The fractional orthogonal derivative

This paper builds on the notion of the so-called orthogonal derivative, where an n-th order derivative is approximated by an integral involving an orthogonal polynomial of degree n. This notion was reviewed in great detail in a paper in J. Approx. Theory (2012) by the author and Koornwinder. Here an approximation of the Weyl or Riemann-Liouville fractional derivative is considered by replacing the n-th derivative by its approximation in the formula for the fractional derivative. In the case of, for instance, Jacobi polynomials an explicit formula for the kernel of this approximate fractional derivative can be given. Next we consider the fractional derivative as a filter and compute the transfer function in the continuous case for the Jacobi polynomials and in the discrete case for the Hahn polynomials. The transfer function in the Jacobi case is a confluent hypergeometric function. A different approach is discussed which starts with this explicit transfer function and then obtains the approximate fractional derivative by taking the inverse Fourier transform. The theory is finally illustrated with an application of a fractional differentiating filter. In particular, graphs are presented of the absolute value of the modulus of the transfer function. These make clear that for a good insight in the behavior of a fractional differentiating filter one has to look for the modulus of its transfer function in a log-log plot, rather than for plots in the time domain.Comment: 32 pages, 7 figures. The section between formula (4.15) and (4.20) is correcte

Differentiation by integration using orthogonal polynomials, a survey

This survey paper discusses the history of approximation formulas for n-th order derivatives by integrals involving orthogonal polynomials. There is a large but rather disconnected corpus of literature on such formulas. We give some results in greater generality than in the literature. Notably we unify the continuous and discrete case. We make many side remarks, for instance on wavelets, Mantica's Fourier-Bessel functions and Greville's minimum R_alpha formulas in connection with discrete smoothing.Comment: 35 pages, 3 figures; minor corrections, subsection 3.11 added; accepted by J. Approx. Theor

Choices Should Have Consequences: Failure to Vaccinate, Harm to Others, and Civil Liability

A parent’s decision not to vaccinate a child may place others at risk if the child becomes infected and exposes others to the disease. Should an individual harmed by an infection transmitted from a child whose parents chose to forgo vaccination have a negligence claim against those parents? While I do not hold a legal degree and therefore cannot speak directly to issues of law, as a physician and ethicist it seems to me that the basic elements that comprise negligence claims—harm, duty, breach of duty, and causation—are met in some cases where parents forgo vaccination

Choices Should Have Consequences: Failure to Vaccinate, Harm to Others, and Civil Liability

A parent’s decision not to vaccinate a child may place others at risk if the child becomes infected and exposes others to the disease. Should an individual harmed by an infection transmitted from a child whose parents chose to forgo vaccination have a negligence claim against those parents? While I do not hold a legal degree and therefore cannot speak directly to issues of law, as a physician and ethicist it seems to me that the basic elements that comprise negligence claims—harm, duty, breach of duty, and causation—are met in some cases where parents forgo vaccination

Pediatric Ethics Guidelines for Hereditary Medullary Thyroid Cancer

<p/> <p>Hereditary medullary thyroid cancer is an aggressive cancer for which there is no standard effective systemic therapy, but which can be prevented through genetic screening and prophylactic thyroidectomy. Although this cancer accounts for roughly 17% of all pediatric thyroid cancers, a significant percentage of affected families do not "accept" screening, while many gene carriers delay or refuse prophylactic thyroid surgery for their children. Current genetic screening practices in medullary thyroid cancer are inadequate; more than 50% of index patients with hereditary medullary thyroid cancer present with a thyroid mass; up to 75% have distant metastasis. These proposed pediatric ethics guidelines focus on two ethical issues that affect at-risk children: (1) how do we identify at-risk children whose RET-positive relative refuses to disclose that they carry the mutation? (2) How do we protect RET-positive children whose parents refuse prophylactic thyroidectomy?</p