Our primary goal in this preamble is to highlight the best of Vasil Popov’s
mathematical achievements and ideas. V. Popov showed his extraordinary talent
for mathematics in his early papers in the (typically Bulgarian) area of approximation
in the Hausdorff metric. His results in this area are very well presented
in the monograph of his advisor Bl. Sendov, “Hausdorff Approximation”.

2000 Mathematics Subject Classification: 46B70, 41A25, 41A17, 26D10.
∗Part of the results were reported at the Conference “Pioneers of Bulgarian Mathematics”,
Sofia, 2006.
Certain types of weighted Peetre K-functionals are characterized by means
of the classical moduli of smoothness taken on a proper linear
transforms of the function. The weights with power-type asymptotic at the
ends of the interval with arbitrary real exponents are considered. This paper
extends the method and results...

2000 Mathematics Subject Classification: 46B70, 41A10, 41A25, 41A27, 41A35, 41A36,
42A10.
The paper presents a method of relating two K-functionals by
means of a continuous linear transform of the function. In particular, a
characterization of various weighted K-functionals by unweighted fixed-step
moduli of smoothness is derived. This is applied in estimating the rate of
convergence of several approximation processes.
Partially supported by grant No. 103/2007 of the National...

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