Bounding smooth solutions of Bezout equations

Given data $f=(f_1,f_2,\dots ,f_n)$ in the holomorphic part $ A= F_+$ of a symmetric Banach\slash topological algebra $ F$ on the unit circle $\mathbb{T}$, we estimate solutions $ g_j\in A$ to the corresponding Bezout equation $\sum _{j=1}^ng_jf_j=1$ in terms of the lower spectral parameter δ, $0< \delta \leq |f(z)|$, and an inversion controlling function $c_1(\delta ,F)$ for the algebra $F$. A scheme developed issues from an analysis of the famous Uchiyama-Wolff proof to the Carleson corona theorem and includes examples of algebras of “smooth” functions, as Beurling-Sobolev, Lipschitz, or Wiener-Dirichlet algebras. There is no real “corona problem” in this setting, the issue is in the growth rate of the upper bound for $\|g\|_{A^n}$ as $\delta \to 0$ and in numerical values of the quantities that occur, which are determined as accurately as possible.

A Novel Technique to Compute the Revisit Time of Satellites and Its Application in Remote Sensing Satellite Optimization Design

This paper proposes a novel technique to compute the revisit time of satellites within repeat ground tracks. Different from the repeat cycle which only depends on the orbit, the revisit time is relevant to the payload of the satellite as well, such as the tilt angle and swath width. The technique is discussed using the Bezout equation and takes the gravitational second zonal harmonic into consideration. The concept of subcycles is defined in a general way and the general concept of “small” offset is replaced by a multiple of the minimum interval on equator when analyzing the revisit time of remote sensing satellites. This technique requires simple calculations with high efficiency. At last, this technique is used to design remote sensing satellites with desired revisit time and minimum tilt angle. When the side-lap, the range of altitude, and desired revisit time are determined, a lot of orbit solutions which meet the mission requirements will be obtained fast. Among all solutions, designers can quickly find out the optimal orbits. Through various case studies, the calculation technique is successfully demonstrated.

On the Bézout equation in the ring of periodic distributions

A corona type theorem is given for the ring D'A(Rd) of periodic distributions in Rd in terms of the sequence of Fourier coefficients of these distributions,which have at most polynomial growth. It is also shown that the Bass stable rank and the topological stable rank of D'A(Rd) are both equal to 1.