In this set of lecture notes we focus on the point-value representation obtained by looking at a particular set of points, the nth roots of unity. In the field of complex numbers C, there are exactly n different solutions to the equation x = 1. We call these solutions the n-th roots of unity. One of these roots of unity is ωn = cos(2π/n)+ i sin(2π/n), and this is called the principal nth root o...

In this set of lecture notes we focus on the point-value representation obtained by looking at a particular set of points, the nth roots of unity. In the field of complex numbers C, there are exactly n different solutions to the equation x = 1. We call these solutions the n-th roots of unity. One of these roots of unity is ωn = cos(2π/n)+ i sin(2π/n), and this is called the principal nth root o...

In a recent paper [11], R.E. Curto, S.H. Lee and J. Yoon asked the following question. Let A be subnormal operator, assume that A2 is quasinormal. Does it follow quasinormal? this paper, we answer question in affirmative. fact, prove more general result nth roots of quasinormal operators are Research on problem has led us to new criterion for semispectral measure half-line spectral, written ter...

*Correspondence: [email protected] Department of Mathematics, Binzhou University, Shandong, 256603, P.R. China Abstract The square iterative roots for strictly monotonic and upper semicontinuous functions with one set-valued point were fully described in (Li et al. in Publ. Math. (Debr.) 75:203-220, 2009). As a continuation, we study both strictly monotonic and nonmonotonic multifunctio...

We discuss the smallest algebraic number field which contains the nth roots of unity and which may be reached from the rational field Q by a sequence of irreducible, radical, Galois extensions. The degree D(n) of this field over Q is φ(m), where m is the smallest multiple of n divisible by each prime factor of φ(m). The prime factors of m/n are precisely the primes not dividing n but which do d...

We develop the nth order Fourier-Bessel series expansion of 1-D functions in the interval (0,α). Hence we establish the sampling theorem for a function with α-bandlimited nth order Hankel transform. The latter statement implies that the function is also Fourier transform αbandlimited. The samples’ locations are given by the roots of nth order Bessel functions. In addition, the sampling distance...

For any field K, a field K(ζn) where ζn is a root of unity (of order n) is called a cyclotomic extension of K. The term cyclotomic means circle-dividing, and comes from the fact that the nth roots of unity divide a circle into arcs of equal length. We will see that the extensions K(ζn)/K have abelian Galois groups and we will look in particular at cyclotomic extensions of Q and finite fields. T...