An algorithm for harmonic decomposition of time-variant signals is derived from a least squares harmonic (LSH) technique. The estimates of harmonic amplitudes and phases are formulated as the solution of a set of linear equations which minimizing mean square error; the signal frequency is modeled by a linear or quadratic polynomial and obtained via a local search over polynomial coefficients. A...

We present several variations on the harmonic Rayleigh–Ritz method. First, we introduce a relative harmonic approach for the standard, generalized, and polynomial eigenproblem. Second, a harmonic extraction method is studied for rightmost eigenvalues of generalized eigenvalue problems. Third, we propose harmonic extraction methods for large eigenvalues of generalized and polynomial eigenproblem...

The harmonic index ) (G H , of a graph G is defined as the sum of weights 1 )) deg( ) (deg( 2   v u of all edges in ) (G E , where deg (u) denotes the degree of a vertex u in V(G). In this paper we define the harmonic polynomial of G as      ) ( 1 ) deg( ) deg( 2 ) , ( G E uv v u x x G H , where   1 0 ) ( ) , ( G H dx x G H . We present explicit formula for the values of harmonic polyn...

We show that every rational knot K of crossing number N admits a polynomial parametrization x = Ta(t), y = Tb(t), z = C(t) where Tk(t) are the Chebyshev polynomials, a = 3 and b + degC = 3N. We show that every rational knot also admits a polynomial parametrization with a = 4. If C(t) = Tc(t) is a Chebyshev polynomial, we call such a knot a harmonic knot. We give the classification of harmonic k...

We prove that if f : Zd → R is harmonic and there exists a polynomial W : Zd → R such that f + W is nonnegative, then f is a polynomial.

We consider the random walks killed at the boundary of the quarter plane, with homogeneous non-zero jump probabilities to the eight nearest neighbors and drift zero in the interior, and which admit a positive harmonic polynomial of degree three. For these processes, we find the asymptotic of the Green functions along all infinite paths of states, and from this we deduce that the Martin compacti...

We compute the best constants of approximation by entire functions of spherical type and by trigonometric polynomials of spherical degree on classes of functions f satisfying the condition ‖ f ‖Lp 1, where p=1 or 2 and is the Laplace operator. © 2007 Elsevier Inc. All rights reserved.

Let J be the Julia set of a conformal dynamics f . Provided that f is polynomial-like we prove that the harmonic measure on J is mutually absolutely continuous with the measure of maximal entropy if and only if f is conformally equivalent to a polynomial. This is no longer true for generalized polynomial-like maps. But for such dynamics the coincidence of classes of these two measures turns out...