Valeriy ich Ivanov

Optimal argument in the sharp Jackson inequality in the space L 2 with hyperbolic weight

In the space L2 on the real axis with hyperbolic weight, the sharp Jackson inequality with optimal argument is proved.

Optimal arguments in jackson’s inequality in the power-weighted space L2(ℝd)

This paper is devoted to the determination of the optimal arguments in the exact Jackson inequality in the space L2 on the Euclidean space with power weight equal to the product of the moduli of the coordinates with nonnegative powers. The optimal arguments are studied depending on the geometry of the spectrum of the approximating entire functions and the neighborhood of zero in the definition of the modulus of continuity. The optimal arguments are obtained in the case where the first skew field is a lpd-ball for 1 ≤p≤ 2, and the second is a parallelepiped.

Some problems of approximation theory in the spacesLp on the line with power weight

In the spaces Lp on the line with power weight, we study approximation of functions by entire functions of exponential type. Using the Dunkl difference-differential operator and the Dunkl transform, we define the generalized shift operator, the modulus of smoothness, and the K-functional. We prove a direct and an inverse theorem of Jackson-Stechkin type and of Bernstein type. We establish the equivalence between the modulus of smoothness and the K-functional.

The Delsarte extremal problem for the Jacobi transform

We give the solution of the Delsarte extremal problem for even entire functions of exponential type that are Jacobi transforms and prove the uniqueness of the extremal function. The quadrature Markov formula on the half-line with zeros of the modified Jacobi function are used.

Approximation in L 2 by partial integrals of the Fourier transform over the eigenfunctions of the Sturm–Liouville operator

For approximations in the space L2(ℝ+) by partial integrals of the Fourier transform over the eigenfunctions of the Sturm–Liouville operator, we prove Jackson’s inequality with exact constant and optimal argument in the modulus of continuity. The optimality of the argument in the modulus of continuity is established using the Gauss quadrature formula on the half-line over the zeros of the eigenfunction of the Sturm–Liouville operator.

Approximation in L 2 by Partial Integrals of the Multidimensional Fourier Transform over the Eigenfunctions of the Sturm–Liouville Operator

For approximations in the space L2(ℝ+d) by partial integrals of the multidimensional Fourier transform over the eigenfunctions of the Sturm–Liouville operator, we prove the Jackson inequality with sharp constant and optimal argument in the modulus of continuity. The multidimensional weight that defines the Sturm–Liouville operator is the product of onedimensional weights. The one-dimensional weights can be, in particular, power and hyperbolic weights with various parameters. The optimality of the argument in the modulus of continuity is established by means of the multidimensional Gauss quadrature formula over zeros of an eigenfunction of the Sturm–Liouville operator. The obtained results are complete; they generalize a number of known results.

Some extremal problems for the Fourier transform on the hyperboloid

We give the solution of the Turán, Fejér, Delsarte, Logan, and Bohman extremal problems for the Fourier transform on the hyperboloid ℍd or Lobachevsky space. We apply the averaging function method over the sphere and the solution of these problems for the Jacobi transform on the half-line.

On the sharpness of Jackson’s inequality in the spaces L p on the half-line with power weight

In the space Lp, 1 ≤ p < 2, on the half-line with power weight, Jackson’s inequality between the value of the best approximation of a function by even entire functions of exponential type and its modulus of continuity defined by means of a generalized shift operator is well known. The question of the sharpness of the inequality remained open. For the constant in Jackson’s inequality, we obtain a lower bound, which proves its sharpness.

Optimal arguments in the Jackson-Stechkin inequality in L 2 (ℝ d ) with Dunkl weight

AbstractThe paper is devoted to the determination of the optimal arguments in the sharp Jackson-Stechkin inequality with modulus of continuity of order r in the space L2(ℝd) with Dunkl weight defined by the root system R and a nonnegative function of multiplicity k. If