Approximation by Quantum Meyer-König-Zeller Fractal Functions
In this paper, a novel class of quantum fractal functions is introduced based on the Meyer-König-Zeller operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>M</mi><mrow><mi>...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-11-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/6/12/704 |
| _version_ | 1827639200894156800 |
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| author | Deependra Kumar Arya K. B. Chand Peter R. Massopust |
| author_facet | Deependra Kumar Arya K. B. Chand Peter R. Massopust |
| author_sort | Deependra Kumar |
| collection | DOAJ |
| description | In this paper, a novel class of quantum fractal functions is introduced based on the Meyer-König-Zeller operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>M</mi><mrow><mi>q</mi><mo>,</mo><mi>n</mi></mrow></msub></semantics></math></inline-formula>. These quantum Meyer-König-Zeller (MKZ) fractal functions employ <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>M</mi><mrow><mi>q</mi><mo>,</mo><mi>n</mi></mrow></msub><mi>f</mi></mrow></semantics></math></inline-formula> as the base function in the iterated function system for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>-fractal functions. For <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>∈</mo><mi>C</mi><mo>(</mo><mi>I</mi><mo>)</mo></mrow></semantics></math></inline-formula>, <i>I</i> closed interval in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">R</mi></semantics></math></inline-formula>, it is shown that a sequence of quantum MKZ fractal functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mrow><mo stretchy="false">{</mo><msubsup><mi>f</mi><mi>n</mi><mrow><mo>(</mo><msub><mi>q</mi><mi>n</mi></msub><mo>,</mo><mi>α</mi><mo>)</mo></mrow></msubsup><mo stretchy="false">}</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mo>∞</mo></msubsup></semantics></math></inline-formula> exists which converges uniformly to <i>f</i> without altering the scaling function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>. The shape of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>f</mi><mi>n</mi><mrow><mo>(</mo><msub><mi>q</mi><mi>n</mi></msub><mo>,</mo><mi>α</mi><mo>)</mo></mrow></msubsup></semantics></math></inline-formula> depends on <i>q</i> as well as the other iterated function system parameters. For <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo>∈</mo><mi>C</mi><mo>(</mo><mi>I</mi><mo>)</mo></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>≥</mo><mi>g</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>, we show that a sequence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mrow><mo stretchy="false">{</mo><msubsup><mi>f</mi><mi>n</mi><mrow><mo>(</mo><msub><mi>q</mi><mi>n</mi></msub><mo>,</mo><mi>α</mi><mo>)</mo></mrow></msubsup><mo stretchy="false">}</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mo>∞</mo></msubsup></semantics></math></inline-formula> exists with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>f</mi><mi>n</mi><mrow><mo>(</mo><msub><mi>q</mi><mi>n</mi></msub><mo>,</mo><mi>α</mi><mo>)</mo></mrow></msubsup><mo>≥</mo><mi>g</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> converging to <i>f</i>. Quantum MKZ fractal versions of some classical Müntz theorems are also presented. For <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>, the box dimension and some approximation-theoretic results of MKZ <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>-fractal functions are investigated in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><mo>(</mo><mi>I</mi><mo>)</mo></mrow></semantics></math></inline-formula>. Finally, MKZ <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>-fractal functions are studied in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mi>p</mi></msup></semantics></math></inline-formula> spaces with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula>. |
| first_indexed | 2024-03-09T16:35:15Z |
| format | Article |
| id | doaj.art-758c42cc6b754f439805cbd8ccd9dcbe |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-09T16:35:15Z |
| publishDate | 2022-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-758c42cc6b754f439805cbd8ccd9dcbe2023-11-24T14:57:14ZengMDPI AGFractal and Fractional2504-31102022-11-0161270410.3390/fractalfract6120704Approximation by Quantum Meyer-König-Zeller Fractal FunctionsDeependra Kumar0Arya K. B. Chand1Peter R. Massopust2Department of Mathematics, Indian Institute of Technology Madras, Chennai 600036, IndiaDepartment of Mathematics, Indian Institute of Technology Madras, Chennai 600036, IndiaCentre of Mathematics, Technical University of Munich (TUM), 85748 Garching b. München, GermanyIn this paper, a novel class of quantum fractal functions is introduced based on the Meyer-König-Zeller operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>M</mi><mrow><mi>q</mi><mo>,</mo><mi>n</mi></mrow></msub></semantics></math></inline-formula>. These quantum Meyer-König-Zeller (MKZ) fractal functions employ <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>M</mi><mrow><mi>q</mi><mo>,</mo><mi>n</mi></mrow></msub><mi>f</mi></mrow></semantics></math></inline-formula> as the base function in the iterated function system for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>-fractal functions. For <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>∈</mo><mi>C</mi><mo>(</mo><mi>I</mi><mo>)</mo></mrow></semantics></math></inline-formula>, <i>I</i> closed interval in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">R</mi></semantics></math></inline-formula>, it is shown that a sequence of quantum MKZ fractal functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mrow><mo stretchy="false">{</mo><msubsup><mi>f</mi><mi>n</mi><mrow><mo>(</mo><msub><mi>q</mi><mi>n</mi></msub><mo>,</mo><mi>α</mi><mo>)</mo></mrow></msubsup><mo stretchy="false">}</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mo>∞</mo></msubsup></semantics></math></inline-formula> exists which converges uniformly to <i>f</i> without altering the scaling function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>. The shape of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>f</mi><mi>n</mi><mrow><mo>(</mo><msub><mi>q</mi><mi>n</mi></msub><mo>,</mo><mi>α</mi><mo>)</mo></mrow></msubsup></semantics></math></inline-formula> depends on <i>q</i> as well as the other iterated function system parameters. For <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo>∈</mo><mi>C</mi><mo>(</mo><mi>I</mi><mo>)</mo></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>≥</mo><mi>g</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>, we show that a sequence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mrow><mo stretchy="false">{</mo><msubsup><mi>f</mi><mi>n</mi><mrow><mo>(</mo><msub><mi>q</mi><mi>n</mi></msub><mo>,</mo><mi>α</mi><mo>)</mo></mrow></msubsup><mo stretchy="false">}</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mo>∞</mo></msubsup></semantics></math></inline-formula> exists with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>f</mi><mi>n</mi><mrow><mo>(</mo><msub><mi>q</mi><mi>n</mi></msub><mo>,</mo><mi>α</mi><mo>)</mo></mrow></msubsup><mo>≥</mo><mi>g</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> converging to <i>f</i>. Quantum MKZ fractal versions of some classical Müntz theorems are also presented. For <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>, the box dimension and some approximation-theoretic results of MKZ <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>-fractal functions are investigated in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><mo>(</mo><mi>I</mi><mo>)</mo></mrow></semantics></math></inline-formula>. Finally, MKZ <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>-fractal functions are studied in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mi>p</mi></msup></semantics></math></inline-formula> spaces with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula>.https://www.mdpi.com/2504-3110/6/12/704fractal interpolation functionquantum meyer-könig-zeller operatorsmooth quantum fractal functionsconstrained approximationmüntz polynomials |
| spellingShingle | Deependra Kumar Arya K. B. Chand Peter R. Massopust Approximation by Quantum Meyer-König-Zeller Fractal Functions Fractal and Fractional fractal interpolation function quantum meyer-könig-zeller operator smooth quantum fractal functions constrained approximation müntz polynomials |
| title | Approximation by Quantum Meyer-König-Zeller Fractal Functions |
| title_full | Approximation by Quantum Meyer-König-Zeller Fractal Functions |
| title_fullStr | Approximation by Quantum Meyer-König-Zeller Fractal Functions |
| title_full_unstemmed | Approximation by Quantum Meyer-König-Zeller Fractal Functions |
| title_short | Approximation by Quantum Meyer-König-Zeller Fractal Functions |
| title_sort | approximation by quantum meyer konig zeller fractal functions |
| topic | fractal interpolation function quantum meyer-könig-zeller operator smooth quantum fractal functions constrained approximation müntz polynomials |
| url | https://www.mdpi.com/2504-3110/6/12/704 |
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