Extremal interpolation with least norm of linear differential operator

Extremal functional interpolation in the mean with least value of thenth derivative for large averaging intervals

“…In the paper by Sharma and Tsimbalario [3], for h = 0, these results were carried over to nth-order linear differential operators with constant coefficients whose characteristic polynomial only has real roots. In [4,5] Shevaldin generalized the above-mentioned result of the author to arbitrary nth-order linear differential operators with constant coefficients. For h > 1, the solution of these problems is not known.…”

“…In order to get a simple representation of Δ j,mh by Δ j,h , we shall avail ourselves of the following expression of Δ j,h in terms of the finite differences of first order defined above (cf. [12,13]):…”

“…Shevaldin defined in [13] (see also [12]) a finite difference operator whose kernel coincides with that of a linear differential operator with constant coefficients. In particular, the differential operator whose kernel is the set of trigonometric polynomials of degree r − 1 is…”

“…We can define a finite difference for f ∈ L p (T) which is identically zero only if f ∈ T r−1 (see [13]) by…”

“…as the modified finite differences ∆ r,h were introduced by Shevaldin [13] (see also [12]) and are defined by…”

On estimating the rate of best trigonometric approximation by a modulus of smoothness

“…In order to get a simple representation of Δ j,mh by Δ j,h , we shall avail ourselves of the following expression of Δ j,h in terms of the finite differences of first order defined above (cf. [12,13]):…”

“…Shevaldin defined in [13] (see also [12]) a finite difference operator whose kernel coincides with that of a linear differential operator with constant coefficients. In particular, the differential operator whose kernel is the set of trigonometric polynomials of degree r − 1 is…”

“…We can define a finite difference for f ∈ L p (T) which is identically zero only if f ∈ T r−1 (see [13]) by…”

An Improved Jackson Inequality for the Best Trigonometric Approximation

“…as the modified finite differences ∆ r,h were introduced by Shevaldin [13] (see also [12]) and are defined by…”