We introduce an orthogonal basis on the square [−1, 1] × [−1, 1] generated by Legendre polynomials on [−1, 1], and define an associated expression for the expansion of a Riemann integrable function. We describe some properties and derive a uniform convergence theorem. We then present several numerical experiments that indicate that our methods are more efficient and have better convergence results than some other methods.

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BackgroundExhaled aerosol patterns, also called aerosol fingerprints, provide clues to the health of the lung and can be used to detect disease-modified airway structures. The key is how to decode the exhaled aerosol fingerprints and retrieve the lung structural information for a non-invasive identification of respiratory diseases.Objective and MethodsIn this study, a CFD-fractal analysis method was developed to quantify exhaled aerosol fingerprints and applied it to one benign and three malign conditions: a tracheal carina tumor, a bronchial tumor, and asthma. Respirations of tracer aerosols of 1 µm at a flow rate of 30 L/min were simulated, with exhaled distributions recorded at the mouth. Large eddy simulations and a Lagrangian tracking approach were used to simulate respiratory airflows and aerosol dynamics. Aerosol morphometric measures such as concentration disparity, spatial distributions, and fractal analysis were applied to distinguish various exhaled aerosol patterns.FindingsUtilizing physiology-based modeling, we demonstrated substantial differences in exhaled aerosol distributions among normal and pathological airways, which were suggestive of the disease location and extent. With fractal analysis, we also demonstrated that exhaled aerosol patterns exhibited fractal behavior in both the entire image and selected regions of interest. Each exhaled aerosol fingerprint exhibited distinct pattern parameters such as spatial probability, fractal dimension, lacunarity, and multifractal spectrum. Furthermore, a correlation of the diseased location and exhaled aerosol spatial distribution was established for asthma.ConclusionAerosol-fingerprint-based breath tests disclose clues about the site and severity of lung diseases and appear to be sensitive enough to be a practical tool for diagnosis and prognosis of respiratory diseases with structural abnormalities.

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