Approximation of the Separation Boundary in the Water-Hydrotrope-Oil Ternary Systems

In this work, we studied the ternary systems water – 2-butoxyethanol – toluene, water – methanol – chloroform and water – methanol – dichloromethane. The separation boundary was experimentally located and approximated by various functions. The main reasons are revealed that prevent a satisfactory approximation of the separation boundaries by empirical functions with a small number of optimized parameters. It is found that the approximation of the separation boundaries in ternary systems can be carried out by polynomial and piecewise smooth functions with an error comparable to the uncertainty of the measurement.

Smoothing approximations for piecewise smooth functions: A probabilistic approach

<p style='text-indent:20px;'>In this article, we present a new approach to construct smoothing approximations for piecewise smooth functions. This approach proposes to formulate any piecewise smooth function as the expectation of a random variable. Based on this formulation, we show that smoothing all elements of a defined space of piecewise smooth functions is equivalent to smooth a single probability distribution. Furthermore, we propose to use the Boltzmann distribution as a smoothing approximation for this probability distribution. Moreover, we present the theoretical results, error estimates, and some numerical examples for this new smoothing method in both one-dimensional and multiple-dimensional cases.</p>

The approximation of piecewise smooth functions by trigonometric Fourier sums

We obtain exact order-of-magnitude estimates of piecewise smooth functions approximation by trigonometric Fourier sums. It is shown that in continuity points Fourier series of piecewise Lipschitz function converges with rate $\ln n/n$. If function $f$ has a piecewise absolutely continuous derivative then it is proven that in continuity points decay order of Fourier series remainder $R_n(f,x)$ for such function is equal to $1/n$. We also obtain exact order-of-magnitude estimates for $q$-times differentiable functions with piecewise smooth $q$-th derivative. In particular, if $f^{(q)}(x)$ is piecewise Lipschitz then $|R_n(f,x)| \le c(x)\frac{\ln n}{n^{q+1}}$ in continuity points of $f^{(q)}(x)$ and $\sup_{x \in [0,2\pi]}|R_n(f,x)| \le \frac{c}{n^q}$. In case when $f^{(q)}(x)$ has piecewise absolutely continuous derivative it is shown that $|R_n(f,x)| \le \frac{c(x)}{n^{q+1}}$ in continuity points of $f^{(q)}(x)$. As a consequence of the last result convergence rate estimate of Fourier series to continuous piecewise linear functions is obtained.