The Elements of Operator Theory

Kubrusly, Carlos S.

doi:10.1007/978-0-8176-4998-2

Cited by 89 publications

(82 citation statements)

References 0 publications

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“…Let be the collection of all scalar-valued complex infinite sequences ( ) in such that ∑ | | < , where and denote, respectively, the sets of all complex and integer numbers; and assume that the function 〈 〉 of into is the usual inner product (as in [12]). An element ( ) -with 〈 〉 -is a -wavelet function, if the sequence ( ) [14]).…”

Section: Wavelet Decomposition Of Level Rmentioning

confidence: 99%

Time series forecasting with the WARIMAX-GARCH method

et al. 2016

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Time series forecasting with the WARIMAX-GARCH method, Neurocomputing, http://dx.doi.org/10. 1016/j.neucom.2016.08.046 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. AbstractIt is well-known that causal forecasting methods that include appropriately chosen Exogenous Variables (EVs) very often present improved forecasting performances over univariate methods. However, in practice, EVs are usually difficult to obtain and in many cases are not available at all. In this paper, a new causal forecasting approach, called Wavelet Auto-Regressive Integrated Moving Average with eXogenous variables and Generalized Auto-Regressive Conditional Heteroscedasticity (WARIMAX-GARCH) method, is proposed to improve predictive performance and accuracy but also to address, at least in part, the problem of unavailable EVs. Basically, the WARIMAX-GARCH method obtains Wavelet "EVs" (WEVs) from Auto-Regressive Integrated Moving Average with eXogenous variables and Generalized Auto-Regressive Conditional Heteroscedasticity (ARIMAX-GARCH) models applied to Wavelet Components (WCs) that are initially determined from the underlying time series. The WEVs are, in fact, treated by the WARIMAX-GARCH method as if they were conventional EVs. Similarly to GARCH and ARIMA-GARCH models, the WARIMAX-GARCH method is suitable for time series exhibiting non-linear characteristics such as conditional variance that depends on past values of observed data. However, unlike those, it can explicitly model frequency domain patterns in the series to help improve predictive performance. An application to a daily time series of dam displacement in Brazil shows the WARIMAX-GARCH method to remarkably outperform the ARIMA-GARCH method, as well as the (multi-layer perceptron) Artificial Neural Network (ANN) and its wavelet version referred to as Wavelet Artificial Neural Network (WANN) as in [1], on statistical measures for both in-sample and out-of-sample forecasting.

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Section: Wavelet Decomposition Of Level Rmentioning

confidence: 99%

Time series forecasting with the WARIMAX-GARCH method

et al. 2016

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“…(i) Therefore, ∈ − ⟹ ∈ and − ⊆ . [Kubrusly(2001)] page 128 ⟨Theorem 3.40⟩, [Haaser and Sullivan(1991)] page 75 ⟨6⋅10, 6⋅11 Propositions⟩, [Bryant(1985)] page 40 ⟨Theorem 3.6, 3.7⟩, [Sutherland(1975) …”

Section: ✎Proofmentioning

confidence: 99%

“…Let be a function in ( , ) ( , ) . [Kubrusly(2001)] page 118 ⟨Theorem 3.30⟩, [Haaser and Sullivan(1991) [Ponnusamy(2002)] pages 94-96 ⟨"2.59. Proposition.…”

mentioning

confidence: 99%

Properties of distance spaces with power triangle inequalities

Greenhoe

2016

Preprint

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Metric spaces provide a framework for analysis and have several very useful properties.Many of these properties follow in part from the triangle inequality. However, there are several applications in which the triangle inequality does not hold but in which we may still like to perform analysis. Properties of distance spaces with power triangle inequalities DANIEL J. GREENHOEAbstract: Metric spaces provide a framework for analysis and have several very useful properties. Many of these properties follow in part from the triangle inequality. However, there are several applications in which the triangle inequality does not hold but in which we may still like to perform analysis. This paper investigates what happens if the triangle inequality is removed all together, leaving what is called a distance space, and also what happens if the triangle inequality is replaced with a much more general two parameter relation, which is herein called the "power triangle inequality". The power triangle inequality represents an uncountably large class of inequalities, and includes the triangle inequality, relaxed triangle inequality, and inframetric inequality as special cases. The power triangle inequality is defined in terms of a function that is herein called the power triangle function. The power triangle function is itself a power mean, and as such is continuous and monotone with respect to its exponential parameter, and also includes the operations of maximum, minimum, mean square, arithmetic mean, geometric mean, and harmonic mean as special cases.2010 Mathematics Subject Classification 54E25 (primary); 54A05,54A20 (secondary)

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“…The best known constructions are the " p type direct sums" (see e.g. [7]) which are natural generalizations of the standard Hilbert space construction corresponding to p = 2. Much more general is the construction introduced by Day (see [3,4]), based on the notion of full function space.…”

Section: Introductionmentioning

confidence: 99%

Infinite Banach direct sums and diagonal $$C_0$$ C 0 -semigroups with applications to a stochastic particle system

Lachowicz

Moszyński

2015

Semigroup Forum

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We show an alternative-axiomatic approach to the direct sums of countably many Banach spaces, called here B-direct sums (BDS). We develop "direct summing" of linear operators and the problems of generating the C 0 -semigroups in such "Banach direct sums of Banach spaces" by "the direct sums of the generators". We study special types of BDS, including M-BDS-the one closely related to Day's construction of a direct sum (M.M. Day, Normed linear spaces, 1973). The main abstract results of the paper concern C 0 -semigroup generation properties for diagonal operators in M-BDS type direct sums. The paper is motivated by stochastic particle systems and the problem of existing C 0 -semigroups defined by the corresponding hierarchies for marginal probability densities. We use our abstract M-BDS results to get a solution of the respective differential equation in the C 0 -semigroup sense.

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The Elements of Operator Theory

Cited by 89 publications

References 0 publications

Time series forecasting with the WARIMAX-GARCH method

Time series forecasting with the WARIMAX-GARCH method

Properties of distance spaces with power triangle inequalities

Infinite Banach direct sums and diagonal $$C_0$$ C 0 -semigroups with applications to a stochastic particle system

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