Abstract
We consider one generalization of functions, which are called as «binary self-similar functi- ons» by Bl. Sendov. In this paper, we analyze the connections of the object of study with well known classes of fractal functions, with the geometry of numerical series, with distributions of random variables with independent random digits of the two-symbol $Q_2$-representation, with theory of fractals. Structural, variational, integral, differential and fractal properties are studied for the functions of this class.
Highlights
ÏÐÎ ÎÄÈÍ ÊËÀÑ ÔÓÍÊÖIÉ Ç ÔÐÀÊÒÀËÜÍÈÌÈ ÂËÀÑÒÈÂÎÑÒßÌÈÓ ðîáîòi ðîçãëÿäà1òüñÿ îäíå óçàãàëüíåííÿ ôóíêöié, ÿêi Áë
Êëþ÷îâi ñëîâà i ôðàçè: Q2-çîáðàæåííÿ ÷èñåë, Q2-öèëiíäð, îñíîâíå ìåòðè÷íå âiäíîøåííÿ, ìíîæèíà íåïîâíèõ ñóì ðÿäó, êâàçiïîêàçíèêîâà ôóíêöiÿ, ñèíãóëÿðíà ôóíêöiÿ, åêñïîíåíöiéíèé ðîçïîäië íà âiäðiçêó, ôóíêöiÿ ç ôðàêòàëüíèìè âëàñòèâîñòÿìè, ôðàêòàëüíà ðîçìiðíiñòü Ãàóñäîðôà-Áåçèêîâè÷à
We analyze the connections of the object of study
Summary
Ó ðîáîòi ðîçãëÿäà1òüñÿ îäíå óçàãàëüíåííÿ ôóíêöié, ÿêi Áë. Ñåíäîâ íàçèâàâ 3⁄4äâîè÷íî ñîáñòâåííî-ïîäîáíûå¿, çäiéñíþ1òüñÿ àíàëiç çâ'ÿçêiâ îá'1êòà âèâ÷åííÿ ç âiäîìèìè êëàñàìè ôðàêòàëüíèõ ôóíêöié, ç ãåîìåòði1þ ÷èñëîâèõ ðÿäiâ, ðîçïîäiëàìè âèïàäêîâèõ âåëè÷èí ç íåçàëåæíèìè âèïàäêîâèìè öèôðàìè äâîñèìâîëüíîãî Q2-çîáðàæåííÿ, ç òåîði1þ ôðàêòàëiâ. Äëÿ ôóíêöié öüîãî êëàñó âèâ÷àþòüñÿ ñòðóêòóðíi, âàðiàöiéíi, iíòåãðàëüíi, äèôåðåíöiàëüíi òà ôðàêòàëüíi âëàñòèâîñòi. Êëþ÷îâi ñëîâà i ôðàçè: Q2-çîáðàæåííÿ ÷èñåë, Q2-öèëiíäð, îñíîâíå ìåòðè÷íå âiäíîøåííÿ, ìíîæèíà íåïîâíèõ ñóì ðÿäó, êâàçiïîêàçíèêîâà ôóíêöiÿ, ñèíãóëÿðíà ôóíêöiÿ, åêñïîíåíöiéíèé ðîçïîäië íà âiäðiçêó, ôóíêöiÿ ç ôðàêòàëüíèìè âëàñòèâîñòÿìè, ôðàêòàëüíà ðîçìiðíiñòü Ãàóñäîðôà-Áåçèêîâè÷à. Ó ðîáîòi [11] Áë. Íèì íàçâàíi äâiéêîâî âëàñíå-ïîäiáíèìè, êîæíà ç ÿêèõ âèçíà÷àëàñü ñâî1þ ÷èñëîâîþ ïîñëiäîâíiñòþ (λk) íåíóëüîâèõ åëåìåíòiâ ∞. Òàêîþ, ùî λk < ∞, i ôóíêöiîíàëüíèìè ñïiââiäíîøåííÿìè: k=1 f (x + 2−k) = λkf (x), x. 2−iai; 2−k + 2−iai , ai ∈ A ≡ {0, 1}
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