Franklin System Research Articles

Some Estimates of Functions of General Franklin System

Some Estimates of Functions of General Franklin System

Accurate and fast quaternion fractional-order Franklin moments for color image analysis

Quaternion type moments (QTM) based on hypergeometric functions and harmonic functions have attracted interest in the imaging field. The Franklin system is a complete orthonormal system consisting of piecewise linear continuous functions with Haar wavelet collocation points. However, due to the lack of efficient computational methods and explicit expressions, research on piecewise polynomials with Haar wavelet collocation points remains underexplored. Furthermore, the impact of magnitude and phase of quaternion on the color image features extracted by QTM has not been comprehensively explored. To address these problems, a fast and accurate computation method for Franklin functions is proposed. Quaternion fractional-order Franklin moments (QFFM) are proposed and using the Minkowski distance as a prior parameter for selecting unit pure quaternions. Experiments are conducted to demonstrate the superior performance of QFFM in image reconstruction and color image classification tasks compared with QTM-based and deep learning-based methods. Moreover, with the help of experiments, it is confirmed the effectiveness of using the Minkowski distance to select magnitude and phase of quaternion.

On uniqueness for series in the general Franklin system

We prove some uniqueness theorems for series in general Franklin systems. In particular, for series in the classical Franklin system our result asserts that if the partial sums $S_{n_i}(x)=\sum_{k=0}^{n_i}a_kf_k(x)$ of a Franklin series $\sum_{k=0}^{\infty}a_kf_k(x)$ converge in measure to an integrable function $f$ and $\sup_i|S_{n_i}(x)|<\infty$, for $x\notin B$, where $B$ is some countable set and $\sup_i(n_i/n_{i-1})<\infty$, then this is the Fourier-Franklin series of $f$. Bibliography: 29 titles.

On the uniqueness of series in the general Franklin system

Доказаны некоторые теоремы единственности для рядов по общей системе Франклина. В частности, для рядов по классической системе Франклина она будет звучать следующим образом: если частичные суммы $S_{n_i}(x)=\sum_{k=0}^{n_i}a_kf_k(x)$ ряда Франклина $\sum_{k=0}^{\infty}a_kf_k(x)$ по мере сходятся к интегрируемой функции $f$ и $\sup_i|S_{n_i}(x)|<\infty$, когда $x\notin B$, где $B$ - некоторое счетное множество и $\sup_i(n_i/n_{i-1})<\infty$, то этот ряд является рядом Фурье-Франклина функции $f$. Библиография: 29 названий.

Evaluation of Detection and Efficacy of Decontamination of Respiratory Pathogens Including SARS-CoV-2 on Basic Military Trainee Gas Masks.

Basic military trainee (BMT) gas mask training poses a potential mechanism of severe acute respiratory syndrome coronavirus-2 (SARS-CoV-2) transmission. After training, gas masks are decontaminated. Insufficient decontamination can lead to viral transmission in the next training class. To our knowledge, the decontamination process has not been validated for efficacy in removing respiratory pathogens such as SARS-CoV-2. Inactivated strains of SARS-CoV-2, influenza A and B, and Bordetella pertussis were separately inoculated onto gas masks in the emitter area (n = 5). Pathogen detection in swabs collected from gas masks was performed by real-time polymerase chain reaction (RT-PCR) using the BioFire® RP2.1 panel and Biomeme Franklin system. For decontamination efficacy experiments, pathogens were inoculated onto gas masks, and contaminated areas were swabbed before and after decontamination, with detection using both PCR platforms. Lastly, 65 gas masks were swabbed after gas mask training, and again after the trainees and guardians decontaminated the masks, to identify the presence of any respiratory pathogen exhaled onto the gas masks. All four pathogens were detected by both PCR platforms. The BioFire® FilmArray® was more sensitive than the Biomeme platform. Decontamination resulted in undetectable levels of all three viruses. B. pertussis was detected on one mask after decontamination. Experiments with live B. pertussis validated that decontamination eliminated all viable bacteria from gas masks. For BMT sampling, all masks were negative for SARS-CoV-2. One mask tested positive for coronavirus 229E. Once decontaminated, all masks tested negative. BMT gas masks can be monitored for the presence of respiratory pathogens using RT-PCR. The decontamination process removed all viable respiratory pathogens tested from the gas masks. This study demonstrates that RT-PCR can be used to conduct pathogen surveillance on BMT gas masks after training and that the current decontamination process is effective to eliminate respiratory viruses including SARS-CoV-2.

Uniqueness of Series by General Franklin System with Convergent Subsequence of Partial Sums

Uniqueness of Series by General Franklin System with Convergent Subsequence of Partial Sums

Uniqueness of series in the Franklin system and the Gevorkyan problems

Uniqueness of series in the Franklin system and the Gevorkyan problems

On a Property of the Franklin System in C[0, 1] and L1[0, 1

A problem posed by J. R. Holub is solved. In particular, it is proved that if $$\left\{ {{{\tilde f}_n}} \right\}$$ is the normalized Franklin system in L1[0, 1], {an} is a monotone sequence converging to zero, and $${\sup\nolimits _{n \in \mathbb{N} }}{\left\| {\sum\nolimits_{k = 0}^n {{a_k}{{\tilde f}_k}} } \right\|_1}\, < \, + \infty $$, then the series $$\sum\nolimits_{n = 0}^\infty {{a_n}{{\tilde f}_n}} $$ converges in L1[0, 1]. A similar result is also obtained for C[0, 1].

On Convergence of Partial Sums of Franklin Series to +∞

In this paper, we prove that if {nk} is an arbitrary increasing sequence of natural numbers such that the ratio nk+1/nk is bounded, then the nk-th partial sum of a series by Franklin system cannot converge to +∞ on a set of positive measure. Also, we prove that if the ratio nk+1/nk is unbounded, then there exists a series by Franklin system, the nk-th partial sum of which converges to +∞ almost everywhere on [0, 1].

On the Convergence of Franklin Series to +∞

It is proved that the partial sums of a series in the Franklin system with numbers 2μ, μ ∈ ℕ, cannot approach +∞ on a set of positive measure. In particular, a Franklin series cannot converge to +∞ on a set of positive measure.

On M*-sets for Series by Franklin System

In this paper, we prove that a set E is an M*-set or an AM*-set for the Franklin system if and only if E contains a nonempty perfect set.

On Uniqueness of Series by General Franklin System

The paper considers the general Franklin system corresponding to a regular by couples partition of the segment [0; 1]. For series by this system, we prove uniqueness theorems and obtain restoration formulas for coefficients, provided that the series converge in measure and satisfy some necessary condition.

The Gibbs phenomenon for general Franklin systems

The paper is devoted to the Gibbs phenomenon for series by general Franklin systems. The general Franklin system corresponding to a given dense sequence of points T = (t n , n ≥ 0) in [0, 1] is a sequence of orthonormal piecewise linear functions with knots from T, that is, the nth function from the system has knots t 0,..., t n . The main result of this paper is that the Gibbs phenomenon for Fourier series by general Franklin systems occurs for almost all points of [0, 1].

On the uniqueness of series in the Franklin system

We prove: a) a uniqueness theorem for everywhere convergent series in the Franklin system; b) a uniqueness theorem for Franklin series that converge in measure, whose least upper bound of the sequence of modules of partial sums is finite everywhere, except possibly on a countable set, and whose coefficients satisfy a certain necessary condition. Bibliography: 16 titles.

On unconditional basis property in the space H 1(R) of a system of Franklin functions with vanishing means

We define a general Franklin system of functions on R with vanishing means, generated by an admissible sequence T. A necessary and sufficient condition on T is found for the corresponding general Franklin system of functions on R with vanishing means to be an unconditional basis in the space H1(R).

On a problem of Gevorkyan for the Franklin system

In 1870 G. Cantor proved that if \(\lim_{N\rightarrow\infty}\sum_{n=-N}^N\,c_{n}e^{inx} = 0\) for every real \(x\), where \(\bar{c}_{n}=c_{n}\) (\(n\in \mathbb{Z}\)), then all coefficients \(c_{n}\) are equal to zero. Later, in 1950 V. Ya. Kozlov proved that there exists a trigonometric series for which a subsequence of its partial sums converges to zero, where not all coefficients of the series are zero. In 2004 G. Gevorkyan raised the issue that if Cantor's result extends to the Franklin system. The conjecture remains open until now. In the present paper we show however that Kozlov's version remains true for Franklin's system.

Uniqueness theorems for series in the Franklin system

Uniqueness theorems for series in the Franklin system

Convergence of Fourier series in classical systems

The following results are proved: • there exists an integrable function such that any subsequence of the Cesàro means of negative order of the Fourier series of this function diverges almost everywhere; • the values of an arbitrary integrable function can be changed on a set (independent of this function) of arbitrarily small measure so that the Fourier series with respect to both the Franklin system and the Haar system of the `modified' function will be absolutely convergent almost everywhere on ; • there exists a continuous function which features an unremovable absolute divergence. Bibliography: 47 titles.

Uniqueness Theorems for Series in the Franklin System

Uniqueness Theorems for Series in the Franklin System

General Franklin system as a basis in

For a general Franklin system generated by an admissible sequence , we obtain necessary and sufficient conditions on under which the corresponding system is a basis or an unconditional basis in .