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Borwein P, Erdelyi T Polynomials and polynomial inequalities
Borwein P, Erdelyi T  Polynomials and polynomial inequalities









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: Polynomials and polynomial inequalities

: Borwein P, Erdelyi T

:

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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: 01.04.2005

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$l_p$ norm      6 48 471
Abel, Niels Henrik      3 61
Algorithms      356371
Algorithms for Chebyshev polynomials      371
Algorithms for counting zeros      364370
Algorithms for polynomial evaluations      363
Algorithms for reversion of power series      362
Algorithms, evaluation of $x^n$      363
Algorithms, fast Fourier transform      359
Algorithms, fast polynomial division      362
Algorithms, fast polynomial expansion      363
Algorithms, fast polynomial multiplication      361
Algorithms, interpolation      360
Algorithms, Newtons method      362 364367
Algorithms, Remez      371
Algorithms, zero finding for polynomials      366370
Alternation set      93
Alternation theorem      94
Apolar polynomials      23 24 25
Arc length of algebraic polynomials      31
Arc length of trigonometric polynomials      35
Bermans formula      166
Bernstein factor      145 150 152 322328
Bernstein polynomials      163164
Bernstein Szego inequality      231 245 259 321323
Bernstein Szego inequality for entire functions of exponential type      245
Bernstein Szego inequality for rational functions on $\mathbb{R}$      323
Bernstein Szego inequality for rational functions on K      322
Bernstein Szego inequality for rational functions on [1,1]      322
Bernstein Szego inequality for trigonometric polynomials      232
Bernstein-type inequality for Chebyshev spaces      206
Bernstein-type inequality for constrained polynomials      420447
Bernstein-type inequality for entire functions of exponential type      245
Bernstein-type inequality for exponential sums      289
Bernstein-type inequality for generalized polynomials      392416
Bernstein-type inequality for generalized polynomials in $L_p$      401417
Bernstein-type inequality for higher derivatives      258
Bernstein-type inequality for nondense Miintz spaces      213 310
Bernstein-type inequality for polynomials      232233 390
Bernstein-type inequality for polynomials in $L_p$      235 390 401417
Bernstein-type inequality for products of Miintz spaces      317
Bernstein-type inequality for rational functions on $\mathbb{R}$      323 329
Bernstein-type inequality for rational functions on D      324
Bernstein-type inequality for rational functions on K      322 327
Bernstein-type inequality for rational functions on [1,1]      323 327
Bernstein-type inequality for self-reciprocal polynomials      339
Bernstein-type inequality for trigonometric polynomials      232
Bernstein-type inequality, bounded      178 182 213214
Bernstein-type inequality, unbounded      154 206217
Bernstein-type inequality, weighted      257
Bessels inequality      46
Best approximation      94
Best approximation to $x^n$      99
Best approximation to $x^{\lambda}$, a problem of Lorentz      108
Best approximation, uniqueness      98
Best approximationby rationals      99
Binomial coefficient      62
Blaschke product      190 324
Blumenthals theorem      78
Bombieris norm      274
Bounded Bernstein-type inequality      see Bernstein-type inequality
Bounded Chebyshev-type inequality      see Chebyshev-type inequality
Bounded linear functionals      50
Budan Fourier theorem      369
Cardano, Girolamo      3
Cartans lemma      350
Cauchy determinant      106
Cauchy indices      367
Cauchy Schwarz inequality      42
Cauchy Schwarz inequality for sequences      46
Cauchys integral formula      14
Cesaro means      165
Chebyshev constants      39
Chebyshev polynomials in Chebyshev spaces      114125
Chebyshev polynomials in rational spaces      see Chebyshev rationals
Chebyshev polynomials, algorithms for      371
Chebyshev polynomials, best approximation to $x^n$      30
Chebyshev polynomials, composition, characterization      33
Chebyshev polynomials, explicit formulas      30 32
Chebyshev polynomials, orthogonality      32
Chebyshev polynomials, reducibility      36
Chebyshev polynomials, second kind      37
Chebyshev polynomials, three-term recursion      32
Chebyshev polynomials, trigonometric on subintervals      235
Chebyshev polynomials, uniqueness      118
Chebyshev rationals      139153
Chebyshev rationals and orthogonality      147
Chebyshev rationals in algebraic rational spaces      142
Chebyshev rationals in trigonometric rational spaces      143
Chebyshev rationals of the first and second kind      141
Chebyshev rationals on the real line      151
Chebyshev rationals, derivative formulas      146
Chebyshev rationals, partial fraction representation      144
Chebyshev space      92
Chebyshev space, dimension on the circle      100
Chebyshev space, functions with prescribed sign changes      100
Chebyshev system      91100
Chebyshev system, extended complete      97
Chebyshev, P.L.      31
Chebyshev-type inequality for entire functions of exponential type      245
Chebyshev-type inequality, bounded      179 182
Chebyshev-type inequality, explicit bounds via Paley Weiner theorem      196
Chebyshevs inequality      235 390
Christoffel function      74
Christoffel function for Miintz spaces      195
Christoffel numbers      67
Christoffel Darboux formula      60
Coefficient bounds for polynomials in special bases      124
Coefficient bounds in nondense rational spaces      153
Coefficient bounds of Markov      248
Comparison theorem      103 120 122 183
Completeness      48 79
Complexity concerns      356371
Consecutive zeros of p'      26
Constrained polynomials      417447
Constrained polynomials, $L_p$ inequalities      422
Constrained polynomials, Bernstein-type inequality      420 425 427
Constrained polynomials, Markov-type inequality      417447
Constrained polynomials, Nikolskii-type      444
Constrained polynomials, Remez-type      443445
Constrained polynomials, Schur-type      436437
Cotes numbers      67
Cubic equations      4
de la Vallee Poussin theorem      99
Denseness      154226
Denseness of Markov spaces      206217
Denseness of Miintz polynomials      171205
Denseness of Miintz rationals      218226
Denseness of polynomials      154170
Derivatives of Markov systems      112
Descartes system      100113
Descartes system, examples      103
Descartes system, lexicographic properties      103
Descartes Rule of Signs      22 102
Divide and conquer      358
Division of polynomials      15 362
dAlembert, Jean le Rond      13
Elementary symmetric function      5
Enestrom Kakeya theorem      12
Erdos inequality      431
Erdos Turan inequality      435
Euler, Leonhard      13
Evaluation of $x^n$      363
Exponential sums      see Miintz polynomials
Exponential sums with nonnegative exponents      294
Exponential sums, a problem of Lorentz      291
Exponential sums, Bernstein-type inequality      291
Exponential sums, Markov-type inequality      294
Exponential sums, Nikolskii-type inequality      289
Exponential sums, Turans inequality      295
Exponential type      196 245
Extended complete Chebyshev system      97
Factor inequalities      260274
Factor inequalities, via Mahlers measure      271273
Factorization      10 36
Fast Fourier Transform      359
Favards theorem      73
Fejer gap      2728
Fejer operators      164
Fejers theorem      165
Fekete point      38
Fekete polynomial      38
Ferrari, Ludovico      3
Ferro, Scipione del      3
Fourier coefficient      53
Fourier series      53
Fundamental theorem of algebra      3
Gamma function      63
Gauss Jacobi quadrature      67 75
Gauss Lucas Theorem      18
Gauss, Carl Friedrich      13
Gaussian hypergeometric series      62
Gegenbauer polynomials      65
Generalized polynomials, $L_p$ inequalities      401407
Generalized polynomials, Bernstein-type inequality      399 407
Generalized polynomials, Markov-type inequality      399 407
Generalized polynomials, Nikolskii-type inequality      394395
Generalized polynomials, Remez-type inequality      393394 414
Generalized polynomials, Schur-type inequality      395
Generalized polynomials, weighted inequalites      407
Girard, Albert      13
Graces complex version of Rolles theorem      25
Graces theorem      18
Gram Schmidt      44
Grams lemma      176
Haar space      92
Haar system      92
Halleys method      365
Hardy space      189
Hellys convergence theorem      71
Hellys selection theorem      71
Hermite interpolation      9
Hermite polynomials      57
Hermite polynomials, explicit formulas      65
Hilbert space      42
Holders Inequality      17 49
Horners rule      8
Hypergeometric differential equation      63
Hypergeometric functions      62
Identity Theorem      15
Inequalities for Miintz polynomials      see Miintz polynomials
Inequalities, Bernstein Szego inequality      see Bernstein Szego inequality
Inequalities, Bernstein-type      see Bernstein-type inequality
Inequalities, Bessels      see Bessels inequality
Inequalities, bounded Bernstein-type      see Bounded Bernstein-type inequality
Inequalities, bounded Chebyshev-type      see Bounded Chebyshev-type inequality
Inequalities, Cauchy Schwarz      see Cauchy Schwarz inequality
Inequalities, Chebyshev-type      see Chebyshev-type inequality
Inequalities, for factors      see Factor inequalities
Inequalities, Holders      see Holders inequality
Inequalities, Lax-type      see Lax
Inequalities, Markov-type      see Markov-type inequality
Inequalities, metric      see Metric inequalities
Inequalities, Minkowskis      see Minkowskis inequality
Inequalities, Nikolskii-type      see Nikolskii-type inequality
Inequalities, Remez-type      see Remez-type inequality
Inequalities, Russaks      see Russak
Inequalities, Schur-type      see Schur-type inequality
Inequalities, triangle      see Triangle inequality
Inequalities, unbounded Bernstein-type      see Unbounded Bernstein-type inequality
Inner product      41
Inner product space      41
Integer-valued polynomials      10
Interpolation, Hermite      see Hermite interpolation
Interpolation, Lagrange      see Lagrange interpolation
Interpolation, Newton      see Newton interpolation
Jacobi polynomials      57 63
Jacobi polynomials, explicit formulas      63
Jensen circles      19
Jensens formula      187
Jensens Inequality      414
Jensens theorem      19
Kernel function      47 132
Kolmogorovs inequality      285
Korovkins theorems      163
Lacunary spaces      308
Lacunary spaces, quasi-Chebyshev polynomials      316
Lagrange interpolation      8
Laguerre polynomials      57 66 130
Laguerre polynomials, explicit formulas      66
Laguerres Theorem      20
Lax-type inequality      438
Lax-type inequality for rationals      329
Lax-type inequality on a half-plane      338
Lax-type inequality, Maliks extension      438
Legendre polynomials      57
Lemniscates of constant modulus      352353
Lexicographic properties      116
Lexicographic properties for Miintz polynomials      120 314
Lexicographic properties for Muntz Legendre polynomials      136
Lexicographic properties for sinh systems      122
Liouvilles theorem      15
Logarithmic capacity      38
Lorentz degree      82
Lorentz degree for polynomials      86
Lorentz degree for trigonometric polynomials      89
Lorentzs problem      108 291
Lucas Theorem      18
m-distribution      57
Mahlers measure      271
Mairhuber theorem      98
Markov system      100
Markov system, closure of nondense      211
Markov system, derivative of      112
Markov Stieltjes inequality      76
Markov-type inequality for $\mathcal{P}_n$      233
Markov-type inequality for $\mathcal{P}_n^c$      255
Markov-type inequality for constrained polynomials      417447
Markov-type inequality for constrained polynomials in $L_p$      422 428429
Markov-type inequality for exponential sums      276280 294295
Markov-type inequality for generalized polynomials      399407 445
Markov-type inequality for generalized polynomials in $L_p$      401407
Markov-type inequality for higher derivatives      248260
Markov-type inequality for monotone polynomials      439441
Markov-type inequality for Muntz polynomials      276279 287288
Markov-type inequality for Muntz polynomials in $L_p$      279280
Markov-type inequality for nonnegative polynomials      420 439
Markov-type inequality for rational functions      336
Markov-type inequality for self-reciprocal polynomials      339
Markov-type inequality for trigonometric polynomials on subintervals      242245
Markov-type inequality in the complex plane      235
Markov-type inequality, weighted      442443
Maximum principal      15
Mergelyans theorem      170
Mesh of zeros      155
Metric inequalities      344355
Metric inequalities for polynomials      345346
Metric inequalities for rational functions      347349
Metric inequalities in $L_p$, $p \leq 1$      52
Minkowskis inequality in $L_p$, $p \geq 1$      49
moment      57
Moment problem      70
Monotone operator theorem      163
Multiplication of polynomials      361
Muntz polynomials, bound for smallest zero      313
Muntz polynomials, bounded Bernstein-type inequality      178 182 213 310 317
Muntz polynomials, lexicographic properties of zeros      116 120
Muntz polynomials, Newmans inequality      276 301
Muntz polynomials, Newmans inequality in $L_p$      279
Muntz polynomials, Newmans problem      317
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