The problem of solving pseudodifferential equations on spheres by collocation with zonal kernels is considered and bounds for the approximation error are established. The bounds are given in terms of the maximum separation distance of the collocation points, the order of the pseudodifferential operator, and the smoothness of the employed zonal kernel. A by-product of the results is an improvement on the previously known convergence order estimates for Lagrange interpolation.
{"title":"Error Bounds for Solving Pseudodifferential Equations on Spheres by Collocation with Zonal Kernels","authors":"Tanya M. Morton, M. Neamtu","doi":"10.1006/jath.2001.3642","DOIUrl":"https://doi.org/10.1006/jath.2001.3642","url":null,"abstract":"The problem of solving pseudodifferential equations on spheres by collocation with zonal kernels is considered and bounds for the approximation error are established. The bounds are given in terms of the maximum separation distance of the collocation points, the order of the pseudodifferential operator, and the smoothness of the employed zonal kernel. A by-product of the results is an improvement on the previously known convergence order estimates for Lagrange interpolation.","PeriodicalId":202056,"journal":{"name":"J. Approx. Theory","volume":"113 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2002-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124365332","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 generalize to several variables both the upper and the lower Gelfond bounds for the least uniform deviation from zero of the quasipolynomials (or Muntz-Legendre polynomials) on the segment [0, 1]. Orthonormal quasipolynomials are also considered.
{"title":"On Multivariate Quasipolynomials of the Minimal Deviation from Zero","authors":"F. Luquin","doi":"10.1006/jath.2001.3582","DOIUrl":"https://doi.org/10.1006/jath.2001.3582","url":null,"abstract":"We generalize to several variables both the upper and the lower Gelfond bounds for the least uniform deviation from zero of the quasipolynomials (or Muntz-Legendre polynomials) on the segment [0, 1]. Orthonormal quasipolynomials are also considered.","PeriodicalId":202056,"journal":{"name":"J. Approx. Theory","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2001-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120492939","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 characterize the orthogonal polynomials in a class of polynomials defined through their generating functions. This led to three new systems of orthogonal polynomials whose generating functions and orthogonality relations involve elliptic functions. The Hamburger moment problems associated with these polynomials are indeterminate. We give infinite families of weight functions in each case. The different polynomials treated in this work are also polynomials in a parameter and as functions of this parameter they are orthogonal with respect to unique measures, which we find explicitly. Through a quadratic transformation we find a new exactly solvable birth and death process with quartic birth and death rates.
{"title":"Some Orthogonal Polynomials Related to Elliptic Functions","authors":"M. Ismail, G. Valent, G. Yoon","doi":"10.1006/jath.2001.3593","DOIUrl":"https://doi.org/10.1006/jath.2001.3593","url":null,"abstract":"We characterize the orthogonal polynomials in a class of polynomials defined through their generating functions. This led to three new systems of orthogonal polynomials whose generating functions and orthogonality relations involve elliptic functions. The Hamburger moment problems associated with these polynomials are indeterminate. We give infinite families of weight functions in each case. The different polynomials treated in this work are also polynomials in a parameter and as functions of this parameter they are orthogonal with respect to unique measures, which we find explicitly. Through a quadratic transformation we find a new exactly solvable birth and death process with quartic birth and death rates.","PeriodicalId":202056,"journal":{"name":"J. Approx. Theory","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2001-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128213141","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}
Let @[email protected]?(@l"j)^~"j"="0 be a sequence of distinct real numbers. The span of {x^@l^"^0, x^@l^"^1, ..., x^@l^"^n} over R is denoted by M"n(@L)@?span{x^@l^"^0, x^@l^"^1, ..., x^@l^"^n}. Elements of M"n(@L) are called Muntz polynomials. The principal result of this paper is the following Markov-type inequality for products of Muntz polynomials. [email protected]@?(@l"j)^~"j"="[email protected]@?(@c"j)^~"j"="0be increasing sequences of nonnegative real numbers. LetK(M"n(@L), M"m(@C))@[email protected]?x(pq)'(x)@?"["0"," "1"]@[email protected]?"["0"," "1"]:[email protected]?M"n(@L),[email protected]?M"m(@C).Then13((m+1)@l"n+(n+1)@c"m)=
设@[email protected]?(@l"j)^~"j"="0是一个不同实数的序列。{x^@l^”^0,x^@l^”^1，…， x^@l^ ^n} / R表示为M ' n(@ l)@?span{x^@l^"^0, x^@l^"^1，…, x ^ @l ^ ^ n}。M ' n(@L)的元素称为蒙兹多项式。本文的主要结果是蒙兹多项式积的马尔可夫型不等式。[email protected]@?(@l"j)^~"j"="[email protected]@?(@c"j)^~"j"="0个递增的非负实数序列。LetK (M“n (@L), M M (@C)) @(邮件保护)? x (pq) (x) @ ?"["0"，" "1"]@[email protected]?“(“0”,“1”]:[电子邮件保护]? M”n (@L),(邮件保护)? M M (@C) .Then13 ((M + 1) @L“n + (n + 1) @C”米)=
{"title":"Markov-Type Inequalities for Products of Müntz Polynomials","authors":"T. Erdélyi","doi":"10.1006/jath.2001.3583","DOIUrl":"https://doi.org/10.1006/jath.2001.3583","url":null,"abstract":"Let @[email protected]?(@l\"j)^~\"j\"=\"0 be a sequence of distinct real numbers. The span of {x^@l^\"^0, x^@l^\"^1, ..., x^@l^\"^n} over R is denoted by M\"n(@L)@?span{x^@l^\"^0, x^@l^\"^1, ..., x^@l^\"^n}. Elements of M\"n(@L) are called Muntz polynomials. The principal result of this paper is the following Markov-type inequality for products of Muntz polynomials. [email protected]@?(@l\"j)^~\"j\"=\"[email protected]@?(@c\"j)^~\"j\"=\"0be increasing sequences of nonnegative real numbers. LetK(M\"n(@L), M\"m(@C))@[email protected]?x(pq)'(x)@?\"[\"0\",\" \"1\"]@[email protected]?\"[\"0\",\" \"1\"]:[email protected]?M\"n(@L),[email protected]?M\"m(@C).Then13((m+1)@l\"n+(n+1)@c\"m)=","PeriodicalId":202056,"journal":{"name":"J. Approx. Theory","volume":"181 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2001-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124530237","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}
For the weight function (1-@?x@?^2)^@m^-^1^/^2 on the unit ball, a closed formula of the reproducing kernel is modified to include the case -1/2 =0 is n^@m^+^(^d^-^1^)^/^2.
{"title":"Representation of Reproducing Kernels and the Lebesgue Constants on the Ball","authors":"Yuan Xu","doi":"10.1006/jath.2001.3597","DOIUrl":"https://doi.org/10.1006/jath.2001.3597","url":null,"abstract":"For the weight function (1-@?x@?^2)^@m^-^1^/^2 on the unit ball, a closed formula of the reproducing kernel is modified to include the case -1/2 =0 is n^@m^+^(^d^-^1^)^/^2.","PeriodicalId":202056,"journal":{"name":"J. Approx. Theory","volume":"101 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2001-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128590389","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}
The Barnes double gamma function G(z) is considered for large argument z. A new integral representation is obtained for logG(z). An asymptotic expansion in decreasing powers of z and uniformly valid for |Argz|<@p is derived from this integral. The expansion is accompanied by an error bound at any order of the approximation. Numerical experiments show that this bound is very accurate for real z. The accuracy of the error bound decreases for increasing Argz.
{"title":"An Asymptotic Expansion of the Double Gamma Function","authors":"Chelo Ferreira, J. López","doi":"10.1006/jath.2001.3578","DOIUrl":"https://doi.org/10.1006/jath.2001.3578","url":null,"abstract":"The Barnes double gamma function G(z) is considered for large argument z. A new integral representation is obtained for logG(z). An asymptotic expansion in decreasing powers of z and uniformly valid for |Argz|<@p is derived from this integral. The expansion is accompanied by an error bound at any order of the approximation. Numerical experiments show that this bound is very accurate for real z. The accuracy of the error bound decreases for increasing Argz.","PeriodicalId":202056,"journal":{"name":"J. Approx. Theory","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2001-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"119052612","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}
Let @s be a finite positive Borel measure supported on an arc @c of the unit circle, such that @s'>0 a.e. on @c. We obtain a theorem about the weak convergence of the corresponding sequence of orthonormal polynomials. Moreover, we prove an analogue of the [email protected]?-Geronimus theorem on strong asymptotics of the orthogonal polynomials on the complement of @c, which completes to its full extent a result of N. I. Akhiezer. The key tool in the proofs is the use of orthogonality with respect to varying measures.
设@s是支撑在单位圆的弧@c上的有限正Borel测度，使得@s'>0 a.e.在@c上。得到了标准正交多项式对应序列的弱收敛性定理。此外，我们证明了[email protected]?-关于@c补上正交多项式的强渐近性的geronimus定理，完整地完成了N. I. Akhiezer的一个结果。证明中的关键工具是对不同测度的正交性的使用。
{"title":"Strong Asymptotic Behavior and Weak Convergence of Polynomials Orthogonal on an Arc of the Unit Circle","authors":"M. Hernández, E. Díaz","doi":"10.1006/jath.2001.3574","DOIUrl":"https://doi.org/10.1006/jath.2001.3574","url":null,"abstract":"Let @s be a finite positive Borel measure supported on an arc @c of the unit circle, such that @s'>0 a.e. on @c. We obtain a theorem about the weak convergence of the corresponding sequence of orthonormal polynomials. Moreover, we prove an analogue of the [email protected]?-Geronimus theorem on strong asymptotics of the orthogonal polynomials on the complement of @c, which completes to its full extent a result of N. I. Akhiezer. The key tool in the proofs is the use of orthogonality with respect to varying measures.","PeriodicalId":202056,"journal":{"name":"J. Approx. Theory","volume":"52 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2001-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120092598","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 introduce an @a-calculus with the help of the generalized Bernoulli polynomials. The parameter @a is the order of a Bessel function of the first kind. The differential @a-calculus can be put in a general context where the concept of supporting function is an important tool for practical purposes. Our somewhat more restrictive point of view has the advantage of permitting a consistent definition of an @a-integral with several interesting properties. It results in the possibility of expressing a remainder, in the aforementioned context, in a completely new form in our case.
{"title":"A Unified Calculus Using the Generalized Bernoulli Polynomials","authors":"C. Frappier","doi":"10.1006/jath.2000.3550","DOIUrl":"https://doi.org/10.1006/jath.2000.3550","url":null,"abstract":"We introduce an @a-calculus with the help of the generalized Bernoulli polynomials. The parameter @a is the order of a Bessel function of the first kind. The differential @a-calculus can be put in a general context where the concept of supporting function is an important tool for practical purposes. Our somewhat more restrictive point of view has the advantage of permitting a consistent definition of an @a-integral with several interesting properties. It results in the possibility of expressing a remainder, in the aforementioned context, in a completely new form in our case.","PeriodicalId":202056,"journal":{"name":"J. Approx. Theory","volume":"97 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2001-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"119175668","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}
The purpose of this paper is to study the approximation of vector-valued mappings defined on a subset of a normed space. We investigate Korovkin-type conditions useful to recognize if a given sequence of linear operators is a so-called approximation process. First, we give a sufficient condition for this sequence to approximate the class of bounded, uniformly continuous functions. Then we present some sufficient and necessary conditions guaranteeing the approximation within the class of unbounded, *weak-to-norm continuous mappings. We also derive some estimates of the rate of convergence. We apply concrete approximation processes to derive representation formulae for semigroups of bounded linear operators.
{"title":"Approximation of *Weak-to-Norm Continuous Mappings","authors":"L. D’Ambrosio","doi":"10.1006/jath.2002.3708","DOIUrl":"https://doi.org/10.1006/jath.2002.3708","url":null,"abstract":"The purpose of this paper is to study the approximation of vector-valued mappings defined on a subset of a normed space. We investigate Korovkin-type conditions useful to recognize if a given sequence of linear operators is a so-called approximation process. First, we give a sufficient condition for this sequence to approximate the class of bounded, uniformly continuous functions. Then we present some sufficient and necessary conditions guaranteeing the approximation within the class of unbounded, *weak-to-norm continuous mappings. We also derive some estimates of the rate of convergence. We apply concrete approximation processes to derive representation formulae for semigroups of bounded linear operators.","PeriodicalId":202056,"journal":{"name":"J. Approx. Theory","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2000-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128452591","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}
Pub Date : 1900-01-01DOI: 10.1016/j.jat.2019.105348
T. A. Bui
{"title":"Hermite pseudo-multipliers on new Besov and Triebel-Lizorkin spaces","authors":"T. A. Bui","doi":"10.1016/j.jat.2019.105348","DOIUrl":"https://doi.org/10.1016/j.jat.2019.105348","url":null,"abstract":"","PeriodicalId":202056,"journal":{"name":"J. Approx. Theory","volume":"90 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120130676","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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