<abstract xmlns="http://www.w3.org/1999/xhtml">The paper presents the investigation of applications of infinite systems of functional equations for modeling functions with complicated local structure that are defined in terms of the nega-˜<italic>Q</italic>-representation. The infinite systems of functional equations<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_tmmp-2019-0024_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>f</mi><mrow><mo>(</mo><mrow><msup><mrow><mover><mi>φ</mi><mo>ˆ</mo></mover></mrow><mi>k</mi></msup><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>=</mo><mover><mi>β</mi><mo>˜</mo></mover><msub><mrow><mi>i</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>,</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>+</mo><mover><mi>p</mi><mo>˜</mo></mover><msub><mrow><mi>i</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>,</mo><mi>k</mi><mo>+</mo><mn>1</mn><mi>f</mi><mrow><mo>(</mo><mrow><msup><mrow><mover><mi>φ</mi><mo>ˆ</mo></mover></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>,</mo></mrow></math><tex-math>f\left( {{{\hat \varphi }^k}(x)} \right) = \tilde \beta {i_{k + 1}},k + 1 + \tilde p{i_{k + 1}},k + 1f\left( {{{\hat \varphi }^{k + 1}}(x)} \right),</tex-math></alternatives></disp-formula>where <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_tmmp-2019-0024_eq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>x</mi><mo>=</mo><msubsup><mrow><mi>Δ</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mn>1</mn></msub><mo>(</mo><mi>x</mi><mo>)</mo><msub><mrow><mi>i</mi></mrow><mn>2</mn></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>…</mo><msub><mrow><mi>i</mi></mrow><mi>n</mi></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>…</mo></mrow><mrow><mo>-</mo><mover><mi>Q</mi><mo>˜</mo></mover></mrow></msubsup></mrow></math><tex-math>x = \Delta _{{i_1}(x){i_2}(x) \ldots {i_n}(x) \ldots }^{ - \tilde Q}</tex-math></alternatives></inline-formula>, and φ ̑ is the shift operator of the <italic>Q</italic>̃-expansion, are investigated. It is proved that the system has a unique solution in the class of determined and bounded on [0, 1] functions. Its analytical presentation is founded. The continuity of the solution is studied. Conditions of its monotonicity and nonmonotonicity, differential, and integral properties are studied. Conditions under which the solution of the system of functional equations is a distribution function of the random variable <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_tmmp-2019-0024_eq_003.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>η</mi><mo>=</mo><msubsup><mrow><mi>Δ</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mn>1</mn></msub><mi> </mi><mi>ξ</mi><mn>2</mn><mo>…</mo><msub><mrow><mi>ξ</mi></mrow><mi>n</mi></msub><mo>…</mo></mrow><mover><mi>Q</mi><mo>˜</mo></mover></msubsup></mrow></math><tex-math>\eta = \Delta _{{\xi _1}\,\xi 2 \ldots {\xi _n} \ldots }^{\tilde Q}</tex-math></alternatives></inline-formula> with independent <italic>Q</italic>̃-symbols are founded.</abstract>

The paper presents the investigation of applications of infinite systems of functional equations for modeling functions with complicated local structure that are defined in terms of the nega-˜Q-representation. The infinite systems of functional equationsf(φˆk(x))=β˜ik+1,k+1+p˜ik+1,k+1f(φˆk+1(x)),f\left( {{{\hat \varphi }^k}(x)} \right) = \tilde \beta {i_{k + 1}},k + 1 + \tilde p{i_{k + 1}},k + 1f\left( {{{\hat \varphi }^{k + 1}}(x)} \right),where x=Δi1(x)i2(x)…in(x)…-Q˜x = \Delta _{{i_1}(x){i_2}(x) \ldots {i_n}(x) \ldots }^{ - \tilde Q}, and φ ̑ is the shift operator of the Q̃-expansion, are investigated. It is proved that the system has a unique solution in the class of determined and bounded on [0, 1] functions. Its analytical presentation is founded. The continuity of the solution is studied. Conditions of its monotonicity and nonmonotonicity, differential, and integral properties are studied. Conditions under which the solution of the system of functional equations is a distribution function of the random variable η=Δξ1 ξ2…ξn…Q˜\eta = \Delta _{{\xi _1}\,\xi 2 \ldots {\xi _n} \ldots }^{\tilde Q} with independent Q̃-symbols are founded.

On One Application of Infinite Systems of Functional Equations in Function Theory

Tatra Mountains Mathematical Publications

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

The paper presents the investigation of applications of infinite systems of functional equations for modeling functions with complicated local structure that...

On One Application of Infinite Systems of Functional Equations in Function Theory

Published Online: Nov 15, 2019

Page range: 117 - 144

Received: Jul 27, 2016

DOI: https://doi.org/10.2478/tmmp-2019-0024

Keywords
Function with complicated local structure, systems of functional equations, singular function, nowhere differentiable function, distribution function, nowhere monotone function, ̃-representation, nega-̃-representation, Lebesgue integral

© 2019 Symon Serbenyuk, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

On One Application of Infinite Systems of Functional Equations in Function Theory

Published Online: Nov 15, 2019

Page range: 117 - 144

Received: Jul 27, 2016

DOI: https://doi.org/10.2478/tmmp-2019-0024

KeywordsFunction with complicated local structure, systems of functional equations, singular function, nowhere differentiable function, distribution function, nowhere monotone function, ̃-representation, nega-̃-representation, Lebesgue integral

© 2019 Symon Serbenyuk, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
Function with complicated local structure, systems of functional equations, singular function, nowhere differentiable function, distribution function, nowhere monotone function, ̃-representation, nega-̃-representation, Lebesgue integral