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The design of a low-pass-frequency filter with the electronic change of the approximation characteristics of resulting responses is presented. The filter also offers the reconnection-less reconfiguration of the order (1st-, 2nd-, 3rd- and 4th-order functions are available). Furthermore, the filter offers the electronic control of the cut-off frequency of the output response. The feature of the electronic change in the approximation characteristics is investigated for the Butterworth, Bessel, Elliptic, Chebyshev and Inverse Chebyshev approximations. The design is verified by PSpice simulations and experimental measurements. The results are also supported by the transient domain response (response to the square waveform), comparison of the group delay, sensitivity analysis and implementation feasibility based on given approximation. The benefit of the proposed electronic change in the approximation characteristics feature (in general signal processing or for sensors in particular) is presented and discussed for an exemplary scenario.

We found analytic approximations for the Bessel function of the first kind J[sub.ν] (x), valid for any real value of x and any value of ν in the interval (−1/2, 3/2). The present approximation is exact for ν=−1/2, ν=1/2, and ν=3/2, where an exact function for each case is well known. The maximum absolute errors for ν near these peculiar values are very small. Throughout the interval, the absolute values remain below 0.05. The structure of the approximate function is defined considering the corresponding power series and asymptotic expansions, and they are quotients of three polynomials of the second degree combined with trigonometrical functions and fractional powers. This is, in some way, the Multipoint Quasi-rational Approximation (MPQA) technique, but now only two variables are considered, x and ν, which is novel, since in all previous publications only the variable x was considered and ν was given. Furthermore, in the case of J[sub.−1/2] (x), J[sub.1/2] (x), and J[sub.3/2] (x), the corresponding exact function was also a condition to be considered and fulfilled. It is important to point out that the zeros of the exact functions and the approximate ones are also almost coincident with small relative errors. Finally, the approximation presented here has the property of preservation of symmetry for ν>0, i.e., when there is a sign change in the variable x , the corresponding change agrees with a similar change in the power series of the exact function.

In this paper, we derive an analytical and explicit approximation for the Bessel function J[sub.2] (x) for positive real x with a maximum absolute error of approximately 0.004, which refines some new published analytic approximation. The absolute errors for large x decrease logarithmically, according to our analysis of these errors in the interval [0,10[sup.3] ]. The power series and the asymptotic series are used in the process to produce this analytic approximation. Additionally, extremely minor relative errors are shown for the values of x at which the zeros of our approximation function and the function J[sub.2] (x) occur. The first positive zero has the biggest relative error equal to 0.00219574. The relative errors then steadily drop, reaching 9.13302 ×10[sup.−7] for the eleventh zero.

In the article, we prove that the double inequalities π e − x 2 ( x + a ) < K 0 ( x ) < π e − x 2 ( x + b ) , 1 + 1 2 ( x + a ) < K 1 ( x ) K 0 ( x ) < 1 + 1 2 ( x + b ) hold for all x > 0 if and only if a ≥ 1 / 4 and b = 0 if a , b ∈ [ 0 , ∞ ) , where K ν ( x ) is the modified Bessel function of the second kind. As applications, we provide bounds for K n + 1 ( x ) / K n ( x ) with n ∈ N and present the necessary and sufficient condition such that the function x ↦ x + p e x K 0 ( x ) is strictly increasing (decreasing) on ( 0 , ∞ ) .

A reproducing kernel Hilbert space (RKHS) approximation problem arising from learning theory is investigated. Some K-functionals and moduli of smoothness with respect to RKHSs are defined with Fourier–Bessel series and Fourier–Bessel transforms, respectively. Their equivalent relation is shown, with which the upper bound estimate for the best RKHS approximation is provided. The convergence rate is bounded with the defined modulus of smoothness, which shows that the RKHS approximation can attain the same approximation ability as that of the Fourier–Bessel series and Fourier–Bessel transform. In particular, it is shown that for a RKHS produced by the Bessel operator, the convergence rate sums up to the bound of a corresponding convolution operator approximation. The investigations show some new applications of Bessel functions. The results obtained can be used to bound the approximation error in learning theory.

In this paper, we derive an analytical and explicit approximation for the Bessel function J2(x) for positive real x with a maximum absolute error of approximately 0.004, which refines some new published analytic approximation. The absolute errors for large x decrease logarithmically, according to our analysis of these errors in the interval [0,103]. The power series and the asymptotic series are used in the process to produce this analytic approximation. Additionally, extremely minor relative errors are shown for the values of x at which the zeros of our approximation function and the function J2(x) occur. The first positive zero has the biggest relative error equal to 0.00219574. The relative errors then steadily drop, reaching 9.13302×10−7 for the eleventh zero.

In the article, we present several sharp bounds for the modified Bessel function of the first kind I 0 ( t ) = ∑ n = 0 ∞ t 2 n 2 2 n ( n ! ) 2 and the Toader-Qi mean T Q ( a , b ) = 2 π ∫ 0 π / 2 a cos 2 θ b sin 2 θ d θ for all t > 0 and a , b > 0 with a ≠ b .

This paper investigates the stability of continued fractions with complex partial denominators and numerators equal to one. Such fractions are an important tool for function approximation and have wide application in physics, engineering, and mathematics. A formula is derived for the relative error of the approximant of a continued fraction, which depends on both the relative errors of the fraction’s elements and the elements themselves. Based on this formula, using the methodology of element sets and their corresponding value sets, conditions are established under which the approximants of continued fractions are stable to perturbations of their elements. Stability sets are constructed, which are sets of admissible values for the fraction’s elements that guarantee bounded errors in the approximants. For each of these sets, an estimate of the relative error that arises from the perturbation of the continued fraction’s elements is obtained. The results are illustrated with an example of a continued fraction that is an expansion of the ratio of Bessel functions of the first kind. A numerical experiment is conducted, comparing two methods for calculating the approximants of a continued fraction: the backward and forward algorithms. The computational stability of the backward algorithm is demonstrated, which corresponds to the theoretical research results. The errors in calculating approximants with this algorithm are close to the unit round-off, regardless of the order of approximation, which demonstrates the advantages of continued fractions in high-precision computation tasks. Another example is a comparative analysis of the accuracy and stability to perturbations of second-order polynomial model and so-called second-order continued fraction model in the problem of wood drying modeling. Experimental studies have shown that the continued fraction model shows better results both in terms of approximation accuracy and stability to perturbations, which makes it more suitable for modeling processes with pronounced asymptotic behavior.

We obtain an accurate analytic approximation for the Bessel function J2(x) using an improved multipoint quasirational approximation technique (MPQA). This new approximation is valid for all real values of the variable x, with a maximum absolute error of approximately 0.009. These errors have been analyzed in the interval from x=0 to x=1000, and we have found that the absolute errors for large x decrease logarithmically. The values of x at which the zeros of the exact function J2(x) and the approximated function J˜2(x) occur are also provided, exhibiting very small relative errors. The largest relative error is for the second zero, with εrel=0.0004, and the relative errors continuously decrease, reaching 0.0001 for the eleventh zero. The procedure to obtain this analytic approximation involves constructing a bridge function that connects the power series with the asymptotic approximation. This is achieved by using rational functions combined with other elementary functions, such as trigonometric and fractional power functions.