We prove a number of inequalities for the mean oscillations O-theta(f, B, I) =3D (1/mu(B) integral(B) vertical bar f(y) - I vertical bar(theta) d mu(y))(1/theta), where theta > 0, B is a ball in a metric space with measure mu satisfying the doubling condition, and the number I is chosen in one of the following ways: I =3D f(x) (x is an element of B), I is the mean value of the function f over the ball B, and I is the best approximation of f by constants in the metric of L-theta(B). These inequalities are used to obtain L-p -estimates (p > 0) of the maximal operators measuring local smoothness, to describe Sobolev-type spaces, and to study the self-improvement property of Poincare-Sobolev-type inequalities.
Strapasson, Joao E.
Jorge, Grasiele C.
Campello, Antonio
Costa, Sueli I. R.
We introduce the notion of degree of imperfection of a code in Z(n) with the l(p) metric, to extend the so-called quasi-perfect codes. Through the establishment of bounds and computational approach, we determine all radii for which there are linear quasi-perfect codes for p =3D 2 and n =3D 2, 3. Numerical results concerning the codes with small degree of imperfection are also presented.
In this paper, we give some sufficient conditions under which perturbations preserve l(p)-localized frames. Using an arbitrary given sequence, we provide a simple way for constructing l(p)-localized sequences.
For any tau >=3D 0, t >=3D 1 and p >=3D 1, the exact value of the James type constant J(X,t) (tau) of the l(p) - l(1) space is investigated. As an application, the exact value of the von Neuman-Jordan type constant of the l(p) - l(1) space can also be obtained.
We study the recovery conditions of weighted mixed l(2)/l(p) (0 < p <=3D 1) minimization for block sparse signal reconstruction from compressed measurements when partial block support information is available. We show that the block p-restricted isometry property (RIP) can ensure the robust recovery. Moreover, we present the sufficient and necessary condition for the recovery by using weighted block p-null space property. The relationship between the block p-RIP and the weighted block p-null space property has been established. Finally, we illustrate our results with a series of numerical experiments. (C) 2018 Elsevier B.V. All rights reserved.
This paper elucidates the underlying structures of l(p)-regularized least squares problems in the nonconvex case of 0 < p < 1. The difference between two formulations is highlighted (which does not occur in the convex case of p =3D 1): 1) an l(p)-constrained optimization (P-c(p)) and 2) an l(p)-penalized (unconstrained) optimization (L-lambda(p)). It is shown that the solution path of (L-lambda(p)) is discontinuous and also a part of the solution path of (P-c(p)). As an alternative to the solution path, a critical path is considered, which is a maximal continuous curve consisting of critical points. Critical paths are piecewise smooth, as can be seen from the viewpoint of the variational method, and generally contain non-optimal points, such as saddle points and local maxima as well as global/local minima. Our study reveals multiplicity (non-monotonicity) in the correspondence between the regularization parameters of (P-c(p)) and (L-lambda(p)). Two particular paths of critical points connecting the origin and an ordinary least squares (OLS) solution are studied further. One is a main path starting at an OLS solution, and the other is a greedy path starting at the origin. Part of the greedy path can be constructed with a generalized Minkowskian gradient. This paper of greedy path leads to a nontrivial close-link between the optimization problem of l(p)-regularized least squares and the greedy method of orthogonal matching pursuit.
In this paper, we show that there exists a positive density subsequence of orthonormal spherical harmonics which achieves the maximal L-p norm growth for 2 < p <=3D 6, therefore giving an example of a Riemannian surface supporting such a subsequence of eigenfunctions. This answers the question proposed by Sogge and Zelditch (Concerning the L-4 Norms of Typical Eigenfunctions on Compact Surfaces. Recent Developments in Geometry and Analysis, Int. Press, Somerville, pp. 407-423, 2012). Furthermore, we provide an explicit lower bound on the density in this example.
In this paper we connect Calderon and Zygmund's notion of L-p-differentiability (Calderon and Zygmund, Proc Natl Acad Sci USA 46: 1385-1389, 1960) with some recent characterizations of Sobolev spaces via the asymptotics of non-local functionals due to Bourgain, Brezis, and Mironescu (Optimal Control and Partial Differential Equations, pp. 439-455, 2001). We showhowthe results of the former can be generalized to the setting of the latter, while the latter results can be strengthened in the spirit of the former. As a consequence of these results we give several new characterizations of Sobolev spaces, a novel condition for whether a function of bounded variation is in the Sobolev space W-1,W-1, and complete the proof of a characterization of the Sobolev spaces recently claimed in (Leoni and Spector, J Funct Anal 261: 2926-2958, 2011; Leoni and Spector, J Funct Anal 266: 1106-1114, 2014).