Tensorial Maclaurin Approximation Bounds and Structural Properties for Mixed-Norm Orlicz–Zygmund Spaces
This article explores two distinct function spaces: Hilbert spaces and mixed-Orlicz–Zygmund spaces with variable exponents. We first examine the relational properties of Hilbert spaces in a tensorial framework, utilizing self-adjoint operators to derive key results. Additionally, we extend a Maclaur...
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2025-03-01
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| author | Waqar Afzal Mujahid Abbas Mutum Zico Meetei Saïd Bourazza |
| author_facet | Waqar Afzal Mujahid Abbas Mutum Zico Meetei Saïd Bourazza |
| author_sort | Waqar Afzal |
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| description | This article explores two distinct function spaces: Hilbert spaces and mixed-Orlicz–Zygmund spaces with variable exponents. We first examine the relational properties of Hilbert spaces in a tensorial framework, utilizing self-adjoint operators to derive key results. Additionally, we extend a Maclaurin-type inequality to the tensorial setting using generalized convex mappings and establish various upper bounds. A non-trivial example involving exponential functions is also presented. Next, we introduce a new function space, the mixed-Orlicz–Zygmund space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="(" close=")"><msub><mo>ℓ</mo><mrow><mi mathvariant="monospace">q</mi><mo>(</mo><mo>·</mo><mo>)</mo></mrow></msub><msup><mo form="prefix">log</mo><mo>β</mo></msup><mfenced separators="" open="(" close=")"><msup><mi mathvariant="monospace">L</mi><mrow><mi mathvariant="monospace">p</mi><mo>(</mo><mo>·</mo><mo>)</mo></mrow></msup></mfenced></mfenced></semantics></math></inline-formula>, which unifies Orlicz–Zygmund spaces of integrability and sequence spaces. We investigate its fundamental properties including separability, compactness, and completeness, demonstrating its significance. This space generalizes the existing structures, reducing to mixed-norm Lebesgue spaces when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>β</mo><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> and to classical Lebesgue spaces when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="monospace">q</mi><mo>=</mo><mo>∞</mo><mo>,</mo><mo>β</mo><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>. Given the limited research on such spaces, our findings contribute valuable insights to the functional analysis. |
| format | Article |
| id | doaj-art-9a40d8d8553b49788dfa731559ebf007 |
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| issn | 2227-7390 |
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| publishDate | 2025-03-01 |
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| spelling | doaj-art-9a40d8d8553b49788dfa731559ebf0072025-08-20T02:42:22ZengMDPI AGMathematics2227-73902025-03-0113691710.3390/math13060917Tensorial Maclaurin Approximation Bounds and Structural Properties for Mixed-Norm Orlicz–Zygmund SpacesWaqar Afzal0Mujahid Abbas1Mutum Zico Meetei2Saïd Bourazza3Abdus Salam School of Mathematical Sciences, Government College University, 68-B, New Muslim Town, Lahore 54600, PakistanAbdus Salam School of Mathematical Sciences, Government College University, 68-B, New Muslim Town, Lahore 54600, PakistanDepartment of Mathematics, College of Science, Jazan University, P.O. Box 114, Jazan 45142, Saudi ArabiaDepartment of Mathematics, College of Science, Jazan University, P.O. Box 114, Jazan 45142, Saudi ArabiaThis article explores two distinct function spaces: Hilbert spaces and mixed-Orlicz–Zygmund spaces with variable exponents. We first examine the relational properties of Hilbert spaces in a tensorial framework, utilizing self-adjoint operators to derive key results. Additionally, we extend a Maclaurin-type inequality to the tensorial setting using generalized convex mappings and establish various upper bounds. A non-trivial example involving exponential functions is also presented. Next, we introduce a new function space, the mixed-Orlicz–Zygmund space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="(" close=")"><msub><mo>ℓ</mo><mrow><mi mathvariant="monospace">q</mi><mo>(</mo><mo>·</mo><mo>)</mo></mrow></msub><msup><mo form="prefix">log</mo><mo>β</mo></msup><mfenced separators="" open="(" close=")"><msup><mi mathvariant="monospace">L</mi><mrow><mi mathvariant="monospace">p</mi><mo>(</mo><mo>·</mo><mo>)</mo></mrow></msup></mfenced></mfenced></semantics></math></inline-formula>, which unifies Orlicz–Zygmund spaces of integrability and sequence spaces. We investigate its fundamental properties including separability, compactness, and completeness, demonstrating its significance. This space generalizes the existing structures, reducing to mixed-norm Lebesgue spaces when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>β</mo><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> and to classical Lebesgue spaces when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="monospace">q</mi><mo>=</mo><mo>∞</mo><mo>,</mo><mo>β</mo><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>. Given the limited research on such spaces, our findings contribute valuable insights to the functional analysis.https://www.mdpi.com/2227-7390/13/6/917self-adjoint operatorsHilbert spacesoperator convexitymixed-variable exponent spacesZygmund space |
| spellingShingle | Waqar Afzal Mujahid Abbas Mutum Zico Meetei Saïd Bourazza Tensorial Maclaurin Approximation Bounds and Structural Properties for Mixed-Norm Orlicz–Zygmund Spaces Mathematics self-adjoint operators Hilbert spaces operator convexity mixed-variable exponent spaces Zygmund space |
| title | Tensorial Maclaurin Approximation Bounds and Structural Properties for Mixed-Norm Orlicz–Zygmund Spaces |
| title_full | Tensorial Maclaurin Approximation Bounds and Structural Properties for Mixed-Norm Orlicz–Zygmund Spaces |
| title_fullStr | Tensorial Maclaurin Approximation Bounds and Structural Properties for Mixed-Norm Orlicz–Zygmund Spaces |
| title_full_unstemmed | Tensorial Maclaurin Approximation Bounds and Structural Properties for Mixed-Norm Orlicz–Zygmund Spaces |
| title_short | Tensorial Maclaurin Approximation Bounds and Structural Properties for Mixed-Norm Orlicz–Zygmund Spaces |
| title_sort | tensorial maclaurin approximation bounds and structural properties for mixed norm orlicz zygmund spaces |
| topic | self-adjoint operators Hilbert spaces operator convexity mixed-variable exponent spaces Zygmund space |
| url | https://www.mdpi.com/2227-7390/13/6/917 |
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