A quantitative mean ergodic theorem for uniformly convex Banach spaces

We initiate the study of effective pointwise ergodic theorems in resource-bounded settings. Classically, the convergence of the ergodic averages for integrable functions can be arbitrarily slow [Ulrich Krengel, 1978]. In contrast, we show that for a class of PSPACE L¹ functions, and a class of PSPACE computable measure-preserving ergodic transformations, the ergodic average exists and is equal to the space average on every EXP random. We establish a partial converse that PSPACE non-randomness can be characterized as non-convergence of ergodic averages. Further, we prove that there is a class of resource-bounded randoms, viz. SUBEXP-space randoms, on which the corresponding ergodic theorem has an exact converse - a point x is SUBEXP-space random if and only if the corresponding effective ergodic theorem holds for x.

We study double averages along orbits for measure-preserving actions of$\mathbb{A}^{{\it\omega}}$, the direct sum of countably many copies of a finite abelian group$\mathbb{A}$. We show an$\text{L}^{p}$norm-variation estimate for these averages, which in particular re-proves their convergence in$\text{L}^{p}$for any finite$p$and for any choice of two$\text{L}^{\infty }$functions. The result is motivated by recent questions on quantifying convergence of multiple ergodic averages.

Let$m\in \mathbb{N}$and$\mathbf{X}=(X,{\mathcal{X}},\unicode[STIX]{x1D707},(T_{\unicode[STIX]{x1D6FC}})_{\unicode[STIX]{x1D6FC}\in \mathbb{R}^{m}})$be a measure-preserving system with an$\mathbb{R}^{m}$-action. We say that a Borel measure$\unicode[STIX]{x1D708}$on$\mathbb{R}^{m}$is weakly equidistributed for$\mathbf{X}$if there exists$A\subseteq \mathbb{R}$of density 1 such that, for all$f\in L^{\infty }(\unicode[STIX]{x1D707})$, we have$$\begin{eqnarray}\lim _{t\in A,t\rightarrow \infty }\int _{\mathbb{R}^{m}}f(T_{t\unicode[STIX]{x1D6FC}}x)\,d\unicode[STIX]{x1D708}(\unicode[STIX]{x1D6FC})=\int _{X}f\,d\unicode[STIX]{x1D707}\end{eqnarray}$$for$\unicode[STIX]{x1D707}$-almost every$x\in X$. Let$W(\mathbf{X})$denote the collection of all$\unicode[STIX]{x1D6FC}\in \mathbb{R}^{m}$such that the$\mathbb{R}$-action$(T_{t\unicode[STIX]{x1D6FC}})_{t\in \mathbb{R}}$is not ergodic. Under the assumption of the pointwise convergence of the double Birkhoff ergodic average, we show that a Borel measure$\unicode[STIX]{x1D708}$on$\mathbb{R}^{m}$is weakly equidistributed for an ergodic system$\mathbf{X}$if and only if$\unicode[STIX]{x1D708}(W(\mathbf{X})+\unicode[STIX]{x1D6FD})=0$for every$\unicode[STIX]{x1D6FD}\in \mathbb{R}^{m}$. Under the same assumption, we also show that$\unicode[STIX]{x1D708}$is weakly equidistributed for all ergodic measure-preserving systems with$\mathbb{R}^{m}$-actions if and only if$\unicode[STIX]{x1D708}(\ell )=0$for all hyperplanes$\ell$of $\mathbb{R}^{m}$. Unlike many equidistribution results in literature whose proofs use methods from harmonic analysis, our results adopt a purely ergodic-theoretic approach.

We show the $L^2$-convergence of continuous time ergodic averages of a product of functions evaluated at return times along polynomials. These averages are the continuous time version of the averages appearing in Furstenberg's proof of Szemerédi’s Theorem. For each average, we show that it is sufficient to prove convergence on special factors, the Host-Kra factors, which have the structure of a nilmanifold. We also give a description of the limit. In particular, if the polynomials are independent over the real numbers then the limit is the product of the integrals. We further show that if the collection of polynomials has “low complexity”, then for every set $E$ of real numbers with positive density and for every $\delta>0$, the set of polynomial return times for the “$\delta$-thickened” set $E_{\delta}$ has bounded gaps. We give bounds for the flow average complexity and show that in some cases the flow average complexity is strictly less than the discrete average complexity.

For a von Neumann algebra \({\mathcal M}\) , we introduce \({\mathcal M}\) -valued Besicovitch sequences and study the norm and individual convergences of the corresponding weighted ergodic averages. The limits of the averages are examined under the condition that the contraction in question is weakly mixing.

The work of Banach [1],2 Mazur [2], and others has emphasized the fundamental role played by convex functions and sets in the study of Banach spaces and (continuous) linear transformations between them. It is natural to suspect that similar considerations underlie the theory of general linear transformations, in which closure rather than the restricted notion of continuity is predominant. In this paper, therefore, we study linear transformations T from a Banach space B1 , to a Banach space B2, where T denotes a single-valued, additive and homogeneous operation with linear domain DT in B1, and linear range RT in B2. Our definition of linearity is purely algebraic and will necessarily imply continuity and total definition (DT = B1) cnly in the case of linear functionals. With these exceptions our terminology conforms to that of Banach. When T is linear the expression K(x) = 11 Tx 11, xe DT is a convex function, and E[K(x) _ a], a > 0, is a convex set, where E[K(x) ? a] denotes the set of points x e DK = DT such that K(x) ? a. An examination of the connections between the closure of T, the closure of E[K(x) < a], and the structure of convex K's yields a number of theorems with interesting applications, including as special cases theorems of Gelfand [3] and Bosanquet and Kestelmann [4] of the Banach-Steinhaus type (cf. Zygmund [5], pp. 97-100). Upon specializing B1 and B2 to Hilbert space we obtain a characterization of the domain of a selfadjoint transformation. A by-product of the analysis is a surprisingly simple proof of Wintner's theorem [6, ?108]: every definite symmetric transformation possesses a self-adjoint extension.

This workshop in real and harmonic analysis surveyed recent investigations in areas including geometric measure theory and restriction theory, multiple ergodic averages, local smoothing estimates, Schrödinger and Hörmander-type oscillatory integral operators, multilinear estimates and analysis on the Hamming cube. In particular, the workshop emphasized the recent solutions of two longstanding problems: the Kakeya phenomenon in dimension three, and almost everywhere convergence of long time averages associated to multiple commuting measure-preserving transformations. The methods presented during the workshop have yielded applications in ergodic theory, number theory, and computer science.

We study the convergence of an iterative projection/reflection algorithm originally proposed for solving what are known as phase retrieval problems in optics. There are two features that frustrate any analysis of iterative methods for solving the phase retrieval problem: nonconvexity and infeasibility. The algorithm that we developed, called relaxed averaged alternating reflections (RAAR), was designed primarily to address infeasibility, though our strategy has advantages for nonconvex problems as well. In the present work we investigate the asymptotic behavior of the RAAR algorithm for the general problem of finding points that achieve the minimum distance between two closed convex sets in a Hilbert space with empty intersection, and for the problem of finding points that achieve a local minimum distance between one closed convex set and a closed prox-regular set, also possibly nonintersecting. The nonconvex theory includes and expands prior results limited to convex sets with nonempty intersection. To place the RAAR algorithm in context, we develop parallel statements about the standard alternating projections algorithm and gradient descent. All of the various algorithms are unified as instances of iterated averaged alternating proximal reflectors applied to a sum of regularized maximal monotone mappings.

The notion of “exceptional family of elments (EFE)” plays a very important role in solving complementarity problems. It has been applied in finite dimensional spaces and Hilbert spaces by many authors. In this paper, by using the generalized projection defined by Alber, we extend this notion from Hilbert spaces to uniformly smooth and uniformly convex Banach spaces, and apply this extension to the study of nonlinear complementarity problems in Banach spaces.

Projections on Convex Sets in Hilbert Space and Spectral Theory: Part I. Projections on Convex Sets: Part II. Spectral Theory

We study the arithmetic analogue of maximal functions on diagonal hypersurfaces. This paper is a natural step following the papers of [13], [14] and [16]. We combine more precise knowledge of oscillatory integrals and exponential sums to generalize the asymptotic formula in Waring’s problem to an approximation formula for the Fourier transform of the solution set of lattice points on hypersurfaces arising in Waring’s problem and apply this result to arithmetic maximal functions and ergodic averages. In sufficiently large dimensions, the approximation formula, l2-maximal theorems and ergodic theorems were previously known. Our contribution is in reducing the dimensional constraint in the approximation formula using recent bounds of Wooley, and improving the range of lp spaces in the maximal and ergodic theorems. We also conjecture the expected range of spaces.

We use a non-commutative generalization of the Banach Principle to show that the classical individual ergodic theorem for subsequences generated by means of uniform sequences can be extended to the von Neumann algebra setting.

The publication of the important work of Rauch and Taylor [J. Funct. Anal. 18 (1975)] started a hole branch of research on wild perturbations of the Laplace–Beltrami operator. Here, we extend certain results and show norm convergence of the resolvent. We consider a (not necessarily compact) manifold with many small balls removed, the number of balls can increase as the radius is shrinking, the number of balls can also be infinite. If the distance of the balls shrinks less fast than the radius, then we show that the Neumann Laplacian converges to the unperturbed Laplacian, i.e., the obstacles vanish. In the Dirichlet case, we consider two cases here: if the balls are too sparse, the limit operator is again the unperturbed one, while if the balls concentrate at a certain region (they become “solid” there), the limit operator is the Dirichlet Laplacian on the complement of the solid region. Norm resolvent convergence in the limit case of homogenisation is treated by Khrabustovskyi and the second author in another article (see also the references therein). Our work is based on a norm convergence result for operators acting in varying Hilbert spaces described in a book from 2012 by the second author.

We prove the L2-convergence for the linear multiple ergodic averages of commuting transformations . The limiting behavior of such averages is controlled by a particular factor, which is an inverse limit of nilsystems. As a corollary we show that the limiting behavior of linear multiple ergodic averages is the same for commuting transformations.

This chapter, in fact, starts our study of functional analysis. Here, we consider linear normed spaces, i.e., sets equipped with the structure of a linear space and a norm (an analog of the notion of the length of a vector). These spaces are one of the main objects of investigation in functional analysis. Banach spaces or spaces of type B (complete with respect to convergence in norm) are the most important class of linear normed spaces. These spaces were introduced by Banach and Wiener. Here, we also consider another important class of spaces — Hilbert spaces — whose norm is defined in terms of a scalar product.