Borwein P, Erdelyi T — Polynomials and polynomial inequalities :: Электронная библиотека попечительского совета мехмата МГУ

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Borwein P, Erdelyi T — Polynomials and polynomial inequalities

Borwein P, Erdelyi T — Polynomials and polynomial inequalities

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Название: Polynomials and polynomial inequalities

Авторы: Borwein P, Erdelyi T

Аннотация:

Polynomials pervade mathematics, virtually every branch of mathematics from algebraic number theory and algebraic geometry to applied analysis and computer science, has a corpus of theory arising from polynomials. The material explored in this book primarily concerns polynomials as they arise in analysis; it focuses on polynomials and rational functions of a single variable. The book is self-contained and assumes at most a senior-undergraduate familiarity with real and complex analysis. After an introduction to the geometry of polynomials and a discussion of refinements of the Fundamental Theorem of Algebra, the book turns to a consideration of various special polynomials. Chebyshev and Descartes systems are then introduced, and Müntz systems and rational systems are examined in detail. Subsequent chapters discuss denseness questions and the inequalities satisfied by polynomials and rational functions. Appendices on algorithms and computational concerns, on the interpolation theorem, and on orthogonality and irrationality conclude the book.

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Рубрика: Математика/Анализ/Продвинутый анализ/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1995

Количество страниц: 473

Добавлена в каталог: 01.04.2005

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Предметный указатель

Muntz polynomials, Nikolskii-type inequality      281 298 317
Muntz polynomials, positive zeros      22
Muntz polynomials, Remez-type inequality      304 307 316
Muntz polynomials, where the sup norm lives      301
Muntz rationals      218—226
Muntz rationals, denseness      218
Muntz space      125—126
Muntz space, lexicographic properties of zeros      120
Muntz space, products of      222 316
Muntz space, quotients of      218—226
Muntz system      125—136
Muntz system, closure      171—205
Muntz system, nondense      303—319
Muntz — Legendre polynomials      125—138
Muntz — Legendre polynomials, definition      126
Muntz — Legendre polynomials, differential recursion      129
Muntz — Legendre polynomials, global estimate of zeros      136
Muntz — Legendre polynomials, integral recursion      132
Muntz — Legendre polynomials, lexicographic properties of zeros      133—136
Muntz — Legendre polynomials, orthogonality      127 132
Muntz — Legendre polynomials, orthonormality      128
Muntz — Legendre polynomials, Rodrigues-type formula      128 131
Muntz — Legendre polynomials, zeros of      133
Muntz’s theorem      171—205
Muntz’s theorem in       171
Muntz’s theorem in       311
Muntz’s theorem in       172
Muntz’s theorem in       186
Muntz’s theorem in C[0,1]      171
Muntz’s theorem in C[a, b]      180 184
Muntz’s theorem on sets of positive measure      303
Muntz’s theorem, another proof      176 192
Muntz’s theorem, closure of span in      178 181 185
Newman’s conjecture on denseness of products      316
Newman’s conjecture on denseness of quotients      220 223
Newman’s inequality      275—279
Newman’s inequality for Muntz polynomials      276
Newman’s inequality for Muntz polynomials in       279
Newman’s inequality on positive intervals      301
Newman’s inequality, an improvement      287
Newton Interpolation      10
Newton’s identities      5
Newton’s method      362 364—367
Newton’s method for $x^{1/2}$       365
Newton’s method in many variables      366
Nikolskii-type inequality for constrained polynomials      444
Nikolskii-type inequality for exponential sums      289
Nikolskii-type inequality for generalized polynomials      394 395
Nikolskii-type inequality for Muntz polynomials      281 298 317
Nikolskii-type inequality for products of Muntz spaces      317
Nondense Muntz spaces      303—319
Nonnegative polynomials      70 85 417 420
Nonnegative trigonometric polynomials      85 409
Norms      471
Norms,       see “ $L_p$

$L_p$

norm”
Norms, supremum      see “Supremum norm”
Orthogonal collection      43
Orthogonal functions      41—56
Orthogonal polynomials      57—79
Orthogonal polynomials as continued fractions      79
Orthogonal polynomials as determinants      76
Orthogonal polynomials, characterization of compact support      77
Orthogonal polynomials, Gegenbauer      see “Gegenbauer polynomials”
Orthogonal polynomials, Hermite      see “Hermite polynomials”
Orthogonal polynomials, interlacing of zeros      61
Orthogonal polynomials, Jacobi      see “Jacobi polynomials”
Orthogonal polynomials, Laguerre      see “Laguerre polynomials”
Orthogonal polynomials, Legendre      see “Legendre polynomials”
Orthogonal polynomials, Muntz — Legendre      see “Muntz — Legendre polynomials”
Orthogonal polynomials, simple real zeros      61
Orthogonal polynomials, ultraspherical      see “Ultraspherical polynomials”
Orthogonal rational functions      147
Orthonormal set      43
Paley — Weiner theorem      196
Parallelogram law      42
Parseval’s identity      48
Partial fraction decomposition      7 144
Pellet’s theorem      16
Polar derivative      20
Polynomials as sums of squares      85 348
Polynomials in $x^{\lambda_n}$       167
Polynomials with integer coefficients      169
Polynomials with nonnegative coefficients      79—90 417
Polynomials with real roots      345 347—348
Polynomials, Bernstein      see “Bernstein polynomials”
Polynomials, Chebyshev      see “Chebyshev polynomials”
Polynomials, Gegenbauer      see “Gegenbauer polynomials”
Polynomials, generalized      see “Generalized polynomials”
Polynomials, growth in the complex plane      239
Polynomials, Hermite      see “Hermite polynomials”
Polynomials, integer valued      see “Integer valued polynomials”
Polynomials, Jacobi      see “Jacobi polynomials”
Polynomials, Laguerre      see “Laguerre polynomials”
Polynomials, Legendre      see “Legendre polynomials”
Polynomials, Muntz      see “Muntz polynomial”
Polynomials, Muntz — Legendre      see “Muntz — Legendre polynomials”
Polynomials, number of real roots      17 137
Polynomials, symmetric      5
Polynomials, trigonometric      see “Trigonometric polynomial”
Polynomials, ultraspherical      see “Ultraspherical polynomials”
Products of Muntz spaces      222 316
Quadratic equations      4

Quartic equations      4
Quasi-Chebyshev polynomials      316 342
Railway track theorem      98
Rakhmanov’s theorem      78
Rational functions, algebraic      139
Rational functions, Chebyshev polynomials of      139—153
Rational functions, coefficient bounds      153
Rational functions, inequalities      see “Inequalities”
Rational functions, trigonometric      139
Rational spaces of algebraic rational functions      139—153 320—321
Rational spaces of trigonometric rational functions      139—153 320—321
Recursive bounds      359
Remez-type inequality for algebraic polynomials      228
Remez-type inequality for constrained polynomials      443 445
Remez-type inequality for generalized polynomials      393 394
Remez-type inequality for generalized polynomials in       401—402
Remez-type inequality for Muntz spaces      307
Remez-type inequality for nondense Muntz spaces      304
Remez-type inequality for products of Muntz spaces      316
Remez-type inequality for trigonometric polynomials      230
Remez-type inequality, pointwise      414
Remez’s algorithm      371
Reproducing kernel      47 132
Reversion of power series      362
Riemann — Lebesgue lemma      54
Riesz representation theorem      50
Riesz — Fischer theorem      50
Riesz’s identity      390
Riesz’s lemma      237
Rising factorial      62
Rolle’s Theorem      25
Rouche’s Theorem      14 16
Russak’s inequalities      336
Salem numbers      6
Schur-type inequality for algebraic polynomials      233
Schur-type inequality for constrained polynomials      436—437
Schur-type inequality for generalized polynomials      395
Schur-type inequality for rational functions      337
Schur-type inequality for trigonometric polynomials      238
Schur’s Theorem      17
Self-reciprocal polynomials      339
Self-reciprocal polynomials, quasi-Chebyshev polynomials      342
Somorjai’s theorem      218
Space, Chebyshev      see “Chebyshev space”
Space, Descartes      see “Descartes space”
Space, Haar      see “Haar space”
Space, Muntz      see “Muntz space”
Space, rational      see “Rational space”
Stieltjes’ theorem      78
Stone — Weierstrass theorem      161
Sums of squares of polynomials      85 348
Supremum norm      6 29 471
Symmetric function      5
Symmetric polynomial      5
Szego’s inequality      391
Szego’s theorem      23 235
Tartaglia, Niccolo      3
Tchebychev      see “Chebyshev”
Three-term recursion      59
Totally positive kernels      110
Transfinite diameter      38
Triangle inequality      42
Trigonometric polynomial      2
Trigonometric polynomials of longest arc length      35
Turan’s inequality      434
Turan’s inequality for exponential sums      295
Ultraspherical polynomials      65
Unbounded Bernstein-type inequality      206—217
Unbounded Bernstein-type inequality, characterization of denseness      207
Unicity theorem      15
Vandermonde determinant      38 103
Variation diminishing property      111
Videnskii’s inequalities      242—245
Walsh’s two circle theorem      20
Weierstrass’ theorem      154—170
Weierstrass’ theorem for Markov systems      155
Weierstrass’ theorem for polynomials      159
Weierstrass’ theorem for polynomials in $x^{\lambda}$       167
Weierstrass’ theorem for polynomials with integer coefficients      169
Weierstrass’ theorem for trigonometric polynomials      165
Weierstrass’ theorem in       169
Weierstrass’ theorem on arcs      170
Weierstrass’ theorem, Stone — Weierstrass theorem      161
wronskian      22
zeros      11—18
Zeros in a disk      369
Zeros in an interval      368
Zeros in Chebyshev spaces      99
Zeros of Chebyshev polynomials      34 116 120 122
Zeros of derivatives of polynomials      18—28
Zeros of integrals of polynomials      24
Zeros of Muntz polynomials      120
Zeros of Muntz-Legendre polynomials      133—136
Zeros of orthogonal polynomials      61
Zeros, algorithms for finding      364—367
Zeros, complexity of      367
Zeros, counting by winding number      370
Zeros, localizing      367
Zeros, maximum number at one      137
Zeros, maximum number of positive      17
Zolotarev      35
Zoomers      218

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