notion of dual Berezin transforms and used it to obtain new characterizations of the Carleson measures for the weighted Bergman spaces over the unit ball in C ⁿ . Continuing our investigation on the Hardy spaces, we obtain new characterizations of the Carleson measures for the Hardy spaces by means of the dual Poisson–Szegö transforms introduced by Koosis. Compared with the results for the weighted Bergman spaces, our results for the Hardy spaces not only show an similarity, but also reveal a new characterization.     

On the average order of a class of arithmetical functions

We consider a class of arithmetical functions generated by Dirichlet series that satisfy a functional equation with multiple gamma factors. We prove one- and two-sided omega theorems for the error terms associated with summatory functions of the type Σ_λn≤xa(n)(x ? λ_n)^?, where ? ≥ 0. In particular, we improve results of Hardy for the circle and Dirichlet divisor problems and results of Szegö and Walfisz for the Piltz divisor problem in algebraic number fields.     

Consecutive minimum phase expansion of physically realizable signals with applications

In digital signal processing, it is a well know fact that a causal signal of finite energy is front loaded if and only if the corresponding analytic signal, or the physically realizable signal, is a minimum phase signal, or an outer function in the complex analysis terminology. Based on this fact, a series expansion method, called unwinding adaptive Fourier decomposition (AFD), to give rise to positive frequency representations with rapid convergence was proposed several years ago. It appears to be a promising positive frequency representation with great potential of applications. The corresponding algorithm, however, is complicated due to consecutive extractions of outer functions involving computation of Hilbert transforms. This paper is to propose a practical algorithm for unwinding AFD that does not depend on computation of Hilbert transform, but, instead, factorizes out the Blaschke product type of inner functions. The proposed method significantly improves applicability of unwinding AFD. As an application, we give the associated Dirac‐type time‐frequency distribution of physically realizable signals. Copyright © 2015 John Wiley & Sons, Ltd.     

On monotonicity of some functionals under rearrangements

We consider the Pólya–Szegö type weighted inequality. We prove this inequality for monotone rearrangement and for Steiner’s symmetrization. In particular we fill the gap in the paper of Brock (Calc Var PDEs 8:15–25, 1999) for 1D case.     

Lyapunov stability for semigroup actions

In this paper we introduce a theory of Lyapunov stability of sets for semigroup actions on Tychonoff spaces. We also present the main properties and the main results relating these new concepts. We generalize several concepts and results of Lyapunov stable sets from Bhatia and Hajek (Local Semi-Dynamical Systems. Lecture Notes in Mathematics, vol. 90. Springer, Berlin, 1969), Bhatia and Szegö (Dynamical Systems: Stability Theory and Applications. Lecture Notes in Mathematics, vol. 35. Springer, Berlin, 1967; and Stability Theory of Dynamical Systems. Springer, Berlin, 1970).     

A rainbow blow‐up lemma

We prove a rainbow version of the blow‐up lemma of Komlós, Sárközy, and Szemerédi for μn‐bounded edge colorings. This enables the systematic study of rainbow embeddings of bounded degree spanning subgraphs. As one application, we show how our blow‐up lemma can be used to transfer the bandwidth theorem of Böttcher, Schacht, and Taraz to the rainbow setting. It can also be employed as a tool beyond the setting of μn‐bounded edge colorings. Kim, Kühn, Kupavskii, and Osthus exploit this to prove several rainbow decomposition results. Our proof methods include the strategy of an alternative proof of the blow‐up lemma given by Rödl and Ruciński, the switching method, and the partial resampling algorithm developed by Harris and Srinivasan.     

Necessary and sufficient conditions for the boundedness of weighted Hardy operators in Hölder spaces

We study one‐ and multi‐dimensional weighted Hardy operators on functions with Hölder‐type behavior. As a main result, we obtain necessary and sufficient conditions on the power weight under which both the left and right hand sided Hardy operators map, roughly speaking, functions with the Hölder behavior only at the singular point to functions differentiable for and bounded after multiplication by a power weight. As a consequence, this implies necessary and sufficient conditions for the boundedness in Hölder spaces due to the corresponding imbeddings. In the multi‐dimensional case we provide, in fact, stronger Hardy inequalities via spherical means. We also separately consider the case of functions with Hölder‐type behavior at infinity (Hölder spaces on the compactified ).