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Papers by Daniel Veronese

Computational & Applied Mathematics, Mar 23, 2023

Journal of Computational and Applied Mathematics, 2020

It was shown recently that given a pair of real sequences {{c n } ∞ n=1 , {d n } ∞ n=1 }, with {d... more It was shown recently that given a pair of real sequences {{c n } ∞ n=1 , {d n } ∞ n=1 }, with {d n } ∞ n=1 a positive chain sequence, we can associate a unique nontrivial probability measure µ on the unit circle, and conversely. In this paper, we consider the set Q(c) of all nontrivial probability measures for which c n = (−1) n c, with c ∈ R. We show that every measure µ ∈ Q(c) can be obtained from a perturbation of a symmetric measure η on [−1, 1]. Moreover, the sequence of orthogonal polynomials associated with µ can be given in terms of a perturbation of symmetric orthogonal polynomials associated with η. We also prove that every measure in Q(c) is quasi-symmetric, that is, there exists a complex valued function q (c) (z) satisfying dµ(z) = −q (c) (z)dµ(z), and such that q (c) (z) → 1 when c → 0. Quadrature rules with quasi-symmetric weights are also considered. Finally, some examples of orthogonal polynomials on the unit circle and its associated quasi-symmetric nontrivial probability measures are obtained.

Among the many well known quadrature formulas one finds those interesting interpolatory quadratur... more Among the many well known quadrature formulas one finds those interesting interpolatory quadrature formulas that take advantage of the properties of orthogonal polynomials Pn or similar polynomials Bn. Here, we consider the interpolatory quadrature rules based on the zeros of the polynomials ψn(x, ξ) = Pn−1(ξ)Pn(x)−Pn(ξ)Pn−1(x), and Gn(x, u) = Bn−1(u)Bn(x)− Bn(u)Bn−1(x) where ξ and u are arbitrary parameters. One of the objective of this dissertation is to study some of the known properties of quadrature rules based on ψn(x, ξ) and consider the analogous properties of the quadrature rules based on Gn(x, u).We also look at the convergence properties of those quadrature rules that serve to approximate integrals of the product of functions k and f, where k is a Lebesgue integrable function and f needs to be a Riemann integrable function.

Computational and Applied Mathematics, 2016

It was shown recently that associated with a pair of real sequences {{c n } ∞ n=1 , {d n } ∞ n=1 ... more It was shown recently that associated with a pair of real sequences {{c n } ∞ n=1 , {d n } ∞ n=1 }, with {d n } ∞ n=1 a positive chain sequence, there exists a unique nontrivial probability measure μ on the unit circle. The Verblunsky coefficients {α n } ∞ n=0 associated with the orthogonal polynomials with respect to μ are given by the relation α n−1 = τ n−1 1 − 2m n − ic n 1 − ic n , n ≥ 1, where τ 0 = 1, τ n = n k=1 (1 − ic k)/(1 + ic k), n ≥ 1 and {m n } ∞ n=0 is the minimal parameter sequence of {d n } ∞ n=1. In this manuscript, we consider this relation and its consequences by imposing some restrictions of sign and periodicity on the sequences {c n } ∞ n=1 and {m n } ∞ n=1. When the sequence {c n } ∞ n=1 is of alternating sign, we use information about the zeros of associated para-orthogonal polynomials to show that there is a gap in the support of the Communicated by Antonio José Silva Neto.

Mathematics of Computation, 2017

Given a non-trivial Borel measure µ on the unit circle T, the corresponding reproducing (or Chris... more Given a non-trivial Borel measure µ on the unit circle T, the corresponding reproducing (or Christoffel-Darboux) kernels with one of the variables fixed at z = 1 constitute a family of so-called para-orthogonal polynomials, whose zeros belong to T. With a proper normalization they satisfy a three-term recurrence relation determined by two sequence of real coefficients, {c n } and {d n }, where {d n } is additionally a positive chain sequence. Coefficients (c n , d n) provide a parametrization of a family of measures related to µ by addition of a mass point at z = 1. In this paper we estimate the location of the extreme zeros (those closest to z = 1) of the para-orthogonal polynomials from the (c n , d n)-parametrization of the measure, and use this information to establish sufficient conditions for the existence of a gap in the support of µ at z = 1. These results are easily reformulated in order to find gaps in the support of µ at any other z ∈ T. We provide also some examples showing that the bounds are tight and illustrating their computational applications.

Journal of Approximation Theory, 2018

Given a nontrivial positive measure µ on the unit circle, the associated Christoffel-Darboux kern... more Given a nontrivial positive measure µ on the unit circle, the associated Christoffel-Darboux kernels are K n (z, w; µ) = n k=0 ϕ k (w; µ) ϕ k (z; µ), n ≥ 0, where ϕ k (•; µ) are the orthonormal polynomials with respect to the measure µ. Let the positive measure ν on the unit circle be given by dν(z) = |G 2m (z)| dµ(z), where G 2m is a conjugate reciprocal polynomial of exact degree 2m. We establish a determinantal formula expressing {K n (z, w; ν)} n≥0 directly in terms of {K n (z, w; µ)} n≥0. Furthermore, we consider the special case of w = 1; it is known that appropriately normalized polynomials K n (z, 1; µ) satisfy a recurrence relation whose coefficients are given in terms of two sets of real parameters {c n (µ)} ∞ n=1 and {g n (µ)} ∞ n=1 , with 0 < g n < 1 for n ≥ 1. The double sequence {(c n (µ), g n (µ))} ∞ n=1 characterizes the measure µ. A natural question about the relation between the parameters c n (µ), g n (µ), associated with µ, and the sequences c n (ν), g n (ν), corresponding to ν, is also addressed. Finally, examples are considered, such as the Geronimus weight (a measure supported on an arc of T), a class of measures given by basic hypergeometric functions, and a class of measures with hypergeometric orthogonal polynomials.

Journal of Mathematical Analysis and Applications, 2017

Resumo.É conhecido (veja [4]) que associado a um par de sequências reais {{c n } ∞ n=1 , {d n } ∞... more Resumo.É conhecido (veja [4]) que associado a um par de sequências reais {{c n } ∞ n=1 , {d n } ∞ n=1 }, com {d n } ∞ n=1 uma sequência encadeada positiva, existe umaúnica medida de probabilidade não trivial µ no círculo unitário. Além disso, os coeficientes de Verblunsky {α n } ∞ n=0 , associados aos polinômios ortogonais com respeito a µ, são dados pela relação α n−1 = τ n−1 1 − 2m n − ic n 1 − ic n , n ≥ 1, onde τ 0 = 1, τ n = n k=1 (1−ic k)/(1+ic k), n ≥ 1 e {m n } ∞ n=0é a sequência de parâmetros minimal para {d n } ∞ n=1. Neste trabalho, impondo algumas restrições sobre o par de sequências reais {{c n } ∞ n=1 , {m n } ∞ n=1 }, mostramos queé possível obter medidas de probabilidade não triviais no círculo unitário cujos respectivos coeficientes de Verblunsky são periódicos.

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics, 2015

Resumo. Recentemente, foi mostrado em [5] que associado ao par de sequências reais {{c n } ∞ n=1 ... more Resumo. Recentemente, foi mostrado em [5] que associado ao par de sequências reais {{c n } ∞ n=1 , {d n } ∞ n=1 }, com {d n } ∞ n=1 uma sequência encadeada positiva, existe umaúnica medida de probabilidade não trivial µ no círculo unitário. Mostrou-se também que os coeficientes de Verblunsky {α n } ∞ n=0 , associados aos polinômios ortogonais com respeito a µ, podem ser relacionados diretamente com tais sequências. Neste trabalho, consideramos esta relação e suas consequências quando impomos uma restrição de sinal sobre a sequência {c n } ∞ n=1. Precisamente, quando a sequência {c n } ∞ n=1 tem uma propriedade de sinal alternante, usamos informações sobre os zeros de certos polinômios para-ortogonais para estimar o suporte da medida associada. Palavras-chave. Polinômios para-ortogonais, Medidas de probabilidade, Sequências encadeadas positivas, Sequências de sinal alternante.

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics, 2015

Applied Numerical Mathematics, 2011

A positive measure ψ defined on [a, b] such that its moments μ n = b a t n dψ(t) exist for n = 0,... more A positive measure ψ defined on [a, b] such that its moments μ n = b a t n dψ(t) exist for n = 0, ±1, ±2,. .. , is called a strong positive measure on [a, b]. If 0 a < b ∞ then the sequence of (monic) polynomials {Q n }, defined by b a t −n+s Q n (t) dψ(t) = 0, s = 0, 1,. .. ,n − 1, is known to exist. We refer to these polynomials as the L-orthogonal polynomials with respect to the strong positive measure ψ. The purpose of this manuscript is to consider some properties of the kernel polynomials associated with these Lorthogonal polynomials. As applications, we consider the quadrature rules associated with these kernel polynomials. Associated eigenvalue problems and numerical evaluation of the nodes and weights of such quadrature rules are also considered.

Journal of Approximation Theory, 2014

The objective of this manuscript is to study directly the Favard type theorem associated with the... more The objective of this manuscript is to study directly the Favard type theorem associated with the three term recurrence formula R n+1 (z) = (1 + ic n+1)z + (1 − ic n+1) R n (z) − 4d n+1 zR n−1 (z), n ≥ 1, with R 0 (z) = 1 and R 1 (z) = (1 + ic 1)z + (1 − ic 1), where {c n } ∞ n=1 is a real sequence and {d n } ∞ n=1 is a positive chain sequence. We establish that there exists an unique nontrivial probability measure µ on the unit circle for which {R n (z) − 2(1 − m n)R n−1 (z)} gives the sequence of orthogonal polynomials. Here, {m n } ∞ n=0 is the minimal parameter sequence of the positive chain sequence {d n } ∞ n=1. The element d 1 of the chain sequence, which does not effect the polynomials R n , has an influence in the derived probability measure µ and hence, in the associated orthogonal polynomials on the unit circle. To be precise, if {M n } ∞ n=0 is the maximal parameter sequence of the chain sequence, then the measure µ is such that M 0 is the size of its mass at z = 1. An example is also provided to completely illustrates the results obtained.

Computational & Applied Mathematics, Mar 23, 2023

Journal of Computational and Applied Mathematics, 2020

It was shown recently that given a pair of real sequences {{c n } ∞ n=1 , {d n } ∞ n=1 }, with {d... more It was shown recently that given a pair of real sequences {{c n } ∞ n=1 , {d n } ∞ n=1 }, with {d n } ∞ n=1 a positive chain sequence, we can associate a unique nontrivial probability measure µ on the unit circle, and conversely. In this paper, we consider the set Q(c) of all nontrivial probability measures for which c n = (−1) n c, with c ∈ R. We show that every measure µ ∈ Q(c) can be obtained from a perturbation of a symmetric measure η on [−1, 1]. Moreover, the sequence of orthogonal polynomials associated with µ can be given in terms of a perturbation of symmetric orthogonal polynomials associated with η. We also prove that every measure in Q(c) is quasi-symmetric, that is, there exists a complex valued function q (c) (z) satisfying dµ(z) = −q (c) (z)dµ(z), and such that q (c) (z) → 1 when c → 0. Quadrature rules with quasi-symmetric weights are also considered. Finally, some examples of orthogonal polynomials on the unit circle and its associated quasi-symmetric nontrivial probability measures are obtained.

Among the many well known quadrature formulas one finds those interesting interpolatory quadratur... more Among the many well known quadrature formulas one finds those interesting interpolatory quadrature formulas that take advantage of the properties of orthogonal polynomials Pn or similar polynomials Bn. Here, we consider the interpolatory quadrature rules based on the zeros of the polynomials ψn(x, ξ) = Pn−1(ξ)Pn(x)−Pn(ξ)Pn−1(x), and Gn(x, u) = Bn−1(u)Bn(x)− Bn(u)Bn−1(x) where ξ and u are arbitrary parameters. One of the objective of this dissertation is to study some of the known properties of quadrature rules based on ψn(x, ξ) and consider the analogous properties of the quadrature rules based on Gn(x, u).We also look at the convergence properties of those quadrature rules that serve to approximate integrals of the product of functions k and f, where k is a Lebesgue integrable function and f needs to be a Riemann integrable function.

Computational and Applied Mathematics, 2016

It was shown recently that associated with a pair of real sequences {{c n } ∞ n=1 , {d n } ∞ n=1 ... more It was shown recently that associated with a pair of real sequences {{c n } ∞ n=1 , {d n } ∞ n=1 }, with {d n } ∞ n=1 a positive chain sequence, there exists a unique nontrivial probability measure μ on the unit circle. The Verblunsky coefficients {α n } ∞ n=0 associated with the orthogonal polynomials with respect to μ are given by the relation α n−1 = τ n−1 1 − 2m n − ic n 1 − ic n , n ≥ 1, where τ 0 = 1, τ n = n k=1 (1 − ic k)/(1 + ic k), n ≥ 1 and {m n } ∞ n=0 is the minimal parameter sequence of {d n } ∞ n=1. In this manuscript, we consider this relation and its consequences by imposing some restrictions of sign and periodicity on the sequences {c n } ∞ n=1 and {m n } ∞ n=1. When the sequence {c n } ∞ n=1 is of alternating sign, we use information about the zeros of associated para-orthogonal polynomials to show that there is a gap in the support of the Communicated by Antonio José Silva Neto.

Mathematics of Computation, 2017

Given a non-trivial Borel measure µ on the unit circle T, the corresponding reproducing (or Chris... more Given a non-trivial Borel measure µ on the unit circle T, the corresponding reproducing (or Christoffel-Darboux) kernels with one of the variables fixed at z = 1 constitute a family of so-called para-orthogonal polynomials, whose zeros belong to T. With a proper normalization they satisfy a three-term recurrence relation determined by two sequence of real coefficients, {c n } and {d n }, where {d n } is additionally a positive chain sequence. Coefficients (c n , d n) provide a parametrization of a family of measures related to µ by addition of a mass point at z = 1. In this paper we estimate the location of the extreme zeros (those closest to z = 1) of the para-orthogonal polynomials from the (c n , d n)-parametrization of the measure, and use this information to establish sufficient conditions for the existence of a gap in the support of µ at z = 1. These results are easily reformulated in order to find gaps in the support of µ at any other z ∈ T. We provide also some examples showing that the bounds are tight and illustrating their computational applications.

Journal of Approximation Theory, 2018

Given a nontrivial positive measure µ on the unit circle, the associated Christoffel-Darboux kern... more Given a nontrivial positive measure µ on the unit circle, the associated Christoffel-Darboux kernels are K n (z, w; µ) = n k=0 ϕ k (w; µ) ϕ k (z; µ), n ≥ 0, where ϕ k (•; µ) are the orthonormal polynomials with respect to the measure µ. Let the positive measure ν on the unit circle be given by dν(z) = |G 2m (z)| dµ(z), where G 2m is a conjugate reciprocal polynomial of exact degree 2m. We establish a determinantal formula expressing {K n (z, w; ν)} n≥0 directly in terms of {K n (z, w; µ)} n≥0. Furthermore, we consider the special case of w = 1; it is known that appropriately normalized polynomials K n (z, 1; µ) satisfy a recurrence relation whose coefficients are given in terms of two sets of real parameters {c n (µ)} ∞ n=1 and {g n (µ)} ∞ n=1 , with 0 < g n < 1 for n ≥ 1. The double sequence {(c n (µ), g n (µ))} ∞ n=1 characterizes the measure µ. A natural question about the relation between the parameters c n (µ), g n (µ), associated with µ, and the sequences c n (ν), g n (ν), corresponding to ν, is also addressed. Finally, examples are considered, such as the Geronimus weight (a measure supported on an arc of T), a class of measures given by basic hypergeometric functions, and a class of measures with hypergeometric orthogonal polynomials.

Journal of Mathematical Analysis and Applications, 2017

Resumo.É conhecido (veja [4]) que associado a um par de sequências reais {{c n } ∞ n=1 , {d n } ∞... more Resumo.É conhecido (veja [4]) que associado a um par de sequências reais {{c n } ∞ n=1 , {d n } ∞ n=1 }, com {d n } ∞ n=1 uma sequência encadeada positiva, existe umaúnica medida de probabilidade não trivial µ no círculo unitário. Além disso, os coeficientes de Verblunsky {α n } ∞ n=0 , associados aos polinômios ortogonais com respeito a µ, são dados pela relação α n−1 = τ n−1 1 − 2m n − ic n 1 − ic n , n ≥ 1, onde τ 0 = 1, τ n = n k=1 (1−ic k)/(1+ic k), n ≥ 1 e {m n } ∞ n=0é a sequência de parâmetros minimal para {d n } ∞ n=1. Neste trabalho, impondo algumas restrições sobre o par de sequências reais {{c n } ∞ n=1 , {m n } ∞ n=1 }, mostramos queé possível obter medidas de probabilidade não triviais no círculo unitário cujos respectivos coeficientes de Verblunsky são periódicos.

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics, 2015

Resumo. Recentemente, foi mostrado em [5] que associado ao par de sequências reais {{c n } ∞ n=1 ... more Resumo. Recentemente, foi mostrado em [5] que associado ao par de sequências reais {{c n } ∞ n=1 , {d n } ∞ n=1 }, com {d n } ∞ n=1 uma sequência encadeada positiva, existe umaúnica medida de probabilidade não trivial µ no círculo unitário. Mostrou-se também que os coeficientes de Verblunsky {α n } ∞ n=0 , associados aos polinômios ortogonais com respeito a µ, podem ser relacionados diretamente com tais sequências. Neste trabalho, consideramos esta relação e suas consequências quando impomos uma restrição de sinal sobre a sequência {c n } ∞ n=1. Precisamente, quando a sequência {c n } ∞ n=1 tem uma propriedade de sinal alternante, usamos informações sobre os zeros de certos polinômios para-ortogonais para estimar o suporte da medida associada. Palavras-chave. Polinômios para-ortogonais, Medidas de probabilidade, Sequências encadeadas positivas, Sequências de sinal alternante.

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics, 2015

Applied Numerical Mathematics, 2011

A positive measure ψ defined on [a, b] such that its moments μ n = b a t n dψ(t) exist for n = 0,... more A positive measure ψ defined on [a, b] such that its moments μ n = b a t n dψ(t) exist for n = 0, ±1, ±2,. .. , is called a strong positive measure on [a, b]. If 0 a < b ∞ then the sequence of (monic) polynomials {Q n }, defined by b a t −n+s Q n (t) dψ(t) = 0, s = 0, 1,. .. ,n − 1, is known to exist. We refer to these polynomials as the L-orthogonal polynomials with respect to the strong positive measure ψ. The purpose of this manuscript is to consider some properties of the kernel polynomials associated with these Lorthogonal polynomials. As applications, we consider the quadrature rules associated with these kernel polynomials. Associated eigenvalue problems and numerical evaluation of the nodes and weights of such quadrature rules are also considered.

Journal of Approximation Theory, 2014

The objective of this manuscript is to study directly the Favard type theorem associated with the... more The objective of this manuscript is to study directly the Favard type theorem associated with the three term recurrence formula R n+1 (z) = (1 + ic n+1)z + (1 − ic n+1) R n (z) − 4d n+1 zR n−1 (z), n ≥ 1, with R 0 (z) = 1 and R 1 (z) = (1 + ic 1)z + (1 − ic 1), where {c n } ∞ n=1 is a real sequence and {d n } ∞ n=1 is a positive chain sequence. We establish that there exists an unique nontrivial probability measure µ on the unit circle for which {R n (z) − 2(1 − m n)R n−1 (z)} gives the sequence of orthogonal polynomials. Here, {m n } ∞ n=0 is the minimal parameter sequence of the positive chain sequence {d n } ∞ n=1. The element d 1 of the chain sequence, which does not effect the polynomials R n , has an influence in the derived probability measure µ and hence, in the associated orthogonal polynomials on the unit circle. To be precise, if {M n } ∞ n=0 is the maximal parameter sequence of the chain sequence, then the measure µ is such that M 0 is the size of its mass at z = 1. An example is also provided to completely illustrates the results obtained.