Characterization of the convergence of weighted averages in a more general setting

2013 ◽  
Vol 50 (1) ◽  
pp. 51-66
Author(s):  
Ferenc Móricz ◽  
Ulrich Stadtmüller
Keyword(s):  
Borel Measure ◽  
Integrable Function ◽  
Sufficient Conditions ◽  
General Setting ◽  
Finite Limit ◽  
Positive Borel Measure ◽  
Weighted Averages ◽  
Locally Integrable Function ◽  
Necessary And Sufficient

Let ν be a positive Borel measure on ℝ̄+:= [0;∞) and let p: ℝ̄+ → ℝ̄+ be a weight function which is locally integrable with respect to ν. We assume that \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $P(t): = \int\limits_0^t {p(u)d\nu (u) \to \infty } andP(t - 0)/P(t) \to 1ast \to \infty .$ \end{document} Let f: ℝ̄+ → ℂ be a locally integrable function with respect to p dν, and define its weighted averages by \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\sigma _t (f;pd\nu ): = \frac{1}{{P(t)}}\int\limits_0^t {f(u)p(u)d\nu (u)} $ \end{document} for large enough t, where P(t) > 0. We prove necessary and sufficient conditions under which the finite limit \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\sigma _t (f;pd\nu ) \to Last \to \infty $ \end{document} exists. This characterization is a unified extension of the results in [5], and it may find application in Probability Theory and Stochastic Processes.

Partially Bounded Solutions of Linear Ordinary Differential Equations

1975 ◽  
Vol 27 (2) ◽  
pp. 366-371
Author(s):  
David Lowell Lovelady
Keyword(s):  
Positive Integer ◽  
Present Author ◽  
Integrable Function ◽  
Analogous Result ◽  
Bounded Solution ◽  
Sufficient Conditions ◽  
Bounded Solutions ◽  
Linear Ordinary Differential Equations ◽  
Locally Integrable Function ◽  
Necessary And Sufficient

Let R, R+, and R- be the intervals (-∞, ∞), [0, ∞), and ( — ∞, 0] respectively. Let m be a positive integer, and let be the algebra of all m X m matrices. Let A be a locally integrable function from R to We propose to study the problems(NH) u‘(t) = f(t) + A﹛t)u﹛t)and(H) v‘(t) =A(t)v(t)in Rm. (H) and (NH) will denote whole-line problems, whereas (H)+, (NH)+, (H)-, and (NH)- will denote corresponding semi-axis problems.In [1] (see also [2, Theorem 1, p. 131]), W. A. Coppel obtained necessary and sufficient conditions for each bounded Continuous/ on R+ to yield at least one bounded solution u of (NH)+. The present author [3] has determined that an analogous result holds for (NH).

Charges solve the truncated complex moment problem

2018 ◽  
Vol 21 (04) ◽  
pp. 1850027
Author(s):  
K. Idrissi ◽  
E. H. Zerouali
Keyword(s):  
Moment Problem ◽  
Borel Measure ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Positive Borel Measure ◽  
General Complex ◽  
A Charge ◽  
Necessary And Sufficient ◽  
Complex Valued ◽  
Charge Formula

Let [Formula: see text], with [Formula: see text] and [Formula: see text], be a given complex-valued sequence. The complex moment problem (respectively, the general complex moment problem) associated with [Formula: see text] consists in determining necessary and sufficient conditions for the existence of a positive Borel measure (respectively, a charge) [Formula: see text] on [Formula: see text] such that [Formula: see text] In this paper, we investigate the notion of recursiveness in the two variable case. We obtain several useful results that we use to deduce new necessary and sufficient conditions for the truncated complex moment problem to admit a solution. In particular, we show that the general complex moment problem always has a solution. A concrete construction of the solution and an illustrating example are also given.

On Weighted Norm Inequalities for Fractional and Singular Integrals

1971 ◽  
Vol 23 (5) ◽  
pp. 907-928 ◽  
Author(s):  
T. Walsh
Keyword(s):  
Singular Integrals ◽  
Integrable Function ◽  
Sufficient Conditions ◽  
Weight Functions ◽  
Weighted Norm ◽  
Variable Kernel ◽  
Norm Inequalities ◽  
General Weight ◽  
Locally Integrable Function ◽  
Necessary And Sufficient

In a recent paper [12] Muckenhoupt and Wheeden have established necessary and sufficient conditions for the validity of norm inequalities of the form ‖ |x|αTƒ ‖q ≦ C‖ |x|αƒ ‖p, where Tƒ denotes a Calderón and Zygmund singular integral of ƒ or a fractional integral with variable kernel. The purpose of the present paper is to prove, by somewhat different methods, similar inequalities for more general weight functions.In what follows, for p ≧ 1, p′ is the exponent conjugate to p, given by l/p + l/p′ = 1. Ω will always denote a locally integrable function on Rn which is homogeneous of degree 0, Ω∼ will denote a measurable function on Rn × Rn such that for each x ∈ Rn, Ω∼(x, .) is locally integrable and homogeneous of degree 0.

scholarly journals On cryptographic properties of (n + 1)-bit S-boxes constructed by known n-bit S-boxes

2020 ◽  
Vol 15 (1) ◽  
pp. 258-265
Author(s):  
Yu Zhou ◽  
Daoguang Mu ◽  
Xinfeng Dong
Keyword(s):  
Sufficient Conditions ◽  
Sum Of Squares ◽  
Basic Component ◽  
Necessary And Sufficient Conditions ◽  
Autocorrelation Functions ◽  
Walsh Spectrum ◽  
Cryptographic Algorithms ◽  
Necessary And Sufficient ◽  
Relationship Of

AbstractS-box is the basic component of symmetric cryptographic algorithms, and its cryptographic properties play a key role in security of the algorithms. In this paper we give the distributions of Walsh spectrum and the distributions of autocorrelation functions for (n + 1)-bit S-boxes in [12]. We obtain the nonlinearity of (n + 1)-bit S-boxes, and one necessary and sufficient conditions of (n + 1)-bit S-boxes satisfying m-order resilient. Meanwhile, we also give one characterization of (n + 1)-bit S-boxes satisfying t-order propagation criterion. Finally, we give one relationship of the sum-of-squares indicators between an n-bit S-box S0 and the (n + 1)-bit S-box S (which is constructed by S0).

Separability criteria based on the Bloch representation of density matrices

2007 ◽  
Vol 7 (7) ◽  
pp. 624-638
Author(s):  
J. de Vicente
Keyword(s):  
Sufficient Conditions ◽  
Quantum Systems ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Pure Product ◽  
Separability Problem ◽  
Arbitrary Dimensions ◽  
Necessary And Sufficient ◽  
Bloch Representation

We study the separability of bipartite quantum systems in arbitrary dimensions using the Bloch representation of their density matrix. This approach enables us to find an alternative characterization of the separability problem, from which we derive a necessary condition and sufficient conditions for separability. For a certain class of states the necessary condition and a sufficient condition turn out to be equivalent, therefore yielding a necessary and sufficient condition. The proofs of the sufficient conditions are constructive, thus providing decompositions in pure product states for the states that satisfy them. We provide examples that show the ability of these conditions to detect entanglement. In particular, the necessary condition is proved to be strong enough to detect bound entangled states.

scholarly journals An Application of Spherical Geometry to Hyperkähler Slices

2020 ◽  
pp. 1-30
Author(s):  
Peter Crooks ◽  
Maarten van Pruijssen
Keyword(s):  
Sufficient Conditions ◽  
Compact Subgroup ◽  
Spherical Geometry ◽  
Spherical Subgroup ◽  
Cotangent Bundles ◽  
Complex Semisimple ◽  
Reductive Subgroup ◽  
Necessary And Sufficient

Abstract This work is concerned with Bielawski’s hyperkähler slices in the cotangent bundles of homogeneous affine varieties. One can associate such a slice with the data of a complex semisimple Lie group  $G$ , a reductive subgroup $H\subseteq G$ , and a Slodowy slice $S\subseteq \mathfrak{g}:=\text{Lie}(G)$ , defining it to be the hyperkähler quotient of $T^{\ast }(G/H)\times (G\times S)$ by a maximal compact subgroup of  $G$ . This hyperkähler slice is empty in some of the most elementary cases (e.g., when $S$ is regular and $(G,H)=(\text{SL}_{n+1},\text{GL}_{n})$ , $n\geqslant 3$ ), prompting us to seek necessary and sufficient conditions for non-emptiness. We give a spherical-geometric characterization of the non-empty hyperkähler slices that arise when $S=S_{\text{reg}}$ is a regular Slodowy slice, proving that non-emptiness is equivalent to the so-called $\mathfrak{a}$ -regularity of $(G,H)$ . This $\mathfrak{a}$ -regularity condition is formulated in several equivalent ways, one being a concrete condition on the rank and complexity of $G/H$ . We also provide a classification of the $\mathfrak{a}$ -regular pairs $(G,H)$ in which $H$ is a reductive spherical subgroup. Our arguments make essential use of Knop’s results on moment map images and Losev’s algorithm for computing Cartan spaces.

Characterization of Low-pass Filters on Local Fields of Positive Characteristic

2016 ◽  
Vol 59 (3) ◽  
pp. 528-541 ◽  
Author(s):  
Qaiser Jahan
Keyword(s):  
Positive Characteristic ◽  
Sufficient Conditions ◽  
Local Fields ◽  
Pass Filter ◽  
Low Pass Filter ◽  
Low Pass ◽  
Necessary And Sufficient ◽  
Low Pass Filters ◽  
The Scaling Function

AbstractIn this article, we give necessary and sufficient conditions on a function to be a low-pass filter on a local field K of positive characteristic associated with the scaling function for multiresolution analysis of L2(K). We use probability and martingale methods to provide such a characterization.

scholarly journals An inverse problem for the non-self-adjoint matrix Sturm-Liouville operator

2018 ◽  
Vol 50 (1) ◽  
pp. 71-102 ◽  
Author(s):  
Natalia Pavlovna Bondarenko
Keyword(s):  
Inverse Problem ◽  
Liouville Operator ◽  
Finite Interval ◽  
Spectral Characteristics ◽  
Sufficient Conditions ◽  
Adjoint Matrix ◽  
Method Of Spectral Mappings ◽  
Sturm Liouville ◽  
Necessary And Sufficient

The inverse problem of spectral analysis for the non-self-adjoint matrix Sturm-Liouville operator on a finite interval is investigated. We study properties of the spectral characteristics for the considered operator, and provide necessary and sufficient conditions for the solvability of the inverse problem. Our approach is based on the constructive solution of the inverse problem by the method of spectral mappings. The characterization of the spectral data in the self-adjoint case is given as a corollary of the main result.

scholarly journals Two Explicit Characterizations of the General Nonnegative-Definite Covariance Matrix Structure for Equality of BLUEs, WLSEs, and LSEs

2020 ◽  
Vol 9 (6) ◽  
pp. 108
Author(s):  
Phil D. Young ◽  
Joshua D. Patrick ◽  
Dean M. Young
Keyword(s):  
Least Squares ◽  
Weighted Least Squares ◽  
Sufficient Conditions ◽  
Covariance Structure ◽  
Least Squares Estimator ◽  
Weighted Least Squares Estimator ◽  
Best Linear Unbiased ◽  
Necessary And Sufficient ◽  
Nonnegative Definite

We provide a new, concise derivation of necessary and sufficient conditions for the explicit characterization of the general nonnegative-definite covariance structure V of a general Gauss-Markov model with E(y) and Var(y) such that the best linear unbiased estimator, the weighted least squares estimator, and the least squares estimator of Xβ are identical. In addition, we derive a representation of the general nonnegative-definite covariance structure V defined above in terms of its Moore-Penrose pseudo-inverse.

scholarly journals Multigenerator Gabor Frames on Local Fields

2018 ◽  
Vol 33 (2) ◽  
pp. 307
Author(s):  
Owais Ahmad ◽  
Neyaz Ahmad Sheikh
Keyword(s):  
Sufficient Conditions ◽  
Complete Characterization ◽  
Necessary And Sufficient Conditions ◽  
Local Fields ◽  
Gabor Frames ◽  
Gabor System ◽  
Necessary And Sufficient

The main objective of this paper is to provide complete characterization of multigenerator Gabor frames on a periodic set $\Omega$ in $K$. In particular, we provide some necessary and sufficient conditions for the multigenerator Gabor system to be a frame for $L^2(\Omega)$. Furthermore, we establish the complete characterizations of multigenerator Parseval Gabor frames.

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