In this paper, a generalised integral called the Laplace integral is defined on unbounded intervals, and some of its properties, including necessary and sufficient condition for differentiating under the integral sign, are discussed. It is also shown that this integral is more general than the Henstock-Kurzweil integral. Finally, the Fourier transform is defined using the Laplace integral, and its well-known properties are established.
{"title":"On the generalisation of Henstock-Kurzweil Fourier transform","authors":"S. Mahanta, S. Ray","doi":"10.7153/jca-2022-20-09","DOIUrl":"https://doi.org/10.7153/jca-2022-20-09","url":null,"abstract":"In this paper, a generalised integral called the Laplace integral is defined on unbounded intervals, and some of its properties, including necessary and sufficient condition for differentiating under the integral sign, are discussed. It is also shown that this integral is more general than the Henstock-Kurzweil integral. Finally, the Fourier transform is defined using the Laplace integral, and its well-known properties are established.","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42074451","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}
Abstract. If f P LpRq it is proved that limSÑ8‖f ́ f ̊ DS‖ “ 0, where DSpxq “ sinpSxq{pπxq is the Dirichlet kernel and ‖f‖ “ supαăβ | şβ α fpxq dx| is the Alexiewicz norm. This gives a symmetric inversion of the Fourier transform on the real line. An asymmetric inversion is also proved. The results also hold for a measure given by dF where F is a continuous function of bounded variation. Such measures need not be absolutely continuous with respect to Lebesgue measure. An example shows there is f P LpRq such that limSÑ8‖f ́ f ̊ DS‖1‰ 0.
摘要如果f P LpRq，则证明了limSñ8‖f́fõDS‖“0，其中DSpxq”sinpSxq｛Pπxq是Dirichlet核“supαăβ|şβαfpxq dx |是Alexewicz范数。这给出了实线上傅立叶变换的对称反演。还证明了非对称反演。结果也适用于dF给出的测度，其中F是有界变差的连续函数。这样的测度不必相对于Lebesgue测度是绝对连续的。一个例子表明其极限为Sñ8½f́f́DS½1‰0。
{"title":"Fourier transform inversion in the Alexiewicz norm","authors":"E. Talvila","doi":"10.7153/jca-2022-19-07","DOIUrl":"https://doi.org/10.7153/jca-2022-19-07","url":null,"abstract":"Abstract. If f P LpRq it is proved that limSÑ8‖f ́ f ̊ DS‖ “ 0, where DSpxq “ sinpSxq{pπxq is the Dirichlet kernel and ‖f‖ “ supαăβ | şβ α fpxq dx| is the Alexiewicz norm. This gives a symmetric inversion of the Fourier transform on the real line. An asymmetric inversion is also proved. The results also hold for a measure given by dF where F is a continuous function of bounded variation. Such measures need not be absolutely continuous with respect to Lebesgue measure. An example shows there is f P LpRq such that limSÑ8‖f ́ f ̊ DS‖1‰ 0.","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42447787","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, we develop a local limit theorem for the Student distribution. We use it to improve the normal approximation of the Student survival function given in Shafiei & Saberali (2015) and to derive asymptotic bounds for the corresponding maximal errors at four levels of approximation. As a corollary, approximations for the percentage points (or quantiles) of the Student distribution are obtained in terms of the percentage points of the standard normal distribution.
{"title":"Refined normal approximations for the Student distribution","authors":"Frédéric Ouimet","doi":"10.7153/jca-2022-20-03","DOIUrl":"https://doi.org/10.7153/jca-2022-20-03","url":null,"abstract":"In this paper, we develop a local limit theorem for the Student distribution. We use it to improve the normal approximation of the Student survival function given in Shafiei & Saberali (2015) and to derive asymptotic bounds for the corresponding maximal errors at four levels of approximation. As a corollary, approximations for the percentage points (or quantiles) of the Student distribution are obtained in terms of the percentage points of the standard normal distribution.","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44862341","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 we have studied the growth of meromorphic solutions of higher order homogeneous and non-homogeneous linear difference equations with entire and meromorphic coefficients. We have extended and improved some results of Zhou and Zheng (2017), Belaidi and Benkarouba (2019) by using (p,q)-order and (p,q)-type.
本文研究了具有全系数和亚纯系数的高阶齐次和非齐次线性差分方程亚纯解的增长问题。我们使用(p,q)阶和(p,q)型扩展和改进了Zhou and Zheng(2017)、Belaidi and Benkarouba(2019)的一些结果。
{"title":"On the growth of meromorphic solutions of homogeneous and non-homogeneous linear difference equations in terms of (p,q)-order","authors":"C. Ghosh, Subhadip Khan, A. Bandyopadhyay","doi":"10.7153/jca-2023-21-07","DOIUrl":"https://doi.org/10.7153/jca-2023-21-07","url":null,"abstract":"In this paper we have studied the growth of meromorphic solutions of higher order homogeneous and non-homogeneous linear difference equations with entire and meromorphic coefficients. We have extended and improved some results of Zhou and Zheng (2017), Belaidi and Benkarouba (2019) by using (p,q)-order and (p,q)-type.","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43837315","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}
{"title":"Uniqueness of non-homogeneous differential polynomials of meromorphic functions sharing a small function","authors":"D. C. Pramanik, Jayanta Roy","doi":"10.7153/jca-2022-19-06","DOIUrl":"https://doi.org/10.7153/jca-2022-19-06","url":null,"abstract":"","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71136488","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}
{"title":"Explicit expressions for some linear Euler-type sums containing harmonic and skew-harmonic numbers","authors":"Ting Zhu","doi":"10.7153/jca-2022-20-07","DOIUrl":"https://doi.org/10.7153/jca-2022-20-07","url":null,"abstract":"","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71136699","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}
{"title":"Coefficient problems of a class of q-starlike functions associated with q-analogue of Al-Oboudi-Al-Qahtani integral operator and nephroid domain","authors":"A. Lasode, T. Opoola","doi":"10.7153/jca-2022-20-04","DOIUrl":"https://doi.org/10.7153/jca-2022-20-04","url":null,"abstract":"","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71136427","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}
{"title":"Path connectedness of Volterra type integral operators on Bergman and Dirichlet type spaces","authors":"Cui Wang","doi":"10.7153/jca-2022-20-05","DOIUrl":"https://doi.org/10.7153/jca-2022-20-05","url":null,"abstract":"","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71136531","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}
{"title":"On the location of zeros of quasi-orthogonal polynomials with applications to some real self-reciprocal polynomials","authors":"V. Botta, Mijael Hancco Suni","doi":"10.7153/jca-2022-19-08","DOIUrl":"https://doi.org/10.7153/jca-2022-19-08","url":null,"abstract":"","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71136066","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}
{"title":"Location of zeros of Lacunary-type polynomials in annular regions","authors":"I. A. Wani, Mohammad Hedayetullah Mir, I. Nazir","doi":"10.7153/jca-2022-20-08","DOIUrl":"https://doi.org/10.7153/jca-2022-20-08","url":null,"abstract":"","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71136707","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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