Haarovy systémy

Mintál, Matúš

One of the principal general onjectives of the approximation theory is a replacement of a complicated mathematical object by another one, which, in a certain sense, is close to given one, and, at the same time, is easier to handle. One of the standard methods of this theory is the approximation of a continuous function by a polynomial. An exten- sion of this method to more general classes of functions leads to approximation by the generalized polynomials. A classical result of Haar of 1917 states that the existence and unicity of an element of best approximation is equivalent to the validity of the so-called Haar condition. In this thesis we shall present several equivalent formulations of the Haar condition and illustrate some of its possible applications with examples. We recall one important application of the Haar condition, the alternation theorem, and demon- strate its usefulness. Finally, we establish basic properties of the operator of the best approximation.

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