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Equioscillation theorem

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The equioscillation theorem is also valid when polynomials are replaced by rational functions: among all rational functions whose numerator has degree
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Notes on how to prove Chebyshev’s equioscillation theorem
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The Chebyshev Equioscillation Theorem by Robert Mayans
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when the merit function is the maximum difference (
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Its discovery is attributed to 14: 651:{\displaystyle \|f-g\|_{\infty }} 204:{\displaystyle \|f-g\|_{\infty }} 1036: 923:minimax approximation algorithms 852: 842: 833: 820: 811: 798: 378: 368: 359: 346: 337: 324: 94: 82: 75:be a continuous function from 1: 1052:. You can help Knowledge by 1009: (archived July 2, 2011) 122:{\displaystyle \mathbb {R} } 1114:Mathematical analysis stubs 470:and denominator has degree 1130: 1099:Theorems about polynomials 1031: 658:if and only if there are 493:, the rational function 211:if and only if there are 951:Golomb, Michael (1962). 906:{\displaystyle \sigma } 432:{\displaystyle \sigma } 25:equioscillation theorem 1048:–related article is a 907: 887: 782: 705: 652: 613: 612:{\displaystyle m-\mu } 587: 586:{\displaystyle n-\nu } 561: 541: 521: 487: 486:{\displaystyle \leq m} 464: 463:{\displaystyle \leq n} 433: 413: 308: 231: 205: 166: 146: 145:{\displaystyle \leq n} 123: 101: 69: 1046:mathematical analysis 908: 888: 783: 706: 653: 614: 588: 562: 542: 522: 520:{\displaystyle g=p/q} 488: 465: 434: 414: 309: 232: 206: 167: 147: 124: 102: 70: 1109:Theorems in analysis 913:is either -1 or +1. 897: 792: 715: 662: 623: 597: 571: 551: 531: 497: 474: 451: 439:is either -1 or +1. 423: 318: 241: 215: 176: 156: 133: 111: 79: 59: 33:continuous functions 230:{\displaystyle n+2} 1104:Numerical analysis 903: 883: 778: 701: 648: 609: 583: 557: 537: 517: 483: 460: 429: 409: 304: 227: 201: 162: 142: 119: 97: 65: 1061: 1060: 560:{\displaystyle q} 540:{\displaystyle p} 165:{\displaystyle g} 152:, the polynomial 68:{\displaystyle f} 1121: 1082: 1075: 1068: 1040: 1033: 990: 989: 987: 986: 980: 974:. 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Retrieved 976:the original 963: 953: 920: 446: 54: 41:uniform norm 24: 18: 37:polynomials 21:mathematics 1093:Categories 985:2022-04-22 933:References 917:Algorithms 788:such that 314:such that 901:σ 879:∞ 875:‖ 868:− 862:‖ 846:− 840:σ 815:− 773:≤ 751:⋯ 722:≤ 696:ν 690:μ 681:− 644:∞ 640:‖ 633:− 627:‖ 607:μ 604:− 581:ν 578:− 478:≤ 455:≤ 427:σ 405:∞ 401:‖ 394:− 388:‖ 372:− 366:σ 341:− 299:≤ 277:⋯ 248:≤ 197:∞ 193:‖ 186:− 180:‖ 137:≤ 51:Statement 45:Chebyshev 921:Several 443:Variants 1005:at the 711:points 527:, with 237:points 16:Theorem 893:where 419:where 35:using 23:, the 1044:This 979:(PDF) 972:(PDF) 1050:stub 754:< 748:< 735:< 593:and 547:and 280:< 274:< 261:< 55:Let 684:min 107:to 31:of 19:In 1095:: 941:^ 929:. 47:. 1081:e 1074:t 1067:v 1056:. 988:. 957:. 871:g 865:f 857:i 853:) 849:1 843:( 837:= 834:) 829:i 825:x 821:( 818:g 812:) 807:i 803:x 799:( 796:f 776:b 768:1 765:+ 762:n 758:x 743:1 739:x 730:0 726:x 719:a 699:} 693:, 687:{ 678:2 675:+ 672:n 669:+ 666:m 636:g 630:f 601:m 575:n 555:q 535:p 515:q 511:/ 507:p 504:= 501:g 481:m 458:n 397:g 391:f 383:i 379:) 375:1 369:( 363:= 360:) 355:i 351:x 347:( 344:g 338:) 333:i 329:x 325:( 322:f 302:b 294:1 291:+ 288:n 284:x 269:1 265:x 256:0 252:x 245:a 225:2 222:+ 219:n 189:g 183:f 160:g 140:n 116:R 95:] 92:b 89:, 86:a 83:[ 63:f

Index

mathematics
approximation
continuous functions
polynomials
uniform norm
Chebyshev
minimax approximation algorithms
Remez algorithm



Lectures on Theory of Approximation
"Notes on how to prove Chebyshev's equioscillation theorem"
the original
Notes on how to prove Chebyshev’s equioscillation theorem
Wayback Machine
The Chebyshev Equioscillation Theorem by Robert Mayans
The de la Vallée-Poussin alternation theorem
Approximation theory by Remco Bloemen
Stub icon
mathematical analysis
stub
expanding it
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t
e
Categories
Theorems about polynomials
Numerical analysis
Theorems in analysis

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