arXiv subjects
Publications and source records attributed to V. L. Sousa.
A Fast algorithm for the Discrete Hartley Transform (DHT) is presented, which resembles radix-2 fast Fourier Transform (FFT). Although fast DHTs are already known, this new approach bring some light about the deep relationship between fast DHT algorithms and a multiplication-free fast algorithm for the Hadamard Transform.
We consider the orthogonal polynomials on $[-1,1]$ with respect to the weight $$ w_c(x)=h(x)(1-x)^α(1+x)^β Ξ_{c}(x), \quad α, β>-1, $$ where $h$ is real analytic and strictly positive on $[-1, 1]$, and $Ξ_{c}$ is a step-like function: $Ξ_{c}(x)=1$ for $x\in [-1, 0)$ and $Ξ_{c}(x)=c^2$, $c>0$, for $x\in [0, 1]$. We obtain strong uniform asymptotics of the monic orthogonal polynomials in $\mathbb{C}$, as well as first terms of the asymptotic expansion of the main parameters (leading coefficients of the orthonormal polynomials and the recurrence coefficients) as $n\to \infty$. In particular, we prove for $w_c$ a conjecture of A. Magnus regarding the asymptotics of the recurrence coefficients. The main focus is on the local analysis at the origin. We study the asymptotics of the Christoffel-Darboux kernel in a neighborhood of the jump and show that the zeros of the orthogonal polynomials no longer exhibit the clock behavior. For the asymptotic analysis we use the steepest descendent method of Deift and Zhou applied to the non-commutative Riemann-Hilbert problems characterizing the orthogonal polynomials. The local analysis at $x=0$ is carried out in terms of the confluent hypergeometric functions. Incidentally, we establish some properties of these functions that may have an independent interest.
In 1995 Magnus posed a conjecture about the asymptotics of the recurrence coefficients of orthogonal polynomials with respect to the weights on [-1,1] of the form $$ (1-x)^α(1+x)^β|x_0 - x|^γ\times a jump at x_0, $$ with $α, β, γ>-1$ and $x_0 \in (-1,1)$. We show rigorously that Magnus' conjecture is correct even in a more general situation, when the weight above has an extra factor, which is analytic in a neighborhood of [-1,1] and positive on the interval. The proof is based on the steepest descendent method of Deift and Zhou applied to the non-commutative Riemann-Hilbert problem characterizing the orthogonal polynomials. A feature of this situation is that the local analysis at $x_0$ has to be carried out in terms of confluent hypergeometric functions.