Quantitative Estimates for Nonlinear Sampling Kantorovich Operators

In this paper, we establish quantitative estimates for nonlinear sampling Kantorovich operators in terms of the modulus of smoothness in the setting of Orlicz spaces. This general frame allows us to directly deduce some quantitative estimates of approximation in $$L^{p}$$ L p -spaces, $$1\le p<\infty $$ 1 ≤ p < ∞ , and in other well-known instances of Orlicz spaces, such as the Zygmung and the exponential spaces. Further, the qualitative order of approximation has been obtained assuming f in suitable Lipschitz classes. The above estimates achieved in the general setting of Orlicz spaces, have been also improved in the $$L^p$$ L p -case, using a direct approach suitable to this context. At the end, we consider the particular cases of the nonlinear sampling Kantorovich operators constructed by using some special kernels.

On the Theory of a Nonlinear Dynamic Circuit Filtering

Stochastic Differential Equations (SDEs) describe physical systems to account for random forcing terms in the evolution of the state trajectory. The noisy sampling mixer, a component of digital wireless communications, can be regarded as a potential case from the dynamical systems’ viewpoint. The universality of the noisy sampling mixer is attributed to the fact that it adopts the structure of a nonlinear SDE and its linearized version becomes a time-varying bilinear SDE. This paper develops a mathematical theory for the nonlinear noisy sampling mixer from the filtering viewpoint. Since the filtering of stochastic systems hinges on the structure of dynamical systems and observation equation set up, we consider three ‘filtering models’. The first model, accounts for a nonlinear SDE coupled with a nonlinear observation equation. In the second model, we consider a bilinear SDE with a linear observation equation to achieve the nonlinear sampling filtering. Note that the bilinear SDE coupled with the linear observation is a consequence of the Carleman linearization to the nonlinear SDE and the nonlinear observation equation. In the third model, we consider a Stratonovich SDE coupled with a nonlinear observation equation. The filtering equation of this paper can be further utilized to guide the design process of the noisy sampling mixer.

Numerical-Sampling-Functionalized Real-Time Index Regulation for Direct k-Domain Calibration in Spectral Domain Optical Coherence Tomography

An index-regulation technique functionalized by numerical sampling for direct calibration of the non-linear wavenumber (k)-domain to a linear domain in spectral domain optical coherence tomography (SD-OCT) is proposed. The objective of the developed method is to facilitate high-resolution identification of microstructures in biomedical imaging. Subjective optical alignments caused by nonlinear sampling of interferograms in the k-domain tend to hinder depth-dependent signal-to-noise ratios (SNR) and axial resolution in SD-OCT. Moreover, the optical-laser-dependent k-domain requires constant recalibrated in accordance with each laser transition, thereby necessitating either hardware or heavy software compensations. As the key feature of the proposed method, a relatively simple software-based k-domain mask calibration technique was developed to enable real-time linear sampling of k-domain interpolations whilst facilitating image observation through use of an index-regulation technique. Moreover, it has been confirmed that dispersion can be simultaneously compensated with noise residuals generated using the proposed technique, and that use of complex numerical or hardware techniques are no longer required. Observed results, such as fall-off, SNR, and axial resolution clearly exhibit the direct impact of the proposed technique, which could help investigators rapidly achieve optical-laser-independent high-quality SD-OCT images.