On the order of summability of the Fourier inversion formula

In this article we show that the order of the point value, in the sense of Łojasiewicz, of a tempered distribution and the order of summability of the pointwise Fourier inversion formula are closely related. Assuming that the order of the point values and certain order of growth at infinity are given for a tempered distribution, we estimate the order of summability of the Fourier inversion formula. For Fourier series, and in other cases, it is shown that if the distribution has a distributional point value of order k, then its Fourier series is e.v. Cesàro summable to the distributional point value of order k+1. Conversely, we also show that if the pointwise Fourier inversion formula is e.v. Cesàro summable of order k, then the distribution is the (k+1)-th derivative of a locally integrable function, and the distribution has a distributional point value of order k+2. We also establish connections between orders of summability and local behavior for other Fourier inversion problems

FUSE-ML: development and external validation of a clinical prediction model for mid-term outcomes after lumbar spinal fusion for degenerative disease

Background: Indications and outcomes in lumbar spinal fusion for degenerative disease are notoriously heterogenous. Selected subsets of patients show remarkable benefit. However, their objective identification is often difficult. Decision-making may be improved with reliable prediction of long-term outcomes for each individual patient, improving patient selection and avoiding ineffective procedures. Methods: Clinical prediction models for long-term functional impairment [Oswestry Disability Index (ODI) or Core Outcome Measures Index (COMI)], back pain, and leg pain after lumbar fusion for degenerative disease were developed. Achievement of the minimum clinically important difference at 12 months postoperatively was defined as a reduction from baseline of at least 15 points for ODI, 2.2 points for COMI, or 2 points for pain severity. Results: Models were developed and integrated into a web-app (https://neurosurgery.shinyapps.io/fuseml/) based on a multinational cohort [N = 817; 42.7% male; mean (SD) age: 61.19 (12.36) years]. At external validation [N = 298; 35.6% male; mean (SD) age: 59.73 (12.64) years], areas under the curves for functional impairment [0.67, 95% confidence interval (CI): 0.59–0.74], back pain (0.72, 95%CI: 0.64–0.79), and leg pain (0.64, 95%CI: 0.54–0.73) demonstrated moderate ability to identify patients who are likely to benefit from surgery. Models demonstrated fair calibration of the predicted probabilities. Conclusions: Outcomes after lumbar spinal fusion for degenerative disease remain difficult to predict. Although assistive clinical prediction models can help in quantifying potential benefits of surgery and the externally validated FUSE-ML tool may aid in individualized risk–benefit estimation, truly impacting clinical practice in the era of “personalized medicine” necessitates more robust tools in this patient population

FUSE-ML: development and external validation of a clinical prediction model for mid-term outcomes after lumbar spinal fusion for degenerative disease

Background: Indications and outcomes in lumbar spinal fusion for degenerative disease are notoriously heterogenous. Selected subsets of patients show remarkable benefit. However, their objective identification is often difficult. Decision-making may be improved with reliable prediction of long-term outcomes for each individual patient, improving patient selection and avoiding ineffective procedures. Methods: Clinical prediction models for long-term functional impairment [Oswestry Disability Index (ODI) or Core Outcome Measures Index (COMI)], back pain, and leg pain after lumbar fusion for degenerative disease were developed. Achievement of the minimum clinically important difference at 12 months postoperatively was defined as a reduction from baseline of at least 15 points for ODI, 2.2 points for COMI, or 2 points for pain severity. Results: Models were developed and integrated into a web-app (https://neurosurgery.shinyapps.io/fuseml/) based on a multinational cohort [N = 817; 42.7% male; mean (SD) age: 61.19 (12.36) years]. At external validation [N = 298; 35.6% male; mean (SD) age: 59.73 (12.64) years], areas under the curves for functional impairment [0.67, 95% confidence interval (CI): 0.59–0.74], back pain (0.72, 95%CI: 0.64–0.79), and leg pain (0.64, 95%CI: 0.54–0.73) demonstrated moderate ability to identify patients who are likely to benefit from surgery. Models demonstrated fair calibration of the predicted probabilities. Conclusions: Outcomes after lumbar spinal fusion for degenerative disease remain difficult to predict. Although assistive clinical prediction models can help in quantifying potential benefits of surgery and the externally validated FUSE-ML tool may aid in individualized risk–benefit estimation, truly impacting clinical practice in the era of “personalized medicine” necessitates more robust tools in this patient population