Fourier analysis and the Theory of Function Spaces are closely connected with several aspects of fractal geometry.

We want to relate three different forms of measuring smoothness on the class of all continuous real functions with compact support:

• Frequency structure in the context of Besov-Triebel-Lizorkin spaces,
• Oscillations between local maximums and minimums of the function,
• Fractal dimensions of the graph, which depend on the number of balls needed to cover the graph.

The diversification of techniques to measure smoothness has been essential in the systematization of the function spaces. In order to achieve that aim, we will:

• Develop embeddings between Besov and oscillation spaces,
• Calculate maximal and minimal fractal dimensions,
• Establish distinction between smoothness and fractal dimensions,
• Point out some fractal properties of the Weierstrass-type functions,
• Search and construct graphs (of continuous functions) which are h-sets.