Preview this item
Preview this item
Checking...

# The Mellin Transformation and Fuchsian Type Partial Differential Equations

Author: Zofia Szmydt; Bogdan Ziemian Dordrecht : Springer Netherlands, 1992. Mathematics and Its Applications (East European Series), 56. eBook : Document : EnglishView all editions and formats This volume provides a systematic introduction to the theory of the multidimensional Mellin transformation in a distributional setting. In contrast to the classical texts on the Mellin and Laplace transformations, this work concentrates on the local properties of the Mellin transforms, i.e. on those properties of the Mellin transforms of distributions u which are preserved under multiplication of u by cut-off functions (of various types). The main part of the book is devoted to the local study of regularity of solutions to linear Fuchsian partial differential operators on a corner, which demonstrates the appearance of non-discrete asymptotic expansions (at the vertex) and of resurgence effects in the spirit of J. Ecalle. The book constitutes a part of a program to use the Mellin transformation as a link between the theory of second micro-localization, resurgence theory and the theory of the generalized Borel transformation. Chapter I contains the basic theorems and definitions of the theory of distributions and Fourier transformations which are used in the succeeding chapters. This material includes proofs which are partially transformed into exercises with hints. Chapter II presents a systematic treatment of the Mellin transform in several dimensions. Chapter III is devoted to Fuchsian-type singular differential equations. For researchers and graduate students interested in differential equations and integral transforms. This book can also be recommended as a graduate text for students of mathematics and engineering.  Read more... (not yet rated) 0 with reviews - Be the first.

## Find a copy in the library

Finding libraries that hold this item...

## Details

Genre/Form: Electronic books Print version: Document, Internet resource Internet Resource, Computer File Zofia Szmydt; Bogdan Ziemian Find more information about: Zofia Szmydt Bogdan Ziemian 9789401124249 9401124248 851388352 1 online resource (xiv, 222 pages). I. Introduction -- §1. Terminology and notation -- §2. Elementary facts on complex topological vector spaces -- Exercise -- §3. A review of basic facts in the theory of distributions -- Exercises -- II. Mellin distributions and the Mellin transformation -- §4. The Fourier and the Fourier-Mellin transformations -- Exercises -- §5. The spaces of Mellin distributions with support in a polyinterval -- Exercises -- §6. Operations of multiplication and differentiation in the space of Mellin distributions -- Exercises -- §7. The Mellin transformation in the space of Mellin distributions -- Exercises -- §8. The structure of Mellin distributions -- Exercises -- §9. Paley-Wiener type theorems for the Mellin transformation -- Exercises -- §10. Mellin transforms of cut-off functions (continued) -- Exercises -- §11. Important subspaces of Mellin distributions -- Exercises -- §12. The modified Cauchy transformation -- Exercises -- III. Fuchsian type singular operators -- §13. Fuchsian type ordinary differential operators -- Exercises -- §14. Elliptic Fuchsian type partial differential equations in spaces % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuaiaacI % cacaWG4bWaaSaaaeaacaWGKbaabaGaamizaiaadIhaaaGaaiykaiaa % dwhacqGH9aqpcaWGMbaaaa!3EE9!$$P(x\frac{d}{{dx}})u = f$$ -- 2. Case of a proper cone -- Exercise -- §15. Fuchsian type partial differential equations in spaces with continuous radial asymptotics -- Appendix. Generalized smooth functions and theory of resurgent functions of Jean Ecalle -- 1. Introduction -- 2. Generalized Taylor expansions -- 3. Algebra of resurgent functions of Jean Ecalle -- 4. Applications -- List of Symbols. Mathematics and Its Applications (East European Series), 56. by Zofia Szmydt, Bogdan Ziemian.

### Abstract:

This volume provides a systematic introduction to the theory of the multidimensional Mellin transformation in a distributional setting. In contrast to the classical texts on the Mellin and Laplace transformations, this work concentrates on the local properties of the Mellin transforms, i.e. on those properties of the Mellin transforms of distributions u which are preserved under multiplication of u by cut-off functions (of various types). The main part of the book is devoted to the local study of regularity of solutions to linear Fuchsian partial differential operators on a corner, which demonstrates the appearance of non-discrete asymptotic expansions (at the vertex) and of resurgence effects in the spirit of J. Ecalle. The book constitutes a part of a program to use the Mellin transformation as a link between the theory of second micro-localization, resurgence theory and the theory of the generalized Borel transformation. Chapter I contains the basic theorems and definitions of the theory of distributions and Fourier transformations which are used in the succeeding chapters. This material includes proofs which are partially transformed into exercises with hints. Chapter II presents a systematic treatment of the Mellin transform in several dimensions. Chapter III is devoted to Fuchsian-type singular differential equations. For researchers and graduate students interested in differential equations and integral transforms. This book can also be recommended as a graduate text for students of mathematics and engineering.

## Reviews

User-contributed reviews

Be the first.

## Similar Items

### Related Subjects:(4)

Confirm this request

You may have already requested this item. Please select Ok if you would like to proceed with this request anyway.

### Primary Entity

<http://www.worldcat.org/oclc/851388352> # The Mellin Transformation and Fuchsian Type Partial Differential Equations
a schema:MediaObject, schema:CreativeWork, schema:Book ;
library:oclcnum "851388352" ;
library:placeOfPublication <http://experiment.worldcat.org/entity/work/data/21001388#Place/dordrecht> ; # Dordrecht
library:placeOfPublication <http://id.loc.gov/vocabulary/countries/ne> ;
schema:about <http://id.worldcat.org/fast/893484> ; # Differential equations, Partial
schema:about <http://id.worldcat.org/fast/975517> ; # Integral transforms
schema:about <http://id.worldcat.org/fast/936061> ; # Functional analysis
schema:bookFormat schema:EBook ;
schema:contributor <http://viaf.org/viaf/56679721> ; # Bogdan Ziemian
schema:creator <http://viaf.org/viaf/12378311> ; # Zofia Szmydt
schema:datePublished "1992" ;
schema:description "This volume provides a systematic introduction to the theory of the multidimensional Mellin transformation in a distributional setting. In contrast to the classical texts on the Mellin and Laplace transformations, this work concentrates on the local properties of the Mellin transforms, i.e. on those properties of the Mellin transforms of distributions u which are preserved under multiplication of u by cut-off functions (of various types). The main part of the book is devoted to the local study of regularity of solutions to linear Fuchsian partial differential operators on a corner, which demonstrates the appearance of non-discrete asymptotic expansions (at the vertex) and of resurgence effects in the spirit of J. Ecalle. The book constitutes a part of a program to use the Mellin transformation as a link between the theory of second micro-localization, resurgence theory and the theory of the generalized Borel transformation. Chapter I contains the basic theorems and definitions of the theory of distributions and Fourier transformations which are used in the succeeding chapters. This material includes proofs which are partially transformed into exercises with hints. Chapter II presents a systematic treatment of the Mellin transform in several dimensions. Chapter III is devoted to Fuchsian-type singular differential equations. For researchers and graduate students interested in differential equations and integral transforms. This book can also be recommended as a graduate text for students of mathematics and engineering."@en ;
schema:description "I. Introduction -- §1. Terminology and notation -- §2. Elementary facts on complex topological vector spaces -- Exercise -- §3. A review of basic facts in the theory of distributions -- Exercises -- II. Mellin distributions and the Mellin transformation -- §4. The Fourier and the Fourier-Mellin transformations -- Exercises -- §5. The spaces of Mellin distributions with support in a polyinterval -- Exercises -- §6. Operations of multiplication and differentiation in the space of Mellin distributions -- Exercises -- §7. The Mellin transformation in the space of Mellin distributions -- Exercises -- §8. The structure of Mellin distributions -- Exercises -- §9. Paley-Wiener type theorems for the Mellin transformation -- Exercises -- §10. Mellin transforms of cut-off functions (continued) -- Exercises -- §11. Important subspaces of Mellin distributions -- Exercises -- §12. The modified Cauchy transformation -- Exercises -- III. Fuchsian type singular operators -- §13. Fuchsian type ordinary differential operators -- Exercises -- §14. Elliptic Fuchsian type partial differential equations in spaces % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuaiaacI % cacaWG4bWaaSaaaeaacaWGKbaabaGaamizaiaadIhaaaGaaiykaiaa % dwhacqGH9aqpcaWGMbaaaa!3EE9!$$P(x\frac{d}{{dx}})u = f$$ -- 2. Case of a proper cone -- Exercise -- §15. Fuchsian type partial differential equations in spaces with continuous radial asymptotics -- Appendix. Generalized smooth functions and theory of resurgent functions of Jean Ecalle -- 1. Introduction -- 2. Generalized Taylor expansions -- 3. Algebra of resurgent functions of Jean Ecalle -- 4. Applications -- List of Symbols."@en ;
schema:exampleOfWork <http://worldcat.org/entity/work/id/21001388> ;
schema:genre "Electronic books"@en ;
schema:inLanguage "en" ;
schema:isPartOf <http://worldcat.org/issn/0169-507X> ; # Mathematics and Its Applications (East European Series),
schema:isPartOf <http://experiment.worldcat.org/entity/work/data/21001388#Series/mathematics_and_its_applications_east_european_series> ; # Mathematics and Its Applications (East European Series) ;
schema:isSimilarTo <http://worldcat.org/entity/work/data/21001388#CreativeWork/> ;
schema:name "The Mellin Transformation and Fuchsian Type Partial Differential Equations"@en ;
schema:productID "851388352" ;
schema:publication <http://www.worldcat.org/title/-/oclc/851388352#PublicationEvent/dordrecht_springer_netherlands_1992> ;
schema:publisher <http://experiment.worldcat.org/entity/work/data/21001388#Agent/springer_netherlands> ; # Springer Netherlands
schema:url <http://dx.doi.org/10.1007/978-94-011-2424-9> ;
schema:url <http://public.eblib.com/choice/publicfullrecord.aspx?p=3103562> ;
schema:workExample <http://dx.doi.org/10.1007/978-94-011-2424-9> ;
schema:workExample <http://worldcat.org/isbn/9789401124249> ;
wdrs:describedby <http://www.worldcat.org/title/-/oclc/851388352> ;
.

### Related Entities

<http://experiment.worldcat.org/entity/work/data/21001388#Agent/springer_netherlands> # Springer Netherlands
a bgn:Agent ;
schema:name "Springer Netherlands" ;
.

<http://experiment.worldcat.org/entity/work/data/21001388#Series/mathematics_and_its_applications_east_european_series> # Mathematics and Its Applications (East European Series) ;
a bgn:PublicationSeries ;
schema:hasPart <http://www.worldcat.org/oclc/851388352> ; # The Mellin Transformation and Fuchsian Type Partial Differential Equations
schema:name "Mathematics and Its Applications (East European Series) ;" ;
.

<http://id.worldcat.org/fast/1012163> # Mathematics
a schema:Intangible ;
schema:name "Mathematics"@en ;
.

<http://id.worldcat.org/fast/893484> # Differential equations, Partial
a schema:Intangible ;
schema:name "Differential equations, Partial"@en ;
.

<http://id.worldcat.org/fast/936061> # Functional analysis
a schema:Intangible ;
schema:name "Functional analysis"@en ;
.

<http://id.worldcat.org/fast/975517> # Integral transforms
a schema:Intangible ;
schema:name "Integral transforms"@en ;
.

rdfs:comment "from Springer" ;
rdfs:comment "(Unlimited Concurrent Users)" ;
.

<http://viaf.org/viaf/12378311> # Zofia Szmydt
a schema:Person ;
schema:familyName "Szmydt" ;
schema:givenName "Zofia" ;
schema:name "Zofia Szmydt" ;
.

<http://viaf.org/viaf/56679721> # Bogdan Ziemian
a schema:Person ;
schema:familyName "Ziemian" ;
schema:givenName "Bogdan" ;
schema:name "Bogdan Ziemian" ;
.

<http://worldcat.org/entity/work/data/21001388#CreativeWork/>
a schema:CreativeWork ;
schema:description "Print version:" ;
schema:isSimilarTo <http://www.worldcat.org/oclc/851388352> ; # The Mellin Transformation and Fuchsian Type Partial Differential Equations
.

<http://worldcat.org/isbn/9789401124249>
a schema:ProductModel ;
schema:isbn "9401124248" ;
schema:isbn "9789401124249" ;
.

<http://worldcat.org/issn/0169-507X> # Mathematics and Its Applications (East European Series),
a bgn:PublicationSeries ;
schema:hasPart <http://www.worldcat.org/oclc/851388352> ; # The Mellin Transformation and Fuchsian Type Partial Differential Equations
schema:issn "0169-507X" ;
schema:name "Mathematics and Its Applications (East European Series)," ;
.

Content-negotiable representations