In a domain having several cylindrical outlets to infinity, the stationary Maxwell system with perfectly conductive boundary conditions is studied. The dielectric permittivity and magnetic permeability are assumed to be arbitrary positive definite matrix-valued functions that slowly stabilize at infinity. The authors introduce the scattering matrix, establish the unique solvability of the problem with radiation conditions at infinity, and describe the asymptotics of solutions.
{"title":"The Maxwell system in nonhomogeneous anisotropic waveguides with slowly stabilizing characteristics of the filling medium","authors":"B. Plamenevskii, A. Poretskii","doi":"10.1090/spmj/1773","DOIUrl":"https://doi.org/10.1090/spmj/1773","url":null,"abstract":"In a domain having several cylindrical outlets to infinity, the stationary Maxwell system with perfectly conductive boundary conditions is studied. The dielectric permittivity and magnetic permeability are assumed to be arbitrary positive definite matrix-valued functions that slowly stabilize at infinity. The authors introduce the scattering matrix, establish the unique solvability of the problem with radiation conditions at infinity, and describe the asymptotics of solutions.","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43624988","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Exponential polynomials satisfying a homogeneous equation of convolution type are called its elementary solutions. The article is devoted to convolution-type operators in the complex domain that generalize the well-known operators of q q -sided convolution and π pi -convolution. The properties of such operators are investigated and the general form of elementary solutions (general elementary solution) of a homogeneous equation of q q -sided convolution type is described.
满足卷积型齐次方程的指数多项式称为它的初等解。本文研究了复域上的卷积型算子，它推广了著名的q - q边卷积算子和π pi -卷积算子。研究了这类算子的性质，给出了一类q - q边卷积型齐次方程初等解的一般形式。
{"title":"General elementary solution of a 𝑞-sided convolution type homogeneous equation","authors":"Yuriy Saranchuk, A. Shishkin","doi":"10.1090/spmj/1774","DOIUrl":"https://doi.org/10.1090/spmj/1774","url":null,"abstract":"Exponential polynomials satisfying a homogeneous equation of convolution type are called its elementary solutions. The article is devoted to convolution-type operators in the complex domain that generalize the well-known operators of \u0000\u0000 \u0000 q\u0000 q\u0000 \u0000\u0000-sided convolution and \u0000\u0000 \u0000 π\u0000 pi\u0000 \u0000\u0000-convolution. The properties of such operators are investigated and the general form of elementary solutions (general elementary solution) of a homogeneous equation of \u0000\u0000 \u0000 q\u0000 q\u0000 \u0000\u0000-sided convolution type is described.","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45564488","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the classical inverse theorems of constructive function theory, structural characteristics of an approximated function are estimated in terms of its best approximations. Most of the known proofs of the inverse theorems utilize Bernstein’s idea to expand the function in polynomials of its best approximation. In the present paper, Bernstein’s proof is modified by using integrals instead of sums. With this modification, it turns out that desired inequalities are based on identities similar to Frullani integrals. The considerations here are quite general, which allows one to obtain analogs of the inverse theorems for functionals in abstract Banach or even seminormed spaces. Then these abstract results are specified and inverse theorems in concrete spaces of functions are deduced, including weighted spaces, with explicit constants.
{"title":"On the constants in abstract inverse theorems of approximation theory","authors":"O. Vinogradov","doi":"10.1090/spmj/1770","DOIUrl":"https://doi.org/10.1090/spmj/1770","url":null,"abstract":"In the classical inverse theorems of constructive function theory, structural characteristics of an approximated function are estimated in terms of its best approximations. Most of the known proofs of the inverse theorems utilize Bernstein’s idea to expand the function in polynomials of its best approximation. In the present paper, Bernstein’s proof is modified by using integrals instead of sums. With this modification, it turns out that desired inequalities are based on identities similar to Frullani integrals. The considerations here are quite general, which allows one to obtain analogs of the inverse theorems for functionals in abstract Banach or even seminormed spaces. Then these abstract results are specified and inverse theorems in concrete spaces of functions are deduced, including weighted spaces, with explicit constants.","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42336313","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The paper describes the topological structure of the maximal ideal space of the algebra of bounded holomorphic functions on a covering of a bordered Riemann surface. Some applications of the obtained results to the theory of bounded operator-valued holomorphic functions on Riemann surfaces are presented.
{"title":"On the maximal ideal spaces of 𝐇^{∞} on coverings of bordered Riemann surfaces","authors":"A. Brudnyi","doi":"10.1090/spmj/1761","DOIUrl":"https://doi.org/10.1090/spmj/1761","url":null,"abstract":"The paper describes the topological structure of the maximal ideal space of the algebra of bounded holomorphic functions on a covering of a bordered Riemann surface. Some applications of the obtained results to the theory of bounded operator-valued holomorphic functions on Riemann surfaces are presented.","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43776472","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is a chapter from the continuation of a survey by the author and N. K. Nikolski published in 1998. It contains two theorems describing when an invariant subspace has an invariant complement and when the angle between two given invariant subspaces is positive. The presentation involves the technique of the coordinate-free functional model.
{"title":"Chapter 7. Angles between invariant subspaces","authors":"V. Vasyunin","doi":"10.1090/spmj/1767","DOIUrl":"https://doi.org/10.1090/spmj/1767","url":null,"abstract":"This paper is a chapter from the continuation of a survey by the author and N. K. Nikolski published in 1998. It contains two theorems describing when an invariant subspace has an invariant complement and when the angle between two given invariant subspaces is positive. The presentation involves the technique of the coordinate-free functional model.","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49126980","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The utility of the (weighted) van der Corput inequalities or of the stationary phase method is illustrated with various examples borrowed from: differentiability issues (Riemann’s function and related); functional analysis on Banach spaces or algebras of analytic functions (composition operators); and local Banach space geometry (Schäffer’s problem).
{"title":"Stationary phase method, powers of functions, and applications to functional analysis","authors":"H. Queffélec, R. Zarouf","doi":"10.1090/spmj/1757","DOIUrl":"https://doi.org/10.1090/spmj/1757","url":null,"abstract":"The utility of the (weighted) van der Corput inequalities or of the stationary phase method is illustrated with various examples borrowed from: differentiability issues (Riemann’s function and related); functional analysis on Banach spaces or algebras of analytic functions (composition operators); and local Banach space geometry (Schäffer’s problem).","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46166546","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
<p>Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a meromorphic function on the complex plane with Nevanlinna characteristic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T left-parenthesis r comma f right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>r</mml:mi> <mml:mo>,</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">T(r,f)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and maximal radial characteristic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ln upper M left-parenthesis t comma f right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>ln</mml:mi> <mml:mo><!-- --></mml:mo> <mml:mi>M</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">ln M(t,f)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M left-parenthesis t comma f right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">M(t,f)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the maximum of the modulus <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue f EndAbsoluteValue"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>f</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">|f|</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on circles centered at zero and of radius <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t"> <mml:semantics> <mml:mi>t</mml:mi> <mml:annotation encoding="application/x-tex">t</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. A number of well-known and widely used results make it possible to estimate from
{"title":"Nevanlinna characteristic and integral inequalities with maximal radial characteristic for meromorphic functions and for differences of subharmonic functions","authors":"B. Khabibullin","doi":"10.1090/spmj/1753","DOIUrl":"https://doi.org/10.1090/spmj/1753","url":null,"abstract":"<p>Let <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f\">\u0000 <mml:semantics>\u0000 <mml:mi>f</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">f</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> be a meromorphic function on the complex plane with Nevanlinna characteristic <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper T left-parenthesis r comma f right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>T</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>r</mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>f</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">T(r,f)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and maximal radial characteristic <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"ln upper M left-parenthesis t comma f right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>ln</mml:mi>\u0000 <mml:mo><!-- --></mml:mo>\u0000 <mml:mi>M</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>t</mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>f</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">ln M(t,f)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, where <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M left-parenthesis t comma f right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>M</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>t</mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>f</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">M(t,f)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is the maximum of the modulus <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"StartAbsoluteValue f EndAbsoluteValue\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo stretchy=\"false\">|</mml:mo>\u0000 </mml:mrow>\u0000 <mml:mi>f</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo stretchy=\"false\">|</mml:mo>\u0000 </mml:mrow>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">|f|</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> on circles centered at zero and of radius <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"t\">\u0000 <mml:semantics>\u0000 <mml:mi>t</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">t</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. A number of well-known and widely used results make it possible to estimate from ","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45679529","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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