{"title":"在沃尔什傅里叶变换的数量级上","authors":"B. L. Ghodadra, V. Fülöp","doi":"10.21136/MB.2019.0075-18","DOIUrl":null,"url":null,"abstract":"For a Lebesgue integrable complex-valued function f defined on R := [0,∞) let f̂ be its Walsh-Fourier transform. The Riemann-Lebesgue lemma says that f̂(y)→ 0 as y → ∞. But in general, there is no definite rate at which the Walsh-Fourier transform tends to zero. In fact, the Walsh-Fourier transform of an integrable function can tend to zero as slowly as we wish. Therefore, it is interesting to know for functions of which subclasses of L(R) there is a definite rate at which the Walsh-Fourier transform tends to zero. We determine this rate for functions of bounded variation on R. We also determine such rate of Walsh-Fourier transform for functions of bounded variation in the sense of Vitali defined on (R) , N ∈ N.","PeriodicalId":45392,"journal":{"name":"Mathematica Bohemica","volume":null,"pages":null},"PeriodicalIF":0.3000,"publicationDate":"2020-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"On the order of magnitude of Walsh-Fourier transform\",\"authors\":\"B. L. Ghodadra, V. Fülöp\",\"doi\":\"10.21136/MB.2019.0075-18\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"For a Lebesgue integrable complex-valued function f defined on R := [0,∞) let f̂ be its Walsh-Fourier transform. The Riemann-Lebesgue lemma says that f̂(y)→ 0 as y → ∞. But in general, there is no definite rate at which the Walsh-Fourier transform tends to zero. In fact, the Walsh-Fourier transform of an integrable function can tend to zero as slowly as we wish. Therefore, it is interesting to know for functions of which subclasses of L(R) there is a definite rate at which the Walsh-Fourier transform tends to zero. We determine this rate for functions of bounded variation on R. We also determine such rate of Walsh-Fourier transform for functions of bounded variation in the sense of Vitali defined on (R) , N ∈ N.\",\"PeriodicalId\":45392,\"journal\":{\"name\":\"Mathematica Bohemica\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.3000,\"publicationDate\":\"2020-10-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematica Bohemica\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.21136/MB.2019.0075-18\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematica Bohemica","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.21136/MB.2019.0075-18","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
On the order of magnitude of Walsh-Fourier transform
For a Lebesgue integrable complex-valued function f defined on R := [0,∞) let f̂ be its Walsh-Fourier transform. The Riemann-Lebesgue lemma says that f̂(y)→ 0 as y → ∞. But in general, there is no definite rate at which the Walsh-Fourier transform tends to zero. In fact, the Walsh-Fourier transform of an integrable function can tend to zero as slowly as we wish. Therefore, it is interesting to know for functions of which subclasses of L(R) there is a definite rate at which the Walsh-Fourier transform tends to zero. We determine this rate for functions of bounded variation on R. We also determine such rate of Walsh-Fourier transform for functions of bounded variation in the sense of Vitali defined on (R) , N ∈ N.