Summability Condition Research Articles

Generalized Carleson embeddings of Müntz spaces

This paper establishes Carleson embeddings of Müntz spaces MΛq into weighted Lebesgue spaces Lp(dμ), where μ is a Borel regular measure on [0,1] satisfying μ([1−ε])≲εβ. In the case β⩾1 we show that such measures are exactly the ones for which Carleson embeddings Lpβ↪Lp(dμ) hold. The case β∈(0,1) is more intricate, but we characterize such measures μ in terms of a summability condition on their moments. Our proof relies on a generalization of Lp estimates à la Gurariy-Macaev in the weighted Lp spaces setting, which we think can be of interest in other contexts.

On weighted generator of triple sequences and its Tauberian conditions

This paper presents a new perspective on the relationship between the (N, p, q, r) method and P-convergence for triple sequences.Our primary goal is to derive Tauberian conditions that dictate the behavior of the weighted generator sequence (zlmn) in relation to the sequences (Pl), (Qm), and (Rn), with the intention of establishing a fresh interpretation. These conditions control the OL- and O-oscillatory properties and establish a connection from ( N, p, q, r) summability to P-convergence, subject to certain restrictions on the weight sequences. Furthermore, we demonstrate that specific cases, such as the OL-condition of Landau type and the O-condition of Hardy type with respect to (Pl), (Qm) and (Rn), serve as Tauberian conditions for (N, p, q, r) summability under additional conditions. Consequently, our findings encompass all the classical Tauberian theorems, including conditions related to slow decrease and slow oscillation in specific contexts.

Lipschitz Transformations and Maurey-Type Non-Homogeneous Integral Inequalities for Operators on Banach Function Spaces

We introduce a method based on Lipschitz pointwise transformations to define a distance on a Banach function space from its norm. We show how some specific lattice geometric properties (p-convexity, p-concavity, p-regularity) or, equivalently, some types of summability conditions (for example, when the terms of the terms in the sums in the range of the operator are restricted to the interval [−1,1]) can be studied by adapting the classical analytical techniques of the summability of operators on Banach lattices, which recalls the work of Maurey. We show a technique to prove new integral dominations (equivalently, operator factorizations), which involve non-homogeneous expressions constructed by pointwise composition with Lipschitz maps. As an example, we prove a new family of integral bounds for certain operators on Lorentz spaces.

Steinhaus Type Theorem for Nörlund-(M, λn) and Nörlund-Euler-(M, λn) Methods of Summability in Non-Archimedean Fields

In the present research paper, an investigation is undertaken of Steinhaus type theorems for Nörlund-(M, λn) and Nörlund-Euler(M, λn) method of summability in K, a complete non-trivially valued Non-Archimedean field. The conditions for statistical summability for those matrices are discussed in such fields K. The consistency of Nörlund-(M, λn) method of summability is investigated when different sequences are used in the summation process. Further, the relation between Nörlund-Euler(M, λn) summable and its statistical summability is also established.

On the diagonal proximal point algorithms

We study the asymptotic behaviour of the diagonal proximal point algorithms governed by a sequence of maximally monotone operators. With a summability condition on the Brézis–Haraux functions of the sequence of operators, we obtain several ergodic and weak convergence results in the monotone and subgradient cases. This approach is of relevance to and motivated by Attouch et al. [Asymptotic behaviour of nonautonomous monotone and subgradient evolution equations. Trans Amer Math Soc. 2018;370:755–790]. Some strong convergence results are presented as well. Our results in this paper include discrete versions of the results in Attouch et al. [Asymptotic behaviour of nonautonomous monotone and subgradient evolution equations. Trans Amer Math Soc. 2018;370:755–790], and also unify and improve many existing results in Attouch et al. [Prox-penalization and splitting methods for constrained variational problems. SIAM J Optim. 2011;21:149–173], Cabot [Proximal point algorithm controlled by a slowly vanishing term: Applications to hierarchical minimization. SIAM J Optim. 2005;15:555–572], Lehdili and Moudafi [Combining the proximal algorithm and Tikhonov regularization. Optimization. 1996;37:239–252], Xu [A regularization method for the proximal point algorithm. J Glob Optim. 2006;36:115–125].

Measurability, spectral densities, and hypertraces in noncommutative geometry

We introduce, in the dual Macaev ideal of compact operators of a Hilbert space, the spectral weight \rho(L) of a positive, self-adjoint operator L having discrete spectrum away from zero. We provide criteria for its measurability and unitarity of its Dixmier traces ( \rho(L) is then called spectral density) in terms of the growth of the spectral multiplicities of L or in terms of the asymptotic continuity of the eigenvalue counting function N_L . Existence of meromorphic extensions and residues of the \zeta -function \zeta_L of a spectral density are provided under summability conditions on spectral multiplicities. The hypertrace property of the states \Omega_L(\cdot)=\mathrm{Tr}_\omega(\cdot\rho(L)) on the norm closure of the Lipschitz algebra \mathcal{A}_L follows if the relative multiplicities of L vanish faster than its spectral gaps or if N_L is asymptotically regular.

Classical flows of vector fields with exponential or sub-exponential summability

We show that vector fields b whose spatial derivative Dxb satisfies a Orlicz summability condition have a spatially continuous representative and are well-posed. For the case of sub-exponential summability, their flows satisfy a Lusin (N) condition in a quantitative form, too. Furthermore, we prove that if Dxb satisfies a suitable exponential summability condition then the flow associated to b has Sobolev regularity, without assuming boundedness of divxb. We then apply these results to the representation and Sobolev regularity of weak solutions of the Cauchy problem for the transport and continuity equations.

Fourier Series Approximation in Besov Spaces

Defined on the top of classical L p -spaces, the Besov spaces of periodic functions are good at encoding the smoothness properties of their elements. These spaces are also characterized in terms of summability conditions on the coefficients in trigonometric series expansions of their elements. In this paper, we study the approximation properties of 2 π -periodic functions in a Besov space under a norm involving the seminorm associated with the space. To achieve our results, we use a summability method presented by a lower triangular matrix with monotonic rows.

On Necessary and Sufficient Conditions for Absolute Matrix Summability

This study gets a new general theorem related to necessary and sufficient conditions for $\varphi-{\mid D,\beta;\delta\mid}_{k}$ summability of the series $\sum a_n \lambda_n$ whenever the series $\sum a_n$ is summable $\varphi-{\mid C,\beta;\delta\mid}$, where $C=(c_{nv})$ and $D=(d_{nv})$ are two positive normal matrices, $k\geq 1$, $\delta\geq 0$ and $-\beta(\delta{k}+k-1)+k > 0$.

Gradient-Push Algorithm for Distributed Optimization With Event-Triggered Communications

Decentralized optimization problems consist of multiple agents connected by a network. The agents have each local cost function, and the goal is to minimize the sum of the functions cooperatively. It requires the agents to communicate with each other, and reducing the cost for communication is desired for a communication-limited environment. Recently, the decentralized gradient descent algorithms involving event-triggered communication have been proposed when the network of the agents is an undirected graph. On the other hand, the network of agents is often directed graph for realistic scenarios whose communication resources are limited. In this work, we first propose a gradient-push algorithm involving event-triggered communication on a directed network. Each agent sends its current states to its neighbors only when the differences between the latest sent states and the current states are larger than thresholds. The convergence of the algorithm is established under suitable decays and summability conditions on a stepsize and triggering thresholds. Numerical experiments are presented to support the effectiveness and the convergence results of the algorithm. More precisely, the numerical results reveals that the proposed algorithm may reduce the communication cost significantly compared to the gradient-push algorithm not involving the event-triggered communication.

Component-by-component construction of randomized rank-1 lattice rules achieving almost the optimal randomized error rate

We study a randomized quadrature algorithm to approximate the integral of periodic functions defined over the high-dimensional unit cube. Recent work by Kritzer, Kuo, Nuyens and Ullrich [J. Approx. Theory 240 (2019), pp. 96–113] shows that rank-1 lattice rules with a randomly chosen number of points and good generating vector achieve almost the optimal order of the randomized error in weighted Korobov spaces, and moreover, that the error is bounded independently of the dimension if the weight parameters,γj\gamma _j, satisfy the summability condition∑j=1∞γj1/α>∞\sum _{j=1}^{\infty }\gamma _j^{1/\alpha }>\infty, whereα\alphais a smoothness parameter. The argument is based on the existence result that at least half of the possible generating vectors yield almost the optimal order of the worst-case error in the same function spaces.In this paper we provide a component-by-component construction algorithm of such randomized rank-1 lattice rules, without any need to check whether the constructed generating vectors satisfy a desired worst-case error bound. Similarly to the above-mentioned work, we prove that our algorithm achieves almost the optimal order of the randomized error and that the error bound is independent of the dimension if the same condition∑j=1∞γj1/α>∞\sum _{j=1}^{\infty }\gamma _j^{1/\alpha }>\inftyholds. We also provide analogous results for tent-transformed lattice rules for weighted half-period cosine spaces and for polynomial lattice rules in weighted Walsh spaces, respectively.

Summability condition and rigidity for finite type maps

We extend a series of results due to Makienko, Dominguez and Sienra on the rigidity of some holomorphic dynamical systems with summable critical values to the setting of finite type maps. We also recover a shorter proof of a transversality theorem of Levin. Our methods are based on the deformation theory introduced by Epstein.

Multi-step inertial forward-backward-half forward algorithm for solving monotone inclusion

In this article, we study a multi-step inertial forward-backward-half forward splitting algorithm to solve the solution of the sum of three operators with two of them being respectively, cocoercive and monotone-Lipschitz continuous. Notably, the convergence of the multi-step inertial forward-backward-half forward algorithm is established under the summability condition formulated on the basis of the iterative sequence in a Hilbert space setting. We also display two applications including problems that are: a composite monotone inclusion problem involving mixtures of linearly composed with parallel-sum type operators, and a composite convex optimization problem. As far as the resolvent of the composite operator is concerned, due to difficulty in calculation, then, the composite monotone inclusion problem is addressed by the proposed multi-step inertial algorithm together with the primal-dual idea. Experiments conducted on an image deblurring problem indicate the superiority and efficiency of the proposed method.

Partial best approximations and the absolute Cesaro summability of multiple Fourier series

The article is devoted to the problem of absolute Cesaro summability of multiple trigonometric Fourier series. Taking a central place in the theory of Fourier series this problem was developed quite widely in the one-dimensional case and the fundamental results of this theory are set forth in the famous monographs by N.K. Bari, A. Zigmund, R. Edwards, B.S. Kashin and A.A. Saakyan [1–4]. In the case of multiple series, the corresponding theory is not so well developed. The multidimensional case has own specifics and the analogy with the one-dimensional case does not always be unambiguous and obvious. In this article, we obtain sufficient conditions for the absolute summability of multiple Fourier series of the function f ∈ Lq(Is) in terms of partial best approximations of this function. Four theorems are proved and four different sufficient conditions for the |C; β¯|λ-summability of the Fourier series of the function f are obtained. In the first theorem, a sufficient condition for the absolute |C; β¯|λ- summability of the Fourier series of the function f is obtained in terms of the partial best approximation of this function which consists of s conditions, in the case when β1 = ... = βs = 1/q'. Other sufficient conditions are obtained for double Fourier series. Sufficient conditions for the |C; β1; β2|λ-summability of the Fourier series of the function f ∈ Lq(I2) are obtained in the cases β1 = 1/q', −1 < β2 < 1/q'(in the second theorem), 1/q'< β1 < +∞, β2 = 1/q', (in the third theorem), −1 < β1 < 1/q', 1/q' < β2 < +∞ (in the fourth theorem).

The approximation of the solution of wave problems by spectral expansions connected with elliptic differential operators

In this research, we investigate the spectral expansions connected with elliptic differential operators in the space of singular distributions, which describes the vibration process made of thin elastic membrane stretched tightly over a circular frame. The sufficient conditions for summability of the spectral expansions connected with wave problems on the disk are obtained by taking into account that the deflection of the membrane during the motion remains small compared to the size of the membrane and for wave propagation problems, the disk is made of some thermally conductive material.

MAXIMAL REGULARITY ESTIMATE FOR A DIFFERENTIAL EQUATION WITH OSCILLATING COEFFICIENTS

MAXIMAL REGULARITY ESTIMATE FOR A DIFFERENTIAL EQUATION WITH OSCILLATING COEFFICIENTS

Soliton Decomposition of the Box-Ball System

AbstractThe box-ball system (BBS) was introduced by Takahashi and Satsuma as a discrete counterpart of the Korteweg-de Vries equation. Both systems exhibit solitons whose shape and speed are conserved after collision with other solitons. We introduce a slot decomposition of ball configurations, each component being an infinite vector describing the number of sizeksolitons in eachk-slot. The dynamics of the components is linear: thekth component moves rigidly at speedk. Let$\zeta $be a translation-invariant family of independent random vectors under a summability condition and$\eta $be the ball configuration with components$\zeta $. We show that the law of$\eta $is translation invariant and invariant for the BBS. This recipe allows us to construct a large family of invariant measures, including product measures and stationary Markov chains with ball density less than$\frac {1}{2}$. We also show that starting BBS with an ergodic measure, the position of a taggedk-soliton at timet, divided bytconverges as$t\to \infty $to an effective speed$v_k$. The vector of speeds satisfies a system of linear equations related with the generalised Gibbs ensemble of conservative laws.

Application of quasi-f -power increasing sequence in absolute ph - |C, a, b; d; l| of infinite series

An increasing quasi-f-power sequence of a wider class has been used to establish a universal theorem on a least set of conditions, which is sufficient for an infinite series to be generalized ph-|C, a, b; d; l| k summable. Further, a set of new and well-known arbitrary results have been obtained by using the main theorem. Considering suitable conditions a previous result has been obtained, which validates the current findings. In this way, Bounded Input Bounded Output(BIBO) stability of impulse has been improved by finding a minimal set of sufficient condition for absolute summability because absolute summable is the necessary and sufficient conditions for BIBO stability.

Strong Riesz summability of Fourier series

The notion of strong summability was introduced by Fekete (Math. És Termesz Ertesitö, 34 (1916), 759-786). Dealing with Nörlund summability of Fourier series Mittal (J. Math. Anal. Appl. 314 (2006), 75-84) has established a result on strong summability. We have established a new result on sufficient condition for strong Riesz summability of Fourier series.

Mean-field model for the junction of two quasi-1-dimensional quantum Coulomb systems

Junctions appear naturally when one studies surface states or transport properties of quasi one dimensional materials such as carbon nanotubes, polymers and quantum wires. These materials can be seen as 1D systems embedded in the 3D space. In this article, we first establish a mean--field description of reduced Hartree--Fock type for a 1D periodic system in the 3D space (a quasi 1D system), the unit cell of which is unbounded. With mild summability condition, we next show that a quasi 1D system in its ground state can be described by a mean--field Hamiltonian. We also prove that the Fermi level of this system is always negative. A junction system is described by two different infinitely extended quasi 1D systems occupying separately half spaces in 3D, where Coulombic electron-electron interactions are taken into account and without any assumption on the commensurability of the periods. We prove the existence of the ground state for a junction system, the ground state is a spectral projector of a mean--field Hamiltonian, and the ground state density is unique.