کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
4614497 | 1339292 | 2016 | 17 صفحه PDF | دانلود رایگان |
We prove that for any circulant matrix C of size n×nn×n with monic characteristic polynomial p(z)p(z), the spectrum of its (n−1)×(n−1)(n−1)×(n−1) submatrix Cn−1Cn−1 consisting of the first n−1n−1 rows and columns of C consists of all critical points of p(z)p(z). Using this fact we provide a simple proof for the Schoenberg conjecture recently proved by R. Pereira and S. Malamud. We also prove full generalization of a higher order Schoenberg-type conjecture proposed by M. de Bruin and A. Sharma and recently proved by W.S. Cheung and T.W. Ng in its original form, i.e. for polynomials whose mass centre of roots equals zero. In this particular case, our inequality is stronger than it was conjectured by de Bruin and Sharma. Some Schmeisser's-like results on majorization of critical point of polynomials are also obtained.
Journal: Journal of Mathematical Analysis and Applications - Volume 439, Issue 2, 15 July 2016, Pages 634–650