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Table Of Integrals Series And Products

Singular Integral Equations. By N.I. Muskhelishvili. Translated fromthe 2nd Russian edition by J.R.M. Radok. Pp. 447. F1. 28.50. 1953. (Noordhoff, Groningen)

Most numerical integration techniques consist of approximating the integrand by a polynomial in a region or regions and then integrating the polynomial exactly. Often a complicated integrand can be factored into a non-negative ''weight'' function and another function better approximated by a polynomial, thus $\int_{a}^{b} g(t)dt = \int_{a}^{b} \omega (t)f(t)dt \approx \sum_{i=1}^{N} w_i f(t_i)$. Hopefully, the quadrature rule ${\{w_j, t_j\}}_{j=1}^{N}$ corresponding to the weight function $\omega$(t) is available in tabulated form, but more likely it is not. We present here two algorithms for generating the Gaussian quadrature rule defined by the weight function when: a) the three term recurrence relation is known for the orthogonal polynomials generated by $\omega$(t), and b) the moments of the weight function are known or can be calculated.

Calculation of Gauss quadrature rules.

Elements of the Theory of Functions and Functional Analysis, Volume 1. Me and Normed Spaces.

http://www.fil.ion.ucl.ac.uk/~wpenny/publications/ridgway11.pdf

The problem of low variance voxels in statistical parametric mapping; a new hat avoids a 'haircut'

The zeros of Bessel functions play an important role in computational mathematics, mathematical physics, and other areas of natural sciences. Studies addressing these zeros (their properties, computational methods) can be found in various sources. This paper offers a detailed overview of the results concerning the real zeros of the Bessel functions of the first and second kinds and general cylinder functions. The author intends to publish several overviews on this subject. In this first publication, works dealing with real zeros are analyzed. Primary emphasis is placed on classical results, which are still important. Some of the most recent publications are also discussed.

Studies on the zeros of Bessel functions and methods for their computation

Two useful estimates are proved for the Fourier transform in the space of square integrable functions on certain classes of functions characterized by the generalized continuity modulus.

Some remarks concerning the Fourier transform in the space L2(ℝn)

The author’s previous work provided a detailed overview of the results concerning the theory and applications of the Rayleigh special function starting from its appearance in science until recent years. Its numerous applications in various areas of mathematics, physics, and other fields were described, and an extensive bibliography was presented. This work overviews the studies not covered in the previous one and addresses new results published in many monographs and journals. Additionally, results concerning the estimation of zeros of some special polynomials and functions closely related to the Rayleigh function are described. The overview embraces the issues addressed in the scientific literature up to the last years.

Overview of some new results concerning the theory and applications of the Rayleigh special function

Some issues concerning the approximation of one-variable functions from the class $\mathbb{L}_2 $ by nth-order partial sums of Fourier-Bessel series are studied. Several theorems are proved that estimate the best approximation of a function characterized by the generalized modulus of continuity.

Some issues concerning approximations of functions by Fourier-Bessel sums

Two estimates useful in applications are proved for the Fourier-Bessel integral transform in L2(ℝ+) as applied to some classes of functions characterized by a generalized modulus of continuity.

M. K. Kerimov

Papers

Studies on the zeros of Bessel functions and methods for their computation

Some remarks concerning the Fourier transform in the space L2(ℝn)

Overview of some new results concerning the theory and applications of the Rayleigh special function

Some issues concerning approximations of functions by Fourier-Bessel sums

On estimates for the Fourier-Bessel integral transform in the space L2(ℝ+)