The classical theory of Gaussian quadrature assumes a positive weight function. We will show that in some cases Gaussian rules can be constructed with respect to an oscillatory weight, yielding methods with complex quadrature nodes and positive weights. These rules are well suited for highly oscillatory integrals because they attain optimal asymptotic order. We show that for the Fourier oscillator this approach yields the numerical method of steepest descent, a method with optimal asymptotic order that has previously been proposed for this class of integrals. However, the approach readily extends to more general kernels, such as Bessel functions that appear as the kernel of the Hankel transform.
We investigate a Gaussian quadrature rule and the corresponding orthogonal polynomials for the oscillatory weight function e iωx on the interval [−1, 1]. We show that such a rule attains high asymptotic order, in the sense that the quadrature error quickly decreases as a function of the frequency ω. However, accuracy is maintained for all values of ω and in particular the rule elegantly reduces to the classical Gauss-Legendre rule as ω → 0. The construction of such rules is briefly discussed, and though not all orthogonal polynomials exist, it is demonstrated numerically that rules with an even number of points are always well defined. We show that these rules are optimal both in terms of asymptotic order as well as in terms of polynomial order.
BackgroundActigraphy could be an objective alternative to clinical ratings of motor activity in bipolar disorder (BD), which is of importance now that increased activity and energy are added as cardinal symptoms of (hypo)mania in the DSM-5 and commonly used rating scales give inadequate information about motor symptoms. To date, most actigraphy studies have been conducted in groups and/or used mean activity levels as the variable of interest. The novelty of this case series is therefore to indicate the potential of actigraphy and non-parametric analysis as an objective and personalized marker of intra-individual activity patterns in different phases of BD. To our knowledge, this is the first case series that provides an objective assessment of non-linear dynamics in within-person activity patterns during acute BD episodes.ResultsWe report on three cases of bipolar I disorder with 24-h actigraphy recordings undertaken during the first few days of two or more separate admissions for an acute illness episode, including admissions for individuals in different phases of BD, or with different levels of severity in the same phase of illness. For each recording, we calculated mean activity levels over 24 h, but especially focused on key measures of variability and complexity in activity. Intra-individual activity patterns were found to be different according to phase of illness, but showed consistency within the same phase. With increasing psychotic symptoms, there was evidence of a lower overall level and greater irregularity in activity. As such, sample entropy (a measure of irregularity) may have particular utility in characterizing mania and psychotic symptoms, while assessment of the distribution of rest versus activity over 24 h may distinguish between phases of BD within an individual.ConclusionsThis case series indicates that objective, intra-individual, real-time recordings of patterns of activity may have clinical impact as a valuable adjunct to clinical observation and symptom ratings. We suggest that actigraphy combined with detailed mathematical analysis provides a biological variable that could become an important tool for developing a personalized approach to diagnostics and treatment monitoring in BD.Electronic supplementary materialThe online version of this article (10.1186/s40345-017-0115-3) contains supplementary material, which is available to authorized users.
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