Получены точные неравенства типа Джексона-Стечкина для ос-редненных с весом модулей непрерывности m-го (m ∈ ℕ) порядка. Для классов функций, определенных при помоши мажорант и укаэанных осредненных величин, вычислены точные эначения
раэличных n-поперечников при выполнении определенных ограничений на мажоранты.
The best rate of approximation of functions on the sphere by spherical polynomials is majorized by recently introduced moduli
of smoothness. The treatment applies to a wide class of Banach spaces of functions.
We consider the weighted Hermite-Fejér interpolation process based on Jacobi nodes for classes of locally continuous functions defined by another Jacobi weight. Necessary and sufficient conditions for the weighted norm boundedness and for the convergence, as well as error estimates of the approximation, are given.
In the paper we construct such second order linear recursive sequences G and H of rational integers that with their terms |a -Gn+1 /H n| < 1/ (\sqrtvDH2n) holds for every positive integer n, where a denotes a real quadratic algebraic integer of discriminant D. An approximating sequence of the form Gn+1 /Hn is also given for a if it is only a real quadratic algebraic number (not an algebraic integer), but in this case the approximating constant is not the best.
) where 1 <
< ∞ was investigated. These results has been generalized to the two-dimensional case and applied to obtain generalizations of the Bernstein inequality for trigonometric polynomials of one and two variables. Also, the rates of convergence of Cesaro and Abel-Poisson means of functions
We obtain estimates of approximations by angle in Hardy and Lp spaces in terms of double Fourier-Vilenkin coefficients. Analogous results are also established for best approximations in
the one-dimensional case.
We present applications of Hermite polynomials in
signal analysis. Among other result, we give a characterization of the
so-called time-frequency window functions in terms of the Hermite--Fourier
coefficients, a Bernstein-type theorem for the best approximations of window
functions by Hermite-functions, time-frequency approximations. Some analogues
for Hankel-transforms will also be considered.