%Bl. Sendov
%Sofia, BULGARIA
%Abstract for the Conference O(h^4), 18-21 August, 1998, Sofia.

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\title{Orthonormal Systems of Fractal Functions}
\author{Bl. Sendov
}

\date{}

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\maketitle

\begin {abstract}

Let $\{\phi_i\}_{i=0}^\infty$ be a orthonormal basis in the Hilbert
space $F$, let $c_i(f) = \langle f,\phi_i\rangle;\; i=0, 1, 2, \dots, n$ be
the Fourier coefficients of the function $f \in F$ and
$$P_n(f;x) = \sum_{i=0}^n c_i(f) \phi_i(x).$$

The error of the approximation of $f$ by $P_n(f;\cdot)$ depend on
$n$ and on the choice of the orthonormal system  $\{\phi_i\}_{i=0}^\infty$.

The aim of this lecture is to demonstrate a method for construction of a
orthonormal basis adapted to a given function $f$, based on a
generalization of the Walsh functions . The advantage of this
adapted orthonormal basis is the improvement of the approximation for
discontinuous functions and fractal functions. The motivation for this study
is the application in the signal and image compression.
\end{abstract}

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