By giving some equivalent definitions of fractional Brauer configuration algebras of type S in some special cases, we construct a fractional Brauer configuration from any monomial algebra. We show that this algebra is isomorphic to the trivial extension of the given monomial algebra. Moreover, we show that there exists a one-to-one correspondence between the isomorphism classes of monomial algebras and the equivalence classes of pairs consisting of a symmetric fractional Brauer configuration algebra of type S with trivial degree function and a given admissible cut over it.
通过给出一些特殊情况下 S 型分数布劳尔配置体的等价定义，我们从任何单项式代数中构造出一个分数布劳尔配置体。我们证明，这个代数与给定单项式代数的微不足道的扩展同构。此外，我们还证明了单项式代数的同构类与由具有三阶度函数的 S 型对称分数布劳尔配置代数和在其上的给定容许割组成的对的等价类之间存在一一对应关系。
{"title":"Trivial extensions of monomial algebras are symmetric fractional Brauer configuration algebras of type S","authors":"Yuming Liu, Bohan Xing","doi":"arxiv-2408.02537","DOIUrl":"https://doi.org/arxiv-2408.02537","url":null,"abstract":"By giving some equivalent definitions of fractional Brauer configuration\u0000algebras of type S in some special cases, we construct a fractional Brauer\u0000configuration from any monomial algebra. We show that this algebra is\u0000isomorphic to the trivial extension of the given monomial algebra. Moreover, we\u0000show that there exists a one-to-one correspondence between the isomorphism\u0000classes of monomial algebras and the equivalence classes of pairs consisting of\u0000a symmetric fractional Brauer configuration algebra of type S with trivial\u0000degree function and a given admissible cut over it.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"60 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141935665","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Recently the notion of post-Hopf algebra was introduced, with the universal enveloping algebra of a post-Lie algebra as the fundamental example. A novel property is that any cocommutative post-Hopf algebra gives rise to a sub-adjacent Hopf algebra with a generalized Grossman-Larson product. By twisting the post-Hopf product, we provide a combinatorial antipode formula for the sub-adjacent Hopf algebra of the universal enveloping algebra of a post-Lie algebra. Relating to such a sub-adjacent Hopf algebra, we also obtain a closed inverse formula for the Oudom-Guin isomorphism in the context of post-Lie algebras. Especially as a byproduct, we derive a cancellation-free antipode formula for the Grossman-Larson Hopf algebra of ordered trees through a concrete tree-grafting expression.
{"title":"On the sub-adjacent Hopf algebra of the universal enveloping algebra of a post-Lie algebra","authors":"Yunnan Li","doi":"arxiv-2408.01345","DOIUrl":"https://doi.org/arxiv-2408.01345","url":null,"abstract":"Recently the notion of post-Hopf algebra was introduced, with the universal\u0000enveloping algebra of a post-Lie algebra as the fundamental example. A novel\u0000property is that any cocommutative post-Hopf algebra gives rise to a\u0000sub-adjacent Hopf algebra with a generalized Grossman-Larson product. By\u0000twisting the post-Hopf product, we provide a combinatorial antipode formula for\u0000the sub-adjacent Hopf algebra of the universal enveloping algebra of a post-Lie\u0000algebra. Relating to such a sub-adjacent Hopf algebra, we also obtain a closed\u0000inverse formula for the Oudom-Guin isomorphism in the context of post-Lie\u0000algebras. Especially as a byproduct, we derive a cancellation-free antipode\u0000formula for the Grossman-Larson Hopf algebra of ordered trees through a\u0000concrete tree-grafting expression.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141935664","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We define the completion of an associative algebra $A$ in a set $M={M_1,dots,M_r}$ of $r$ right $A$-modules in such a way that if $mathfrak asubseteq A$ is an ideal in a commutative ring $A$ the completion $A$ in the (right) module $A/mathfrak a$ is $hat A^Msimeq hat A^{mathfrak a}.$ This works by defining $hat A^M$ as a formal algebra determined up to a computation in a category called GMMP-algebras. From deformation theory we get that the computation results in a formal algebra which is the prorepresenting hull of the noncommutative deformation functor, and this hull is unique up to isomorphism.
{"title":"Countably Generated Matrix Algebras","authors":"Arvid Siqveland","doi":"arxiv-2408.01034","DOIUrl":"https://doi.org/arxiv-2408.01034","url":null,"abstract":"We define the completion of an associative algebra $A$ in a set\u0000$M={M_1,dots,M_r}$ of $r$ right $A$-modules in such a way that if $mathfrak\u0000asubseteq A$ is an ideal in a commutative ring $A$ the completion $A$ in the\u0000(right) module $A/mathfrak a$ is $hat A^Msimeq hat A^{mathfrak a}.$ This\u0000works by defining $hat A^M$ as a formal algebra determined up to a computation\u0000in a category called GMMP-algebras. From deformation theory we get that the\u0000computation results in a formal algebra which is the prorepresenting hull of\u0000the noncommutative deformation functor, and this hull is unique up to\u0000isomorphism.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"41 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141935668","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, a matrix is said to be prime if the row and column of this matrix are both prime numbers. We establish various necessary and sufficient conditions for developing matrices into the sum of tensor products of prime matrices. For example, if the diagonal of a matrix blocked evenly are pairwise commutative, it yields such a decomposition. The computational complexity of multiplication of these algorithms is shown to be $O(n^{5/2})$. In the section 5, a decomposition is proved to hold if and only if every even natural number greater than 2 is the sum of two prime numbers.
{"title":"Factorization of a prime matrix in even blocks","authors":"Haoming Wang","doi":"arxiv-2408.00627","DOIUrl":"https://doi.org/arxiv-2408.00627","url":null,"abstract":"In this paper, a matrix is said to be prime if the row and column of this\u0000matrix are both prime numbers. We establish various necessary and sufficient\u0000conditions for developing matrices into the sum of tensor products of prime\u0000matrices. For example, if the diagonal of a matrix blocked evenly are pairwise\u0000commutative, it yields such a decomposition. The computational complexity of\u0000multiplication of these algorithms is shown to be $O(n^{5/2})$. In the section\u00005, a decomposition is proved to hold if and only if every even natural number\u0000greater than 2 is the sum of two prime numbers.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141885193","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A ring $S(X,mathcal{A})$ of real valued $mathcal{A}$-measurable functions defined over a measurable space $(X,mathcal{A})$ is called a $chi$-ring if for each $Ein mathcal{A} $, the characteristic function $chi_{E}in S(X,mathcal{A})$. The set $mathcal{U}_X$ of all $mathcal{A}$-ultrafilters on $X$ with the Stone topology $tau$ is seen to be homeomorphic to an appropriate quotient space of the set $mathcal{M}_X$ of all maximal ideals in $S(X,mathcal{A})$ equipped with the hull-kernel topology $tau_S$. It is realized that $(mathcal{U}_X,tau)$ is homeomorphic to $(mathcal{M}_S,tau_S)$ if and only if $S(X,mathcal{A})$ is a Gelfand ring. It is further observed that $S(X,mathcal{A})$ is a Von-Neumann regular ring if and only if each ideal in this ring is a $mathcal{Z}_S$-ideal and $S(X,mathcal{A})$ is Gelfand when and only when every maximal ideal in it is a $mathcal{Z}_S$-ideal. A pair of topologies $u_mu$-topology and $m_mu$-topology, are introduced on the set $S(X,mathcal{A})$ and a few properties are studied.
{"title":"Structure spaces and allied problems on a class of rings of measurable functions","authors":"Soumajit Dey, Sudip Kumar Acharyya, Dhananjoy Mandal","doi":"arxiv-2408.00505","DOIUrl":"https://doi.org/arxiv-2408.00505","url":null,"abstract":"A ring $S(X,mathcal{A})$ of real valued $mathcal{A}$-measurable functions\u0000defined over a measurable space $(X,mathcal{A})$ is called a $chi$-ring if\u0000for each $Ein mathcal{A} $, the characteristic function $chi_{E}in\u0000S(X,mathcal{A})$. The set $mathcal{U}_X$ of all $mathcal{A}$-ultrafilters on\u0000$X$ with the Stone topology $tau$ is seen to be homeomorphic to an appropriate\u0000quotient space of the set $mathcal{M}_X$ of all maximal ideals in\u0000$S(X,mathcal{A})$ equipped with the hull-kernel topology $tau_S$. It is\u0000realized that $(mathcal{U}_X,tau)$ is homeomorphic to\u0000$(mathcal{M}_S,tau_S)$ if and only if $S(X,mathcal{A})$ is a Gelfand ring.\u0000It is further observed that $S(X,mathcal{A})$ is a Von-Neumann regular ring if\u0000and only if each ideal in this ring is a $mathcal{Z}_S$-ideal and\u0000$S(X,mathcal{A})$ is Gelfand when and only when every maximal ideal in it is a\u0000$mathcal{Z}_S$-ideal. A pair of topologies $u_mu$-topology and\u0000$m_mu$-topology, are introduced on the set $S(X,mathcal{A})$ and a few\u0000properties are studied.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141885099","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study simplicity of $C^*$-algebras arising from self-similar groups of $mathbb{Z}_2$-multispinal type, a generalization of the Grigorchuk case whose simplicity was first proved by L. Clark, R. Exel, E. Pardo, C. Starling, and A. Sims in 2019, and we prove results generalizing theirs. Our first main result is a sufficient condition for simplicity of the Steinberg algebra satisfying conditions modeled on the behavior of the groupoid associated to the first Grigorchuk group. This closely resembles conditions found by B. Steinberg and N. Szak'acs. As a key ingredient we identify an infinite family of $2-(2q-1,q-1,q/2-1)$-designs, where $q$ is a positive even integer. We then deduce the simplicity of the associated $C^*$-algebra, which is our second main result. Results of similar type were considered by B. Steinberg and N. Szak'acs in 2021, and later by K. Yoshida, but their methods did not follow the original methods of the five authors.
我们研究由$mathbb{Z}_2$-multispinal类型的自相似群产生的$C^*$-代数的简单性，这是格里高丘克情况的广义化，其简单性由L. Clark、R. Exel、E. Pardo、C. Starling和A.Sims于2019年首次证明，我们证明了他们的结果的广义化。我们的第一个主要结果是斯坦伯格代数简单性的充分条件，它满足以与第一个格里高丘克群相关联的类群的行为为模型的条件。这与 B. Steinberg 和 N. Szak'acs 发现的条件非常相似。Szak'acs 发现的条件。作为关键要素，我们确定了$2-(2q-1,q-1,q/2-1)$设计的无穷系列，其中$q$为正偶数整数。然后，我们推导出相关$C^*$代数的简单性，这是我们的第二个主要结果。B. Steinberg 和 N.Szak'acs 在 2021 年以及后来的 K. Yoshida 也考虑过类似的结果，但他们的方法并没有沿用五位作者最初的方法。
{"title":"Simplicity of $*$-algebras of non-Hausdorff $mathbb{Z}_2$-multispinal groupoids","authors":"C. Farsi, N. S. Larsen, J. Packer, N. Thiem","doi":"arxiv-2408.00442","DOIUrl":"https://doi.org/arxiv-2408.00442","url":null,"abstract":"We study simplicity of $C^*$-algebras arising from self-similar groups of\u0000$mathbb{Z}_2$-multispinal type, a generalization of the Grigorchuk case whose\u0000simplicity was first proved by L. Clark, R. Exel, E. Pardo, C. Starling, and A.\u0000Sims in 2019, and we prove results generalizing theirs. Our first main result\u0000is a sufficient condition for simplicity of the Steinberg algebra satisfying\u0000conditions modeled on the behavior of the groupoid associated to the first\u0000Grigorchuk group. This closely resembles conditions found by B. Steinberg and\u0000N. Szak'acs. As a key ingredient we identify an infinite family of\u0000$2-(2q-1,q-1,q/2-1)$-designs, where $q$ is a positive even integer. We then\u0000deduce the simplicity of the associated $C^*$-algebra, which is our second main\u0000result. Results of similar type were considered by B. Steinberg and N.\u0000Szak'acs in 2021, and later by K. Yoshida, but their methods did not follow\u0000the original methods of the five authors.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"99 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141885100","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
James East, Victoria Gould, Craig Miller, Thomas Quinn-Gregson
The left and right diameters of a monoid are topological invariants defined in terms of suprema of lengths of derivation sequences with respect to finite generating sets for the universal left or right congruences. We compute these parameters for the endomorphism monoid $End(C)$ of a chain $C$. Specifically, if $C$ is infinite then the left diameter of $End(C)$ is 2, while the right diameter is either 2 or 3, with the latter equal to 2 precisely when $C$ is a quotient of $C{setminus}{z}$ for some endpoint $z$. If $C$ is finite then so is $End(C),$ in which case the left and right diameters are 1 (if $C$ is non-trivial) or 0.
{"title":"Diameters of endomorphism monoids of chains","authors":"James East, Victoria Gould, Craig Miller, Thomas Quinn-Gregson","doi":"arxiv-2408.00416","DOIUrl":"https://doi.org/arxiv-2408.00416","url":null,"abstract":"The left and right diameters of a monoid are topological invariants defined\u0000in terms of suprema of lengths of derivation sequences with respect to finite\u0000generating sets for the universal left or right congruences. We compute these\u0000parameters for the endomorphism monoid $End(C)$ of a chain $C$. Specifically,\u0000if $C$ is infinite then the left diameter of $End(C)$ is 2, while the right\u0000diameter is either 2 or 3, with the latter equal to 2 precisely when $C$ is a\u0000quotient of $C{setminus}{z}$ for some endpoint $z$. If $C$ is finite then so\u0000is $End(C),$ in which case the left and right diameters are 1 (if $C$ is\u0000non-trivial) or 0.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141885178","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
An extension $Bsubset A$ of finite dimensional algebras is bounded if the $B$-$B$-bimodule $A/B$ is $B$-tensor nilpotent, its projective dimension is finite and $mathrm{Tor}_i^B(A/B, (A/B)^{otimes_B j})=0$ for all $i, jgeq 1$. We show that for a bounded extension $Bsubset A$, the algebras $A$ and $B$ are singularly equivalent of Morita type with level. Additively, under some conditions, their stable categories of Gorenstein projective modules and Gorenstein defect categories are equivalent, respectively. Some homological conjectures are also investigated for bounded extensions, including Auslander-Reiten conjecture, finististic dimension conjecture, Fg condition, Han's conjecture, and Keller's conjecture. Applications to trivial extensions and triangular matrix algebras are given. In course of proofs, we give some handy criteria for a functor between module categories induces triangle functors between stable categories of Gorenstein projective modules and Gorenstein defect categories, which generalise some known criteria and hence, might be of independent interest.
{"title":"Categorical properties and homological conjectures for bounded extensions of algebras","authors":"Yongyun Qin, Xiaoxiao Xu, Jinbi Zhang, Guodong Zhou","doi":"arxiv-2407.21480","DOIUrl":"https://doi.org/arxiv-2407.21480","url":null,"abstract":"An extension $Bsubset A$ of finite dimensional algebras is bounded if the\u0000$B$-$B$-bimodule $A/B$ is $B$-tensor nilpotent, its projective dimension is\u0000finite and $mathrm{Tor}_i^B(A/B, (A/B)^{otimes_B j})=0$ for all $i, jgeq 1$.\u0000We show that for a bounded extension $Bsubset A$, the algebras $A$ and $B$ are\u0000singularly equivalent of Morita type with level. Additively, under some\u0000conditions, their stable categories of Gorenstein projective modules and\u0000Gorenstein defect categories are equivalent, respectively. Some homological\u0000conjectures are also investigated for bounded extensions, including\u0000Auslander-Reiten conjecture, finististic dimension conjecture, Fg condition,\u0000Han's conjecture, and Keller's conjecture. Applications to trivial extensions\u0000and triangular matrix algebras are given. In course of proofs, we give some handy criteria for a functor between module\u0000categories induces triangle functors between stable categories of Gorenstein\u0000projective modules and Gorenstein defect categories, which generalise some\u0000known criteria and hence, might be of independent interest.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"78 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141871572","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the algebra of upper triangular matrices endowed with a group grading and a homogeneous involution over an infinite field. We compute the asymptotic behaviour of its (graded) star-codimension sequence. It turns out that the asymptotic growth of the sequence is independent of the grading and the involution under consideration, depending solely on the size of the matrix algebra. This independence of the group grading also applies to the graded codimension sequence of the associative algebra of upper triangular matrices.
{"title":"On the asymptotic behaviour of the graded-star-codimension sequence of upper triangular matrices","authors":"Diogo Diniz, Felipe Yukihide Yasumura","doi":"arxiv-2408.00087","DOIUrl":"https://doi.org/arxiv-2408.00087","url":null,"abstract":"We study the algebra of upper triangular matrices endowed with a group\u0000grading and a homogeneous involution over an infinite field. We compute the\u0000asymptotic behaviour of its (graded) star-codimension sequence. It turns out\u0000that the asymptotic growth of the sequence is independent of the grading and\u0000the involution under consideration, depending solely on the size of the matrix\u0000algebra. This independence of the group grading also applies to the graded\u0000codimension sequence of the associative algebra of upper triangular matrices.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"44 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141885027","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We define Sobolev spaces $H^{mathfrak{s}}(K_q)$ over a local field $K_q$ of finite characteristic $p>0$, where $q=p^c$ for a prime $p$ and $cin mathbb{N}$. This paper introduces novel fractal functions, such as the Weierstrass type and 3-adic Cantor type, as intriguing examples within these spaces and a few others. Employing prime elements, we develop a Multi-Resolution Analysis (MRA) and examine wavelet expansions, focusing on the orthogonality of both basic and fractal wavelet packets at various scales. We utilize convolution theory to construct Haar wavelet packets and demonstrate the orthogonality of all discussed wavelet packets within $H^{mathfrak{s}}(K_q)$, enhancing the analytical capabilities of these Sobolev spaces.
{"title":"Constructing Multiresolution Analysis via Wavelet Packets on Sobolev Space in Local Fields","authors":"Manish Kumar","doi":"arxiv-2408.00028","DOIUrl":"https://doi.org/arxiv-2408.00028","url":null,"abstract":"We define Sobolev spaces $H^{mathfrak{s}}(K_q)$ over a local field $K_q$ of\u0000finite characteristic $p>0$, where $q=p^c$ for a prime $p$ and $cin\u0000mathbb{N}$. This paper introduces novel fractal functions, such as the\u0000Weierstrass type and 3-adic Cantor type, as intriguing examples within these\u0000spaces and a few others. Employing prime elements, we develop a\u0000Multi-Resolution Analysis (MRA) and examine wavelet expansions, focusing on the\u0000orthogonality of both basic and fractal wavelet packets at various scales. We\u0000utilize convolution theory to construct Haar wavelet packets and demonstrate\u0000the orthogonality of all discussed wavelet packets within\u0000$H^{mathfrak{s}}(K_q)$, enhancing the analytical capabilities of these Sobolev\u0000spaces.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141885028","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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