R. Bojanic

Slowly varying functions and asymptotic relations

Abstract : A survey of basic properties of slowly varying functions is given. The notion of quasi monotone functions is introduced and it is shown that a quasi monotone slowly varying function can be represented as a quotient of two non decreasing functions. The same problem of representation is also considered for some subclasses of quasi monotone slowly varying functions. (Author)

Rate of convergence of Bernstein polynomials for functions with derivatives of bounded variation

Abstract For any x ϵ (0, 1) we first prove that if ƒ x (t) ≡ ¦t − x¦ on [0, 1] then the Bernstein polynomials of ƒx satisfy the asymptotic relation ∑ k = 0 n ¦ k n − x¦( k n ) x k (1 − x) n − k = (2x (1 − x) π ) 1 2 1 √n + O( 1 n ) . This asymptotic relation is then used to study the rate of convergence of Bernstein polynomials for functions with derivatives of bounded variation. An estimate of the rate of convergence is given. This estimate is asymptotically the best possible at points where ƒ′ is continuous.

A note on approximation by Bernstein polynomials

3. J . W. Milnor, On the cobordism ring ti*, and a complex analogue. I, Amer. J. Math. 82 (1960), 505-521. 4. S. P. Novikov, Homotopy properties of Thorn complexes, Mat . Sb. (N.S.) 57 (99) (1962), 407-442. (Russian) 5. R. Thorn, Quelques propriétés globales des variétés differentiable. Comment. Math. Helv. 28 (1954), 17-86. 6. G. W. Whitehead, Generalized homology theories, Trans. Amer. Math . Soc. 102 (1962), 227-283.

An estimate for the rate of convergence of the eigenfunction expansions of functions of bounded variation

A quantitative version of Titchmarshs theorem is given concerning the eigenfunction expansion of functions of bounded variation

An Estimate for the Rate of Convergence of Convolution Products of Sequences

Suppose that the series

On the Approximation of Bounded Functions with Discontinuities of the First Kind by Generalized Shepard Operators

\sum _{k = 0}^\infty p_k x^k

A Unified Theory of Regularly Varying Sequences.

has a positive radius of convergence R and suppose that the sequence of positive numbers

On the rate of convergence of Fourier-Legendre series of functions of bounded variation

(a_n )

Rate of convergence of Hermite-Fejér polynomials for functions with derivatives of bounded variation

satisfies the condition

A Class of L 1 -Convergence

a_n / a_{n + 1} = \lambda + O(\delta _n )