On an extremal problem for nonnegative trigonometric polynomials and the characterization of positive quadrature formulas with Chebyshev weight function

1982 ◽  
Vol 39 (1-3) ◽  
pp. 107-116 ◽  
Author(s):  
F. Peherstorfer
Keyword(s):  
Weight Function ◽  
Extremal Problem ◽  
Trigonometric Polynomials ◽  
Quadrature Formulas ◽  
Positive Quadrature Formulas

scholarly journals A certain class of quadratures with even Tchebychev weights

2012 ◽  
Vol 20 (1) ◽  
pp. 447-458
Author(s):  
Zlatko Udovičić ◽  
Mirna Udovičić
Keyword(s):  
Weight Function ◽  
Algebraic Degree ◽  
Quadrature Formulas ◽  
Approximate Computation ◽  
Degree Of Exactness

Abstract We are considering the quadrature formulas of “practical type” (with five knots) for approximate computation of integral [xxx] where w(·) denotes (even) Tchebychev weight function. We prove that algebraic degree of exactness of those formulas can not be greater than five. We also determined some admissible nodes and compared proposed formula with some other quadrature formulas.

scholarly journals An extremal problem for trigonometric polynomials

1999 ◽  
Vol 127 (1) ◽  
pp. 211-216 ◽  
Author(s):  
J. Marshall Ash ◽  
Michael Ganzburg
Keyword(s):  
Extremal Problem ◽  
Trigonometric Polynomials

A Quadrature Formula for a Hypersingular Integral Containing the Weight Function of Jacobi Polynomials with Complex Exponents

2020 ◽  
pp. 94-100
Author(s):  
A.V. Sahakyan
Keyword(s):  
Weight Function ◽  
Integral Equations ◽  
Quadrature Formula ◽  
Jacobi Polynomials ◽  
Quadrature Formulas ◽  
Hölder Continuous ◽  
Approximate Methods ◽  
Hypersingular Integral ◽  
Hypersingular Integral Equations ◽  
True Value

Although the concept of a hypersingular integral was introduced by Hadamard at the beginning of the 20th century, it began to be put into practical use only in the second half of the century. The theory of hypersingular integral equations has been widely developed in recent decades and this is due to the fact that they describe the governing equations of many applied problems in various fields: elasticity theory, fracture mechanics, wave diffraction theory, electrodynamics, nuclear physics, geophysics, theory vibrator antennas, aerodynamics, etc. It is analytically possible to calculate the hypersingular integral only for a very narrow class of functions; therefore, approximate methods for calculating such an integral are always in the field of view of researchers and are a rapidly developing area of computational mathematics. There are a very large number of papers devoted to this subject, in which various approaches are proposed both to approximate calculation of the hypersingular integral and to the solution of hypersingular integral equations, mainly taking into account the specifics of the behavior of the densi-ty of the hypersingular integral. In this paper, quadrature formulas are obtained for a hypersingular integral whose density is the product of the Hölder continuous function on the closed interval [–1, 1], and weight function of the Jacobi polynomials . It is assumed that the exponents α and β can be arbitrary complex numbers that satisfy the condition of non-negativity of the real part. The numerical examples show the convergence of the quadrature formula to the true value of the hypersingular integral. The possibility of applying the mechanical quadrature method to the solution of various, including hypersingular, integral equations is indicated.

Certain extremal problem for nonnegative trigonometric polynomials

1990 ◽  
Vol 47 (1) ◽  
pp. 10-20 ◽  
Author(s):  
V. V. Arestov ◽  
V. P. Kondrat'ev
Keyword(s):  
Extremal Problem ◽  
Trigonometric Polynomials

scholarly journals The Bojanov-Naidenov problem for trigonometric polynomials and periodic splines

2019 ◽  
Vol 27 (1) ◽  
pp. 3
Author(s):  
E.V. Asadova ◽  
V.A. Kofanov
Keyword(s):  
Extremal Problem ◽  
Fixed Interval ◽  
Trigonometric Polynomials ◽  
Minimal Defect ◽  
Periodic Spline

For given $n, r \in \mathbb{N}$; $p, A > 0$ and any fixed interval $[a,b] \subset \mathbb{R}$ we solve the extremal problem $\int\limits_a^b |x(t)|^q dt \rightarrow \sup$, $q \geqslant p$, over sets of trigonometric polynomials $T$ of order $\leqslant n$ and $2\pi$-periodic splines $s$ of order $r$ and minimal defect with knots at the points $k\pi / n$, $k \in \mathbb{Z}$, such that $\| T \| _{p, \delta} \leqslant A \| \sin n (\cdot) \|_{p, \delta} \leqslant A \| \varphi_{n,r} \|_{p, \delta}$, $\delta \in (0, \pi / n]$, where $\| x \|_{p, \delta} := \sup \{ \| x \|_{L_p[a,b]} \colon a, b \in \mathbb{R}, 0 < b - a < \delta\}$ and $\varphi_{n, r}$ is the $(2\pi / n)$-periodic spline of Euler of order $r$. In particular, we solve the same problem for the intermediate derivatives $x^{(k)}$, $k = 1, ..., r-1$, with $q \geqslant 1$.

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