Self-Affine Singular and Nowhere Monotone Functions Related to the Q-Representation of Real Numbers
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We study functional, differential, integral, self-affine, and fractal properties of continuous functions from a finite-parameter family of functions with a continual set of “peculiarities.” Almost all functions in this family are singular (their derivative is equal to zero almost everywhere in a sense of the Lebesgue measure) or nowhere monotone and, in particular, not differentiable. We consider various approaches to the definition of these functions (by using a system of functional equations, projectors of the symbols of various representations, distributions of random variables, etc.).
KeywordsMonotone Function Ukrainian National Academy Singular Function Bernoulli Convolution Singular Distribution
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