Orthogonal polynomials of several variables:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge [u.a.]
Cambridge University Press
2001
|
| Ausgabe: | 1. publication |
| Schriftenreihe: | Encyclopedia of Mathematics and Its Applications
81 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XV, 390 Seiten |
| ISBN: | 0521800439 |
Internformat
MARC
| LEADER | 00000nam a2200000 cb4500 | ||
|---|---|---|---|
| 001 | BV013750537 | ||
| 003 | DE-604 | ||
| 005 | 20210520 | ||
| 007 | t | ||
| 008 | 010530s2001 |||| 00||| eng d | ||
| 020 | |a 0521800439 |9 0-521-80043-9 | ||
| 035 | |a (OCoLC)247746764 | ||
| 035 | |a (DE-599)BVBBV013750537 | ||
| 040 | |a DE-604 |b ger |e rakwb | ||
| 041 | 0 | |a eng | |
| 049 | |a DE-29T |a DE-355 |a DE-703 |a DE-824 |a DE-739 |a DE-11 |a DE-188 | ||
| 050 | 0 | |a QA404.5 | |
| 082 | 0 | |a 515.55 | |
| 084 | |a SK 470 |0 (DE-625)143241: |2 rvk | ||
| 084 | |a SK 680 |0 (DE-625)143252: |2 rvk | ||
| 100 | 1 | |a Dunkl, Charles F. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Orthogonal polynomials of several variables |c Charles F. Dunkl, University of Virginia, Yuan Xu, University of Oregon |
| 250 | |a 1. publication | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge University Press |c 2001 | |
| 300 | |a XV, 390 Seiten | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Encyclopedia of Mathematics and Its Applications |v 81 | |
| 650 | 4 | |a Orthogonale Polynome - Mehrere Variable | |
| 650 | 4 | |a Functions of several real variables | |
| 650 | 4 | |a Orthogonal polynomials | |
| 650 | 0 | 7 | |a Orthogonale Polynome |0 (DE-588)4172863-4 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Mehrere Variable |0 (DE-588)4277015-4 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Orthogonale Polynome |0 (DE-588)4172863-4 |D s |
| 689 | 0 | 1 | |a Mehrere Variable |0 (DE-588)4277015-4 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Xu, Yuan |e Verfasser |4 aut | |
| 830 | 0 | |a Encyclopedia of Mathematics and Its Applications |v 81 |w (DE-604)BV000903719 |9 81 | |
| 856 | 4 | 2 | |m GBV Datenaustausch |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009399414&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-009399414 | ||
Datensatz im Suchindex
| _version_ | 1804128570328481792 |
|---|---|
| adam_text | IMAGE 1
ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS
ORTHOGONAL POLYNOMIALS OF
SEVERAL VARIABLES
CHARLES F. DUNKL UNIVERSITY OF VIRGINIA
YUAN XU
UNIVERSITY OF OREGON
C A M B R I D GE UNIVERSITY PRESS
IMAGE 2
CONTENTS
PREFACE PAGE XIII
1 BACKGROUND 1
1.1 THE GAMMA AND BETA FUNCTIONS 1
1.2 HYPERGEOMETRIC SERIES 3
1.3 ORTHOGONAL POLYNOMIALS OF ONE VARIABLE 7
1.3.1 GENERAL PROPERTIES 7
1.3.2 THREE TERM RECURRENCE 9
1.4 CLASSICAL ORTHOGONAL POLYNOMIALS 13
1.4.1 HERMITE POLYNOMIALS 14
1.4.2 LAGUERRE POLYNOMIALS 15
1.4.3 GEGENBAUER POLYNOMIALS 17
1.4.4 JACOBI POLYNOMIALS 21
1.5 MODIFIED CLASSICAL POLYNOMIALS 23
1.5.1 GENERALIZED HERMITE POLYNOMIALS 25
1.5.2 GENERALIZED GEGENBAUER POLYNOMIALS 26
1.5.3 A LIMITING RELATION 28
1.6 NOTES 28
2 EXAMPLES OF ORTHOGONAL POLYNOMIALS IN SEVERAL VARIABLES 30
2.1 NOTATION AND PRELIMINARY 30
2.2 SPHERICAL HARMONICS 33
2.3 CLASSICAL ORTHOGONAL POLYNOMIALS 37
2.3.1 MULTIPLE JACOBI POLYNOMIALS ON THE CUBE 37
2.3.2 CLASSICAL ORTHOGONAL POLYNOMIALS ON THE UNIT BALL 38
2.3.3 CLASSICAL ORTHOGONAL POLYNOMIALS ON THE SIMPLEX 46
2.3.4 MULTIPLE HERMITE POLYNOMIALS ON R D 49
2.3.5 MULTIPLE LAGUERRE POLYNOMIALS ON IR+ 51
VN
*
IMAGE 3
VIII
CONTENTS
2.4 OTHER EXAMPLES OF ORTHOGONAL POLYNOMIALS 51
2.4.1 TWO GENERAL FAMILIES OF ORTHOGONAL POLYNOMIALS 51
2.4.2 A METHOD FOR GENERATING ORTHOGONAL POLYNOMIALS OF TWO VARIABLES 55
2.4.3 DISC POLYNOMIALS 57
2.5 VAN DER CORPUT-SCHAAKE INEQUALITY 58
2.6 NOTES 60
3 GENERAL PROPERTIES OF ORTHOGONAL POLYNOMIALS IN SEVERAL VARIABLES 63
3.1 MOMENT FUNCTIONALS AND ORTHOGONAL POLYNOMIALS IN SEVERAL VARIABLES
64
3.1.1 DEFINITION OF ORTHOGONAL POLYNOMIALS 64
3.1.2 ORTHOGONAL POLYNOMIALS AND MOMENT MATRICES 69
3.1.3 THE MOMENT PROBLEM 72
3.2 THE THREE TERM RELATION 75
3.2.1 DEFINITION AND BASIC PROPERTIES 75
3.2.2 FAVARD S THEOREM 79
3.2.3 CENTRALLY SYMMETRIE INTEGRALS 82
3.2.4 EXAMPLES 85
3.3 JACOBI MATRICES AND COMMUTING OPERATORS 88
3.4 FURTHER PROPERTIES OF THE THREE TERM RELATION 95
3.4.1 RECURRENCE FORMULA 95
3.4.2 GENERAL SOLUTIONS OF THE THREE-TERM RELATION 102
3.5 REPRODUCING KERNELS AND FOURIER ORTHOGONAL SERIES 105
3.5.1 REPRODUCING KERNEIS 106
3.5.2 FOURIER ORTHOGONAL SERIES 110
3.6 COMMON ZEROS OF ORTHOGONAL POLYNOMIALS IN SEVERAL VARIABLES 113
3.7 GAUSSIAN CUBATURE FORMULAE 117
3.7.1 CHARACTERIZATION OF GAUSSIAN CUBATURE FORMULAE 118
3.7.2 EXAMPLES OF GAUSSIAN CUBATURE FORMULAE 123
3.8 ORTHOGONAL POLYNOMIALS ON THE UNIT SPHERE 126
3.8.1 ORTHOGONAL STRUETURES ON S D AND ON B D 126
3.8.2 ORTHOGONAL STRUETURE ON B D AND ON S D+M 132
3.9 NOTES 134
4 ROOT SYSTEMS AND COXETER GROUPS 137
4.1 INTRODUCTION AND OVERVIEW 137
4.2 ROOT SYSTEMS 139
4.2.1 T Y P E A I -I 142
IMAGE 4
CONTENTS
IX
4.2.2 T Y P E SS D 143
4.2.3 TYPE J 2 (M) 144
4.2.4 T Y P E D D 144
4.2.5 TYPE ^3 145
4.2.6 T Y P E F 4 146
4.2.7 OTHERTYPES 146
4.2.8 MISCELLANEOUS RESULTS 146
4.3 INVARIANT POLYNOMIALS 147
4.3.1 TYPE AD-I INVARIANTS 149
4.3.2 TYPE B D INVARIANTS 150
4.3.3 TYPE D D INVARIANTS 150
4.3.4 TYPE J 2 (M) INVARIANTS 151
4.3.5 TYPE H3 INVARIANTS 151
4.3.6 TYPE F 4 INVARIANTS 151
4.4 DIFFERENTIAL-DIFFERENCE OPERATORS 151
4.5 THE INTERTWINING OPERATOR 157
4.6 THE K- ANALOG OF THE EXPONENTIAL 167
4.7 INVARIANT DIFFERENTIAL OPERATORS 169
4.8 NOTES 173
5 SPHERICAL HARMONICS ASSOCIATED WITH REFLECTION GROUPS 175
5.1 FT-HARMONIC POLYNOMIALS 175
5.2 INNER PRODUCTS ON POLYNOMIALS 185
5.3 REPRODUCING KERNELS AND THE POISSON KERNEL 189
5.4 INTEGRATION OF THE INTERTWINING OPERATOR 192
5.5 EXAMPLE: ABELIAN GROUP Z$ 197
5.6 EXAMPLE: DIHEDRAL GROUPS 205
5.6.1 AN ORTHONORMAL BASIS OF H N (H^ SS) 206
5.6.2 CAUCHY AND POISSON KERNEIS 214
5.7 THE FOURIER TRANSFORM 216
5.8 NOTES 224
6 CLASSICAL AND GENERALIZED CLASSICAL ORTHOGONAL POLYNOMIALS 225
6.1 GENERALIZED CLASSICAL ORTHOGONAL POLYNOMIALS ON THE BALL 225 6.1.1
DEFINITION AND DIFFERENTIAL-DIFFERENCE EQUATIONS 225
6.1.2 BASES, REPRODUCING KERNEIS, AND THE FUNK-HECKE FORMULA 231 6.2
ORTHOGONAL POLYNOMIALS ON THE SIMPLEX 236
6.2.1 GENERAL WEIGHT FUNCTIONS ON T D 236
6.2.2 GENERALIZED CLASSICAL ORTHOGONAL POLYNOMIALS 239
*
IMAGE 5
X CONTENTS
6.3 GENERALIZED HERMITE POLYNOMIALS 243
6.4 GENERALIZED LAGUERRE POLYNOMIALS 249
6.5 NOTES 253
7 SUMMABILITY OF ORTHOGONAL EXPANSIONS 255
7.1 GENERAL RESULTS ON ORTHOGONAL EXPANSIONS 255
7.1.1 UNIFORM CONVERGENCE OF PARTIAL SUMS 255
7.1.2 CESAERO MEANS OF THE ORTHOGONAL EXPANSION 259
7.2 ORTHOGONAL EXPANSION ON THE SPHERE 262
7.3 ORTHOGONAL EXPANSION ON THE BALL 266
7.4 ORTHOGONAL EXPANSIONS ON THE SIMPLEX 271
7.5 ORTHOGONAL EXPANSION OF LAGUERRE AND HERMITE POLYNOMIALS 274
7.6 MULTIPLE JACOBI EXPANSION 279
7.7 NOTES 284
8 ORTHOGONAL POLYNOMIALS ASSOCIATED WITH SYMMETRIE GROUPS 287
8.1 INTRODUCTION 287
8.2 PARTITIONS, COMPOSITIONS AND ORDERINGS 288
8.3 COMMUTING SELF-ADJOINT OPERATORS 289
8.4 THE DUAL POLYNOMIAL BASIS 292
8.5 SD INVARIANT SUBSPACES 299
8.6 DEGREE CHANGING RECURRENCES 304
8.7 NORM FORMULAE 308
8.7.1 HOOK LENGTH PRODUETS AND THE PAIRING NORM 308
8.7.2 THE BIORTHOGONAL TYPE NORM 311
8.7.3 THE TORUS INNER PRODUET 313
8.7.4 NORMALIZING CONSTANTS 316
8.8 SYMMETRIE FUNCTIONS AND JACK POLYNOMIALS 321
8.9 MISCELLANEOUS TOPICS 330
8.10 NOTES 335
9 ORTHOGONAL POLYNOMIALS ASSOCIATED WITH OCTAHEDRAL GROUPS AND
APPLICATIONS 337
9.1 INTRODUCTION 337
9.2 OPERATORS OF TYPE B 338
9.3 POLYNOMIAL EIGENFUNCTIONS OF TYPE B 341
9.4 GENERALIZED BINOMIAL COEFFICIENTS 350
9.5 HERMITE POLYNOMIALS OF TYPE B 359
9.6 CALOGERO-SUTHERLAND SYSTEMS 360
IMAGE 6
CONTENTS XI
9.6.1 THE SIMPLE HARMONIC OSCILLATOR
9.6.2 ROOT SYSTEMS AND THE LAPLACIAN 9.6.3 TYPE A MODEIS ON THE LINE
9.6.4 TYPE A MODEIS ON THE CIRCLE 9.6.5 TYPE B MODEIS ON THE LINE 9.7
NOTES
BIBLIOGRAPHY AUTHOR INDEX SYMBOL INDEX SUBJECT INDEX
361 362 363 365 368
370 372 384
387 389
|
| any_adam_object | 1 |
| author | Dunkl, Charles F. Xu, Yuan |
| author_facet | Dunkl, Charles F. Xu, Yuan |
| author_role | aut aut |
| author_sort | Dunkl, Charles F. |
| author_variant | c f d cf cfd y x yx |
| building | Verbundindex |
| bvnumber | BV013750537 |
| callnumber-first | Q - Science |
| callnumber-label | QA404 |
| callnumber-raw | QA404.5 |
| callnumber-search | QA404.5 |
| callnumber-sort | QA 3404.5 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 470 SK 680 |
| ctrlnum | (OCoLC)247746764 (DE-599)BVBBV013750537 |
| dewey-full | 515.55 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.55 |
| dewey-search | 515.55 |
| dewey-sort | 3515.55 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 1. publication |
| format | Book |
| fullrecord | <?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>01868nam a2200457 cb4500</leader><controlfield tag="001">BV013750537</controlfield><controlfield tag="003">DE-604</controlfield><controlfield tag="005">20210520 </controlfield><controlfield tag="007">t</controlfield><controlfield tag="008">010530s2001 |||| 00||| eng d</controlfield><datafield tag="020" ind1=" " ind2=" "><subfield code="a">0521800439</subfield><subfield code="9">0-521-80043-9</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(OCoLC)247746764</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-599)BVBBV013750537</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-604</subfield><subfield code="b">ger</subfield><subfield code="e">rakwb</subfield></datafield><datafield tag="041" ind1="0" ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="049" ind1=" " ind2=" "><subfield code="a">DE-29T</subfield><subfield code="a">DE-355</subfield><subfield code="a">DE-703</subfield><subfield code="a">DE-824</subfield><subfield code="a">DE-739</subfield><subfield code="a">DE-11</subfield><subfield code="a">DE-188</subfield></datafield><datafield tag="050" ind1=" " ind2="0"><subfield code="a">QA404.5</subfield></datafield><datafield tag="082" ind1="0" ind2=" "><subfield code="a">515.55</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">SK 470</subfield><subfield code="0">(DE-625)143241:</subfield><subfield code="2">rvk</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">SK 680</subfield><subfield code="0">(DE-625)143252:</subfield><subfield code="2">rvk</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Dunkl, Charles F.</subfield><subfield code="e">Verfasser</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Orthogonal polynomials of several variables</subfield><subfield code="c">Charles F. Dunkl, University of Virginia, Yuan Xu, University of Oregon</subfield></datafield><datafield tag="250" ind1=" " ind2=" "><subfield code="a">1. publication</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="a">Cambridge [u.a.]</subfield><subfield code="b">Cambridge University Press</subfield><subfield code="c">2001</subfield></datafield><datafield tag="300" ind1=" " ind2=" "><subfield code="a">XV, 390 Seiten</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="b">n</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="b">nc</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="490" ind1="1" ind2=" "><subfield code="a">Encyclopedia of Mathematics and Its Applications</subfield><subfield code="v">81</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Orthogonale Polynome - Mehrere Variable</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Functions of several real variables</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Orthogonal polynomials</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Orthogonale Polynome</subfield><subfield code="0">(DE-588)4172863-4</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Mehrere Variable</subfield><subfield code="0">(DE-588)4277015-4</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="689" ind1="0" ind2="0"><subfield code="a">Orthogonale Polynome</subfield><subfield code="0">(DE-588)4172863-4</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="0" ind2="1"><subfield code="a">Mehrere Variable</subfield><subfield code="0">(DE-588)4277015-4</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="0" ind2=" "><subfield code="5">DE-604</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Xu, Yuan</subfield><subfield code="e">Verfasser</subfield><subfield code="4">aut</subfield></datafield><datafield tag="830" ind1=" " ind2="0"><subfield code="a">Encyclopedia of Mathematics and Its Applications</subfield><subfield code="v">81</subfield><subfield code="w">(DE-604)BV000903719</subfield><subfield code="9">81</subfield></datafield><datafield tag="856" ind1="4" ind2="2"><subfield code="m">GBV Datenaustausch</subfield><subfield code="q">application/pdf</subfield><subfield code="u">http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009399414&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA</subfield><subfield code="3">Inhaltsverzeichnis</subfield></datafield><datafield tag="999" ind1=" " ind2=" "><subfield code="a">oai:aleph.bib-bvb.de:BVB01-009399414</subfield></datafield></record></collection> |
| id | DE-604.BV013750537 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T18:51:20Z |
| institution | BVB |
| isbn | 0521800439 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009399414 |
| oclc_num | 247746764 |
| open_access_boolean | |
| owner | DE-29T DE-355 DE-BY-UBR DE-703 DE-824 DE-739 DE-11 DE-188 |
| owner_facet | DE-29T DE-355 DE-BY-UBR DE-703 DE-824 DE-739 DE-11 DE-188 |
| physical | XV, 390 Seiten |
| publishDate | 2001 |
| publishDateSearch | 2001 |
| publishDateSort | 2001 |
| publisher | Cambridge University Press |
| record_format | marc |
| series | Encyclopedia of Mathematics and Its Applications |
| series2 | Encyclopedia of Mathematics and Its Applications |
| spelling | Dunkl, Charles F. Verfasser aut Orthogonal polynomials of several variables Charles F. Dunkl, University of Virginia, Yuan Xu, University of Oregon 1. publication Cambridge [u.a.] Cambridge University Press 2001 XV, 390 Seiten txt rdacontent n rdamedia nc rdacarrier Encyclopedia of Mathematics and Its Applications 81 Orthogonale Polynome - Mehrere Variable Functions of several real variables Orthogonal polynomials Orthogonale Polynome (DE-588)4172863-4 gnd rswk-swf Mehrere Variable (DE-588)4277015-4 gnd rswk-swf Orthogonale Polynome (DE-588)4172863-4 s Mehrere Variable (DE-588)4277015-4 s DE-604 Xu, Yuan Verfasser aut Encyclopedia of Mathematics and Its Applications 81 (DE-604)BV000903719 81 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009399414&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Dunkl, Charles F. Xu, Yuan Orthogonal polynomials of several variables Encyclopedia of Mathematics and Its Applications Orthogonale Polynome - Mehrere Variable Functions of several real variables Orthogonal polynomials Orthogonale Polynome (DE-588)4172863-4 gnd Mehrere Variable (DE-588)4277015-4 gnd |
| subject_GND | (DE-588)4172863-4 (DE-588)4277015-4 |
| title | Orthogonal polynomials of several variables |
| title_auth | Orthogonal polynomials of several variables |
| title_exact_search | Orthogonal polynomials of several variables |
| title_full | Orthogonal polynomials of several variables Charles F. Dunkl, University of Virginia, Yuan Xu, University of Oregon |
| title_fullStr | Orthogonal polynomials of several variables Charles F. Dunkl, University of Virginia, Yuan Xu, University of Oregon |
| title_full_unstemmed | Orthogonal polynomials of several variables Charles F. Dunkl, University of Virginia, Yuan Xu, University of Oregon |
| title_short | Orthogonal polynomials of several variables |
| title_sort | orthogonal polynomials of several variables |
| topic | Orthogonale Polynome - Mehrere Variable Functions of several real variables Orthogonal polynomials Orthogonale Polynome (DE-588)4172863-4 gnd Mehrere Variable (DE-588)4277015-4 gnd |
| topic_facet | Orthogonale Polynome - Mehrere Variable Functions of several real variables Orthogonal polynomials Orthogonale Polynome Mehrere Variable |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009399414&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000903719 |
| work_keys_str_mv | AT dunklcharlesf orthogonalpolynomialsofseveralvariables AT xuyuan orthogonalpolynomialsofseveralvariables |