Journal of Approximation Theory最新文献

{"title":"The norm of the Cesàro operator minus the identity and related operators acting on decreasing sequences","authors":"Santiago Boza ,&nbsp;Javier Soria","doi":"10.1016/j.jat.2023.105911","DOIUrl":"10.1016/j.jat.2023.105911","url":null,"abstract":"<div><p>Recently, several authors have considered the problem of determining optimal norm inequalities for discrete Hardy-type operators (like Cesàro or Copson). In this work, we obtain sharp bounds for the norms of the difference of the Cesàro operator with either the identity or the shift, when they are restricted to the cone of decreasing sequences in <span><math><msup><mrow><mi>ℓ</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span> (which is closely related to the previously mentioned estimates). Finally, we also address the case of weighted inequalities and find an interesting contrast between the norms of these two difference operators.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45046241","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Approximation by linear combinations of translates in invariant Banach spaces of tempered distributions via Tauberian conditions","authors":"Hans G. Feichtinger ,&nbsp;Anupam Gumber","doi":"10.1016/j.jat.2023.105908","DOIUrl":"10.1016/j.jat.2023.105908","url":null,"abstract":"<div><p><span><span>This paper describes an approximation theoretic approach<span> to the problem of completeness of a set of translates of a “Tauberian generator”, which is an integrable function<span> whose Fourier transform does not vanish. This is achieved by the construction of </span></span></span>finite rank operators<span>, whose range is contained in the linear span of the translates of such a generator, and which allow uniform approximation of the identity operator over compact sets of certain Banach spaces </span></span><span><math><mrow><mo>(</mo><mi>B</mi><mo>,</mo><mspace></mspace><msub><mrow><mo>‖</mo><mspace></mspace><mi>⋅</mi><mspace></mspace><mo>‖</mo></mrow><mrow><mi>B</mi></mrow></msub><mo>)</mo></mrow></math></span>. The key assumption is availability of a double module structure on <span><math><mrow><mo>(</mo><mi>B</mi><mo>,</mo><mspace></mspace><msub><mrow><mo>‖</mo><mspace></mspace><mi>⋅</mi><mspace></mspace><mo>‖</mo></mrow><mrow><mi>B</mi></mrow></msub><mo>)</mo></mrow></math></span><span><span>, meaning the availability of sufficiently many smoothing operators (via convolution) and also pointwise multipliers, allowing localization of its elements. This structure is shared by a wide variety of function spaces and allows us to make explicit use of the Riesz–Kolmogorov Theorem characterizing </span>compact subsets in such Banach spaces.</span></p><p>The construction of these operators is universal with respect to large families of such Banach spaces, i.e. they do not depend on any further information concerning the particular Banach space. As a corollary we conclude that the linear span of the set of the translates of such a Tauberian generator is dense in any such space <span><math><mrow><mo>(</mo><mi>B</mi><mo>,</mo><mspace></mspace><msub><mrow><mo>‖</mo><mspace></mspace><mi>⋅</mi><mspace></mspace><mo>‖</mo></mrow><mrow><mi>B</mi></mrow></msub><mo>)</mo></mrow></math></span><span>. Our work has been inspired by a completeness result of V. Katsnelson which was formulated in the context of specific Hilbert spaces within this family and Gaussian generators.</span></p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42317479","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Boundary value problems of potential theory for the exterior ball and the approximation and ergodic behaviour of the solutions","authors":"P.L. Butzer,&nbsp;R.L. Stens","doi":"10.1016/j.jat.2023.105916","DOIUrl":"10.1016/j.jat.2023.105916","url":null,"abstract":"<div><p><span>The paper is concerned with the interconnection of the boundary behaviour of the solutions of the exterior Dirichlet, Neumann and Robin problems of harmonic analysis for the unit ball in </span><span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> with the corresponding behaviour of the associated ergodic inverse problems for the entire space. Rates of approximation play a basic role.</p><p><span><span><span>The solutions themselves are evaluated by means of Fourier expansions with respect to </span>spherical harmonics<span>. In case of the first two problems, the basis for the investigation of the approximation and ergodic behaviour is the theory of semigroups of linear operators mapping a </span></span>Banach space </span><em>X</em> into itself. The connection between the semigroup property and the major premise of Huygens’ principle is emphasized.</p><p><span>Another tool is a Drazin-like inverse operator </span><span><math><msup><mrow><mi>A</mi></mrow><mrow><mi>ad</mi></mrow></msup></math></span><span> for the infinitesimal generator </span><span><math><mi>A</mi></math></span><span> of a semigroup that arises naturally in ergodic theory. This operator is a closed, not necessarily bounded, operator. It was introduced in a paper with U. Westphal (Butzer and Westphal, 1970/71) and extended to a generalized setting with J.J. Koliha (Butzer and Koliha, 2009).</span></p><p>Unlike the latter two problems, the solution of Robin’s problem does not have the semigroup property and therefore the semigroup methods applied to Dirichlet’s and Neumann’s problem do not work. The authors give several hints how to overcome these difficulties.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49193160","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A modified Christoffel function and its asymptotic properties","authors":"Jean B. Lasserre","doi":"10.1016/j.jat.2023.105955","DOIUrl":"10.1016/j.jat.2023.105955","url":null,"abstract":"<div><p>We introduce a certain variant (or regularization) <span><math><msubsup><mrow><mover><mrow><mi>Λ</mi></mrow><mrow><mo>̃</mo></mrow></mover></mrow><mrow><mi>n</mi></mrow><mrow><mi>μ</mi></mrow></msubsup></math></span> of the standard Christoffel function <span><math><msubsup><mrow><mi>Λ</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>μ</mi></mrow></msubsup></math></span> associated with a measure <span><math><mi>μ</mi></math></span> on a compact set <span><math><mrow><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></mrow></math></span>. Its reciprocal is now a sum-of-squares polynomial in the variables <span><math><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>ɛ</mi><mo>)</mo></mrow></math></span>, <span><math><mrow><mi>ɛ</mi><mo>&gt;</mo><mn>0</mn></mrow></math></span>. It shares the same dichotomy property of the standard Christoffel function, that is, the growth with <span><math><mi>n</mi></math></span> of its inverse is at most polynomial inside and exponential outside the support of the measure. Its distinguishing and crucial feature states that for fixed <span><math><mrow><mi>ɛ</mi><mo>&gt;</mo><mn>0</mn></mrow></math></span><span>, and under weak assumptions, </span><span><math><mrow><msub><mrow><mo>lim</mo></mrow><mrow><mi>n</mi><mo>→</mo><mi>∞</mi></mrow></msub><msup><mrow><mi>ɛ</mi></mrow><mrow><mo>−</mo><mi>d</mi></mrow></msup><msubsup><mrow><mover><mrow><mi>Λ</mi></mrow><mrow><mo>̃</mo></mrow></mover></mrow><mrow><mi>n</mi></mrow><mrow><mi>μ</mi></mrow></msubsup><mrow><mo>(</mo><mi>ξ</mi><mo>,</mo><mi>ɛ</mi><mo>)</mo></mrow><mo>=</mo><mi>f</mi><mrow><mo>(</mo><msub><mrow><mi>ζ</mi></mrow><mrow><mi>ɛ</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> where <span><math><mi>f</mi></math></span> (assumed to be continuous) is the unknown density of <span><math><mi>μ</mi></math></span><span> w.r.t. Lebesgue measure on </span><span><math><mi>Ω</mi></math></span>, and <span><math><mrow><msub><mrow><mi>ζ</mi></mrow><mrow><mi>ɛ</mi></mrow></msub><mo>∈</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>∞</mi></mrow></msub><mrow><mo>(</mo><mi>ξ</mi><mo>,</mo><mi>ɛ</mi><mo>)</mo></mrow></mrow></math></span> (and so <span><math><mrow><mi>f</mi><mrow><mo>(</mo><msub><mrow><mi>ζ</mi></mrow><mrow><mi>ɛ</mi></mrow></msub><mo>)</mo></mrow><mo>≈</mo><mi>f</mi><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow></mrow></math></span> when <span><math><mrow><mi>ɛ</mi><mo>&gt;</mo><mn>0</mn></mrow></math></span> is small). This is in contrast with the standard Christoffel function where if <span><math><mrow><msub><mrow><mo>lim</mo></mrow><mrow><mi>n</mi><mo>→</mo><mi>∞</mi></mrow></msub><msup><mrow><mi>n</mi></mrow><mrow><mi>d</mi></mrow></msup><msubsup><mrow><mi>Λ</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>μ</mi></mrow></msubsup><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow></mrow></math></span> exists, it is of the form <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow><mo>/</mo><msub><mrow><mi>ω</mi></mrow><mrow><mi>E</m","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43100526","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On dynamics of asymptotically minimal polynomials","authors":"Turgay Bayraktar ,&nbsp;Melike Efe","doi":"10.1016/j.jat.2023.105956","DOIUrl":"10.1016/j.jat.2023.105956","url":null,"abstract":"<div><p><span>We study dynamical properties of asymptotically extremal polynomials associated with a non-polar planar compact set </span><span><math><mi>E</mi></math></span><span>. In particular, we prove that if the zeros of such polynomials are uniformly bounded then their Brolin measures converge weakly to the equilibrium measure of </span><span><math><mi>E</mi></math></span>. In addition, if <span><math><mi>E</mi></math></span> is regular and the zeros of such polynomials are sufficiently close to <span><math><mi>E</mi></math></span><span> then we show that the filled Julia sets<span> converge to polynomial convex hull of </span></span><span><math><mi>E</mi></math></span> in the Klimek topology.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48829613","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Harmonic analysis of little q-Legendre polynomials","authors":"Stefan Kahler","doi":"10.1016/j.jat.2023.105946","DOIUrl":"https://doi.org/10.1016/j.jat.2023.105946","url":null,"abstract":"<div><p><span><span><span><span>Many classes of orthogonal polynomials<span> satisfy a specific linearization property giving rise to a polynomial hypergroup structure, which offers an elegant and fruitful link to Fourier analysis, </span></span>harmonic analysis and </span>functional analysis. From the opposite point of view, this allows regarding certain </span>Banach algebras as </span><span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-algebras, associated with underlying orthogonal polynomials. The individual behavior strongly depends on these underlying polynomials. We study the little <span><math><mi>q</mi></math></span>-Legendre polynomials, which are orthogonal with respect to a discrete measure. We will show that their <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span><span>-algebras have the property that every element can be approximated by linear combinations<span> of idempotents. This particularly implies that these </span></span><span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-algebras are weakly amenable (i.e., every bounded derivation into the dual module is an inner derivation), which is known to be shared by any <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span><span>-algebra of a locally compact group<span>; in the polynomial hypergroup context, weak amenability is rarely satisfied and of particular interest because it corresponds to a certain property of the derivatives of the underlying polynomials and their (Fourier) expansions w.r.t. the polynomial basis. To our knowledge, the little </span></span><span><math><mi>q</mi></math></span>-Legendre polynomials yield the first example of a polynomial hypergroup whose <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span><span>-algebra is weakly amenable and right character amenable but fails to be amenable. As a crucial tool, we establish certain uniform boundedness<span> properties of the characters. Our strategy relies on the Fourier transformation on hypergroups, the Plancherel isomorphism, continued fractions, character estimations and asymptotic behavior.</span></span></p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50183872","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Harmonic analysis of little q-Legendre polynomials","authors":"Stefan Kahler","doi":"10.1016/j.jat.2023.105946","DOIUrl":"https://doi.org/10.1016/j.jat.2023.105946","url":null,"abstract":"","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44408303","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Spectral properties of a class of Moran measures on R2","authors":"Zhi-Hui Yan","doi":"10.1016/j.jat.2023.105914","DOIUrl":"https://doi.org/10.1016/j.jat.2023.105914","url":null,"abstract":"<div><p>Given a pair <span><math><mrow><mo>(</mo><mi>R</mi><mo>,</mo><mi>D</mi><mo>)</mo></mrow></math></span>, where <span><math><mrow><mi>R</mi><mo>=</mo><msubsup><mrow><mrow><mo>{</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>}</mo></mrow></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>∞</mi></mrow></msubsup></mrow></math></span> is a sequence of expanding matrix (i.e., all the eigenvalues of <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> have modulus strictly greater than 1), and <span><math><mrow><mi>D</mi><mo>=</mo><msubsup><mrow><mrow><mo>{</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>}</mo></mrow></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>∞</mi></mrow></msubsup><mo>⊆</mo><msup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math></span><span>. It is well known that there exists an infinite convolution generated by </span><span><math><mrow><mo>(</mo><mi>R</mi><mo>,</mo><mi>D</mi><mo>)</mo></mrow></math></span> which satisfies <span><span><span><math><mrow><msub><mrow><mi>μ</mi></mrow><mrow><mi>R</mi><mo>,</mo><mi>D</mi></mrow></msub><mo>≔</mo><msub><mrow><mi>δ</mi></mrow><mrow><msubsup><mrow><mi>R</mi></mrow><mrow><mn>1</mn></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><msub><mrow><mi>D</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>∗</mo><msub><mrow><mi>δ</mi></mrow><mrow><msup><mrow><mrow><mo>(</mo><msub><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msub><msub><mrow><mi>R</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub><mo>∗</mo><mo>⋯</mo><mspace></mspace><mo>,</mo></mrow></math></span></span></span>we say that <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mi>R</mi><mo>,</mo><mi>D</mi></mrow></msub></math></span> is a <em>Moran measure</em> if it convergent to a probability measure with compact support in a weak sense, where <span><math><mrow><msub><mrow><mi>δ</mi></mrow><mrow><mi>E</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>#</mi><mi>E</mi></mrow></mfrac><msub><mrow><mo>∑</mo></mrow><mrow><mi>d</mi><mo>∈</mo><mi>E</mi></mrow></msub><msub><mrow><mi>δ</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></math></span> is the uniformly discrete measure on <span><math><mi>E</mi></math></span><span>. In this paper, we consider the spectral properties of the Moran measure </span><span><math><msub><mrow><mi>μ</mi></mrow><mrow><mi>R</mi><mo>,</mo><mi>D</mi></mrow></msub></math></span> with <span><math><mrow><mi>R</mi><mo>=</mo><mfenced><mrow><mtable><mtr><mtd><msub><mrow><mi>b</mi></mrow><mrow><mn>1</mn></mrow></msub></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><msub><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></msub></mtd></mtr></mtable></mrow></mfenced></mrow></math></span>, and <span><math><mrow><mi>#</mi><msub><mrow><mi>D</mi></mr","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50198336","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Density results and trace operator in weighted Sobolev spaces defined on the half-line, equipped with power weights","authors":"Radosław Kaczmarek ,&nbsp;Agnieszka Kałamajska","doi":"10.1016/j.jat.2023.105896","DOIUrl":"10.1016/j.jat.2023.105896","url":null,"abstract":"<div><p>We study properties of <span><math><mrow><msubsup><mrow><mi>W</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>1</mn><mo>,</mo><mi>p</mi></mrow></msubsup><mrow><mo>(</mo><msub><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>,</mo><msup><mrow><mi>t</mi></mrow><mrow><mi>β</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span> — the completion of <span><math><mrow><msubsup><mrow><mi>C</mi></mrow><mrow><mn>0</mn></mrow><mrow><mi>∞</mi></mrow></msubsup><mrow><mo>(</mo><msub><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>)</mo></mrow></mrow></math></span><span> in the power-weighted Sobolev spaces </span><span><math><mrow><msup><mrow><mi>W</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>p</mi></mrow></msup><mrow><mo>(</mo><msub><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>,</mo><msup><mrow><mi>t</mi></mrow><mrow><mi>β</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span>, where <span><math><mrow><mi>β</mi><mo>∈</mo><mi>R</mi></mrow></math></span>. Among other results, we obtain the analytic characterization of <span><math><mrow><msubsup><mrow><mi>W</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>1</mn><mo>,</mo><mi>p</mi></mrow></msubsup><mrow><mo>(</mo><msub><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>,</mo><msup><mrow><mi>t</mi></mrow><mrow><mi>β</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span> for all <span><math><mrow><mi>β</mi><mo>∈</mo><mi>R</mi></mrow></math></span>. Our analysis is based on the precise study of the two trace operators: <span><math><mrow><mi>T</mi><msup><mrow><mi>r</mi></mrow><mrow><mn>0</mn></mrow></msup><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>≔</mo><msub><mrow><mo>lim</mo></mrow><mrow><mi>t</mi><mo>→</mo><mn>0</mn></mrow></msub><mi>u</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>T</mi><msup><mrow><mi>r</mi></mrow><mrow><mi>∞</mi></mrow></msup><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>≔</mo><msub><mrow><mo>lim</mo></mrow><mrow><mi>t</mi><mo>→</mo><mi>∞</mi></mrow></msub><mi>u</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></math></span><span>, which leads to the analysis of the asymptotic behavior of functions from </span><span><math><mrow><msubsup><mrow><mi>W</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>1</mn><mo>,</mo><mi>p</mi></mrow></msubsup><mrow><mo>(</mo><msub><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>,</mo><msup><mrow><mi>t</mi></mrow><mrow><mi>β</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span><span><span> near zero or infinity. The obtained statements can contribute to the proper formulation of Boundary Value Problems<span> in ODEs, or </span></span>PDEs<span><span> with the radial symmetries. We can also apply our results to some questions in the complex </span>interpolation theory, raised by Cwikel and Einav (2019), which we discuss within the particular case of Sobolev spaces </span></span><span><math><mrow><msup><mrow><mi>W</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>p</mi></mrow>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46007064","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Approximation by sums of shifts and dilations of a single function and neural networks","authors":"K. Shklyaev","doi":"10.1016/j.jat.2023.105915","DOIUrl":"10.1016/j.jat.2023.105915","url":null,"abstract":"<div><p>We find sufficient conditions on a function <span><math><mi>f</mi></math></span> to ensure that sums of functions of the form <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>α</mi><mi>x</mi><mo>−</mo><mi>θ</mi><mo>)</mo></mrow></mrow></math></span>, where <span><math><mrow><mi>α</mi><mo>∈</mo><mi>A</mi><mo>⊂</mo><mi>R</mi></mrow></math></span> and <span><math><mrow><mi>θ</mi><mo>∈</mo><mi>Θ</mi><mo>⊂</mo><mi>R</mi></mrow></math></span>, are dense in the real spaces <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span><span><span> on the real line or its compact subsets. That is, we consider </span>linear combinations in which all coefficients are 1. As a corollary we deduce results on density of sums of functions </span><span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>w</mi><mi>⋅</mi><mi>x</mi><mo>−</mo><mi>θ</mi><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><mi>w</mi><mo>∈</mo><mi>W</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></mrow></math></span>, <span><math><mrow><mi>θ</mi><mo>∈</mo><mi>Θ</mi><mo>⊂</mo><mi>R</mi></mrow></math></span> in <span><math><mrow><mi>C</mi><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span> in the topology of uniform convergence on compact subsets.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46707846","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}