A Note on Function Space and Boundedness of the General Fractional Integral in Continuous Time Random Walk

The general fractional calculus becomes popular in continuous time random walk recently. However, the boundedness condition of the general fractional integral is one of the fundamental problems. It wasn’t given yet. In this short communication, the classical norm space is used, and a general boundedness theorem is presented. Finally, various long–tailed waiting time probability density functions are suggested in continuous time random walk since the general fractional integral is well defined.

Asymptotic pointwise error estimates for reconstructing shift-invariant signals with generators in a hybrid-norm space

Sampling and reconstruction of signals in a shift-invariant space are generally studied under the requirement that the generator is in a stronger Wiener amalgam space, and the error estimates are usually given in the sense of $L_{p,{1 / \omega }}$ L p , 1 / ω -norm. Since we often need to reflect the local characteristics of reconstructing error, the asymptotic pointwise error estimates for nonuniform and average sampling in a non-decaying shift-invariant space are discussed under the assumption that the generator is in a hybrid-norm space. Based on Lemma 2.1–Lemma 2.6, we first rewrite the iterative reconstruction algorithms for two kinds of average sampling functionals and prove their convergence. Then, the asymptotic pointwise error estimates are presented for two algorithms under the case that the average samples are corrupted by noise.

Average sampling in mixed shift-invariant subspaces with generators in hybrid-norm spaces

This paper mainly studies the average sampling and reconstruction in shift-invariant subspaces of mixed Lebesgue spaces $L^{p,q}(\mathbb{R}^{d+1})$, under the condition that the generator $\varphi$ of the shift-invariant subspace belongs to a hybrid-norm space of mixed form, which is weaker than the usual assumption of Wiener amalgam space and allows to control the orders $p,q$. First, the sampling stability for two kinds of average sampling functionals are established. Then, we give the corresponding iterative approximation projection algorithms with exponential convergence for recovering the time-varying shift-invariant signals from the average samples.

Inverse Function Theorem. Part I1

Summary In this article we formalize in Mizar [1], [2] the inverse function theorem for the class of C 1 functions between Banach spaces. In the first section, we prove several theorems about open sets in real norm space, which are needed in the proof of the inverse function theorem. In the next section, we define a function to exchange the order of a product of two normed spaces, namely 𝔼 ↶ ≂ (x, y) ∈ X × Y ↦ (y, x) ∈ Y × X, and formalized its bijective isometric property and several differentiation properties. This map is necessary to change the order of the arguments of a function when deriving the inverse function theorem from the implicit function theorem proved in [6]. In the third section, using the implicit function theorem, we prove a theorem that is a necessary component of the proof of the inverse function theorem. In the last section, we finally formalized an inverse function theorem for class of C 1 functions between Banach spaces. We referred to [9], [10], and [3] in the formalization.

Robust Autoregression with Exogenous Input Model for System Identification and Predicting

Autoregression with exogenous input (ARX) is a widely used model to estimate the dynamic relationships between neurophysiological signals and other physiological parameters. Nevertheless, biological signals, such as electroencephalogram (EEG), arterial blood pressure (ABP), and intracranial pressure (ICP), are inevitably contaminated by unexpected artifacts, which may distort the parameter estimation due to the use of the L2 norm structure. In this paper, we defined the ARX in the Lp (p ≤ 1) norm space with the aim of resisting outlier influence and designed a feasible iteration procedure to estimate model parameters. A quantitative evaluation with various outlier conditions demonstrated that the proposed method could estimate ARX parameters more robustly than conventional methods. Testing with the resting-state EEG with ocular artifacts demonstrated that the proposed method could predict missing data with less influence from the artifacts. In addition, the results on ICP and ABP data further verified its efficiency for model fitting and system identification. The proposed Lp-ARX may help capture system parameters reliably with various input and output signals that are contaminated with artifacts.

Completeness of Sequence Spaces Generated by an Orlicz Function

In this paper, we discuss about completeness property of Orlicz sequence spaces defined by an Orlicz function. Orlicz sequence space is generalization of p-summable sequence space, for every which is also an Orlicz sequence space. Based on the property of convergence sequence on norm space, we define $\Phi$-convergence sequence on Orlicz sequence space. Moreover, we define $\Phi$-Cauchy sequence and $\Phi$-complete on Orlicz sequence space. In this paper, we show the relationship between the (ordinary) convergent sequence, $\Phi$-convergent and $\Phi$-Cauchy sequences. Finally, it will also be shown that Orlicz sequence space is Banach space and $\Phi$-complete space.

Monotone iterative technique for non-autonomous semilinear differential equations with nonlocal condition

The objective of this article is to discuss the existence and uniqueness of mild solutions for a class of non-autonomous semilinear differential equations with nonlocal condition via monotone iterative method with upper and lower solutions in an ordered complete norm space X, using evolution system and measure of noncompactness.