Кіріспе
Көптамалар тізбегі
Polynomial sequence
The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as and They can be defined in several equivalent ways, one of which starts with trigonometric functions:
The Chebyshev polynomials of the first kind are defined by:
Similarly, the Chebyshev polynomials of the second kind are defined by:
That these expressions define polynomials in may not be obvious at first sight but follows by rewriting and using de Moivre's formula or by using the angle sum formulas for and repeatedly. For example, the double angle formulas, which follow directly from the angle sum formulas, may be used to obtain and , which are respectively a polynomial in and a polynomial in multiplied by Hence and
An important and convenient property of the Tn(x) is that they are orthogonal with respect to the inner product:
and Un(x) are orthogonal with respect to another, analogous inner product, given below. The Chebyshev polynomials Tn are polynomials with the largest possible leading coefficient whose absolute value on the interval is bounded by 1. They are also the "extremal" polynomials for many other properties. In 1952, Cornelius Lanczos showed that the Chebyshev polynomials are important in approximation theory for the solution of linear systems; the roots of Tn(x), which are also called Chebyshev nodes, are used as matching points for optimizing polynomial interpolation. The resulting interpolation polynomial minimizes the problem of Runge's phenomenon and provides an approximation that is close to the best polynomial approximation to a continuous function under the maximum norm, also called the "minimax" criterion. This approximation leads directly to the method of Clenshaw–Curtis quadrature. These polynomials were named after Pafnuty Chebyshev. The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebycheff, Tchebyshev (French) or Tschebyschow (German).
The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as and They can be defined in several equivalent ways, one of which starts with trigonometric functions:
The Chebyshev polynomials of the first kind are defined by:
Similarly, the Chebyshev polynomials of the second kind are defined by:
That these expressions define polynomials in may not be obvious at first sight but follows by rewriting and using de Moivre's formula or by using the angle sum formulas for and repeatedly. For example, the double angle formulas, which follow directly from the angle sum formulas, may be used to obtain and , which are respectively a polynomial in and a polynomial in multiplied by Hence and
An important and convenient property of the Tn(x) is that they are orthogonal with respect to the inner product:
and Un(x) are orthogonal with respect to another, analogous inner product, given below. The Chebyshev polynomials Tn are polynomials with the largest possible leading coefficient whose absolute value on the interval is bounded by 1. They are also the "extremal" polynomials for many other properties. In 1952, Cornelius Lanczos showed that the Chebyshev polynomials are important in approximation theory for the solution of linear systems; the roots of Tn(x), which are also called Chebyshev nodes, are used as matching points for optimizing polynomial interpolation. The resulting interpolation polynomial minimizes the problem of Runge's phenomenon and provides an approximation that is close to the best polynomial approximation to a continuous function under the maximum norm, also called the "minimax" criterion. This approximation leads directly to the method of Clenshaw–Curtis quadrature. These polynomials were named after Pafnuty Chebyshev. The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebycheff, Tchebyshev (French) or Tschebyschow (German).
Чебышев көптамалары – косинус және синус функцияларымен байланысты көптамалардың екі тізбегі, олар және деп белгіленеді. Оларды бірнеше эквивалентті тәсілмен анықтауға болады, олардың бірі тригонометриялық функциялардан басталады:
Polynomial sequence
The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as and They can be defined in several equivalent ways, one of which starts with trigonometric functions:
The Chebyshev polynomials of the first kind are defined by:
Similarly, the Chebyshev polynomials of the second kind are defined by:
That these expressions define polynomials in may not be obvious at first sight but follows by rewriting and using de Moivre's formula or by using the angle sum formulas for and repeatedly. For example, the double angle formulas, which follow directly from the angle sum formulas, may be used to obtain and , which are respectively a polynomial in and a polynomial in multiplied by Hence and
An important and convenient property of the Tn(x) is that they are orthogonal with respect to the inner product:
and Un(x) are orthogonal with respect to another, analogous inner product, given below. The Chebyshev polynomials Tn are polynomials with the largest possible leading coefficient whose absolute value on the interval is bounded by 1. They are also the "extremal" polynomials for many other properties. In 1952, Cornelius Lanczos showed that the Chebyshev polynomials are important in approximation theory for the solution of linear systems; the roots of Tn(x), which are also called Chebyshev nodes, are used as matching points for optimizing polynomial interpolation. The resulting interpolation polynomial minimizes the problem of Runge's phenomenon and provides an approximation that is close to the best polynomial approximation to a continuous function under the maximum norm, also called the "minimax" criterion. This approximation leads directly to the method of Clenshaw–Curtis quadrature. These polynomials were named after Pafnuty Chebyshev. The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebycheff, Tchebyshev (French) or Tschebyschow (German).
The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as and They can be defined in several equivalent ways, one of which starts with trigonometric functions:
The Chebyshev polynomials of the first kind are defined by:
Similarly, the Chebyshev polynomials of the second kind are defined by:
That these expressions define polynomials in may not be obvious at first sight but follows by rewriting and using de Moivre's formula or by using the angle sum formulas for and repeatedly. For example, the double angle formulas, which follow directly from the angle sum formulas, may be used to obtain and , which are respectively a polynomial in and a polynomial in multiplied by Hence and
An important and convenient property of the Tn(x) is that they are orthogonal with respect to the inner product:
and Un(x) are orthogonal with respect to another, analogous inner product, given below. The Chebyshev polynomials Tn are polynomials with the largest possible leading coefficient whose absolute value on the interval is bounded by 1. They are also the "extremal" polynomials for many other properties. In 1952, Cornelius Lanczos showed that the Chebyshev polynomials are important in approximation theory for the solution of linear systems; the roots of Tn(x), which are also called Chebyshev nodes, are used as matching points for optimizing polynomial interpolation. The resulting interpolation polynomial minimizes the problem of Runge's phenomenon and provides an approximation that is close to the best polynomial approximation to a continuous function under the maximum norm, also called the "minimax" criterion. This approximation leads directly to the method of Clenshaw–Curtis quadrature. These polynomials were named after Pafnuty Chebyshev. The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebycheff, Tchebyshev (French) or Tschebyschow (German).
Бірінші түрдегі Чебышев көптамалары былай анықталады:
Polynomial sequence
The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as and They can be defined in several equivalent ways, one of which starts with trigonometric functions:
The Chebyshev polynomials of the first kind are defined by:
Similarly, the Chebyshev polynomials of the second kind are defined by:
That these expressions define polynomials in may not be obvious at first sight but follows by rewriting and using de Moivre's formula or by using the angle sum formulas for and repeatedly. For example, the double angle formulas, which follow directly from the angle sum formulas, may be used to obtain and , which are respectively a polynomial in and a polynomial in multiplied by Hence and
An important and convenient property of the Tn(x) is that they are orthogonal with respect to the inner product:
and Un(x) are orthogonal with respect to another, analogous inner product, given below. The Chebyshev polynomials Tn are polynomials with the largest possible leading coefficient whose absolute value on the interval is bounded by 1. They are also the "extremal" polynomials for many other properties. In 1952, Cornelius Lanczos showed that the Chebyshev polynomials are important in approximation theory for the solution of linear systems; the roots of Tn(x), which are also called Chebyshev nodes, are used as matching points for optimizing polynomial interpolation. The resulting interpolation polynomial minimizes the problem of Runge's phenomenon and provides an approximation that is close to the best polynomial approximation to a continuous function under the maximum norm, also called the "minimax" criterion. This approximation leads directly to the method of Clenshaw–Curtis quadrature. These polynomials were named after Pafnuty Chebyshev. The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebycheff, Tchebyshev (French) or Tschebyschow (German).
The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as and They can be defined in several equivalent ways, one of which starts with trigonometric functions:
The Chebyshev polynomials of the first kind are defined by:
Similarly, the Chebyshev polynomials of the second kind are defined by:
That these expressions define polynomials in may not be obvious at first sight but follows by rewriting and using de Moivre's formula or by using the angle sum formulas for and repeatedly. For example, the double angle formulas, which follow directly from the angle sum formulas, may be used to obtain and , which are respectively a polynomial in and a polynomial in multiplied by Hence and
An important and convenient property of the Tn(x) is that they are orthogonal with respect to the inner product:
and Un(x) are orthogonal with respect to another, analogous inner product, given below. The Chebyshev polynomials Tn are polynomials with the largest possible leading coefficient whose absolute value on the interval is bounded by 1. They are also the "extremal" polynomials for many other properties. In 1952, Cornelius Lanczos showed that the Chebyshev polynomials are important in approximation theory for the solution of linear systems; the roots of Tn(x), which are also called Chebyshev nodes, are used as matching points for optimizing polynomial interpolation. The resulting interpolation polynomial minimizes the problem of Runge's phenomenon and provides an approximation that is close to the best polynomial approximation to a continuous function under the maximum norm, also called the "minimax" criterion. This approximation leads directly to the method of Clenshaw–Curtis quadrature. These polynomials were named after Pafnuty Chebyshev. The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebycheff, Tchebyshev (French) or Tschebyschow (German).
Сонымен қатар, екінші түрдегі Чебышев көптамалары былай анықталады:
Polynomial sequence
The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as and They can be defined in several equivalent ways, one of which starts with trigonometric functions:
The Chebyshev polynomials of the first kind are defined by:
Similarly, the Chebyshev polynomials of the second kind are defined by:
That these expressions define polynomials in may not be obvious at first sight but follows by rewriting and using de Moivre's formula or by using the angle sum formulas for and repeatedly. For example, the double angle formulas, which follow directly from the angle sum formulas, may be used to obtain and , which are respectively a polynomial in and a polynomial in multiplied by Hence and
An important and convenient property of the Tn(x) is that they are orthogonal with respect to the inner product:
and Un(x) are orthogonal with respect to another, analogous inner product, given below. The Chebyshev polynomials Tn are polynomials with the largest possible leading coefficient whose absolute value on the interval is bounded by 1. They are also the "extremal" polynomials for many other properties. In 1952, Cornelius Lanczos showed that the Chebyshev polynomials are important in approximation theory for the solution of linear systems; the roots of Tn(x), which are also called Chebyshev nodes, are used as matching points for optimizing polynomial interpolation. The resulting interpolation polynomial minimizes the problem of Runge's phenomenon and provides an approximation that is close to the best polynomial approximation to a continuous function under the maximum norm, also called the "minimax" criterion. This approximation leads directly to the method of Clenshaw–Curtis quadrature. These polynomials were named after Pafnuty Chebyshev. The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebycheff, Tchebyshev (French) or Tschebyschow (German).
The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as and They can be defined in several equivalent ways, one of which starts with trigonometric functions:
The Chebyshev polynomials of the first kind are defined by:
Similarly, the Chebyshev polynomials of the second kind are defined by:
That these expressions define polynomials in may not be obvious at first sight but follows by rewriting and using de Moivre's formula or by using the angle sum formulas for and repeatedly. For example, the double angle formulas, which follow directly from the angle sum formulas, may be used to obtain and , which are respectively a polynomial in and a polynomial in multiplied by Hence and
An important and convenient property of the Tn(x) is that they are orthogonal with respect to the inner product:
and Un(x) are orthogonal with respect to another, analogous inner product, given below. The Chebyshev polynomials Tn are polynomials with the largest possible leading coefficient whose absolute value on the interval is bounded by 1. They are also the "extremal" polynomials for many other properties. In 1952, Cornelius Lanczos showed that the Chebyshev polynomials are important in approximation theory for the solution of linear systems; the roots of Tn(x), which are also called Chebyshev nodes, are used as matching points for optimizing polynomial interpolation. The resulting interpolation polynomial minimizes the problem of Runge's phenomenon and provides an approximation that is close to the best polynomial approximation to a continuous function under the maximum norm, also called the "minimax" criterion. This approximation leads directly to the method of Clenshaw–Curtis quadrature. These polynomials were named after Pafnuty Chebyshev. The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebycheff, Tchebyshev (French) or Tschebyschow (German).
Бұл өрнектер көптамаларды анықтайтыны алғашқыда көрінбесе де, де Муавр формуласын қолдану немесе және үшін бұрыш қосындысы формулаларын қайталап жазу арқылы оны көрсетуге болады. Мысалы, бұрыш қосындысы формулаларынан тікелей шығатын екі есе бұрыш формулаларын, және алуға болады, бұл сәйкесінше x-тің полиномы және x-ке көбейтілген полином болып табылады. Осылайша және .
Polynomial sequence
The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as and They can be defined in several equivalent ways, one of which starts with trigonometric functions:
The Chebyshev polynomials of the first kind are defined by:
Similarly, the Chebyshev polynomials of the second kind are defined by:
That these expressions define polynomials in may not be obvious at first sight but follows by rewriting and using de Moivre's formula or by using the angle sum formulas for and repeatedly. For example, the double angle formulas, which follow directly from the angle sum formulas, may be used to obtain and , which are respectively a polynomial in and a polynomial in multiplied by Hence and
An important and convenient property of the Tn(x) is that they are orthogonal with respect to the inner product:
and Un(x) are orthogonal with respect to another, analogous inner product, given below. The Chebyshev polynomials Tn are polynomials with the largest possible leading coefficient whose absolute value on the interval is bounded by 1. They are also the "extremal" polynomials for many other properties. In 1952, Cornelius Lanczos showed that the Chebyshev polynomials are important in approximation theory for the solution of linear systems; the roots of Tn(x), which are also called Chebyshev nodes, are used as matching points for optimizing polynomial interpolation. The resulting interpolation polynomial minimizes the problem of Runge's phenomenon and provides an approximation that is close to the best polynomial approximation to a continuous function under the maximum norm, also called the "minimax" criterion. This approximation leads directly to the method of Clenshaw–Curtis quadrature. These polynomials were named after Pafnuty Chebyshev. The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebycheff, Tchebyshev (French) or Tschebyschow (German).
The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as and They can be defined in several equivalent ways, one of which starts with trigonometric functions:
The Chebyshev polynomials of the first kind are defined by:
Similarly, the Chebyshev polynomials of the second kind are defined by:
That these expressions define polynomials in may not be obvious at first sight but follows by rewriting and using de Moivre's formula or by using the angle sum formulas for and repeatedly. For example, the double angle formulas, which follow directly from the angle sum formulas, may be used to obtain and , which are respectively a polynomial in and a polynomial in multiplied by Hence and
An important and convenient property of the Tn(x) is that they are orthogonal with respect to the inner product:
and Un(x) are orthogonal with respect to another, analogous inner product, given below. The Chebyshev polynomials Tn are polynomials with the largest possible leading coefficient whose absolute value on the interval is bounded by 1. They are also the "extremal" polynomials for many other properties. In 1952, Cornelius Lanczos showed that the Chebyshev polynomials are important in approximation theory for the solution of linear systems; the roots of Tn(x), which are also called Chebyshev nodes, are used as matching points for optimizing polynomial interpolation. The resulting interpolation polynomial minimizes the problem of Runge's phenomenon and provides an approximation that is close to the best polynomial approximation to a continuous function under the maximum norm, also called the "minimax" criterion. This approximation leads directly to the method of Clenshaw–Curtis quadrature. These polynomials were named after Pafnuty Chebyshev. The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebycheff, Tchebyshev (French) or Tschebyschow (German).
Tn(x) көптамаларының маңызды және ыңғайлы қасиеті – олар келесідей берілген ішкі көбейтуге қатысты ортогоналды:
Polynomial sequence
The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as and They can be defined in several equivalent ways, one of which starts with trigonometric functions:
The Chebyshev polynomials of the first kind are defined by:
Similarly, the Chebyshev polynomials of the second kind are defined by:
That these expressions define polynomials in may not be obvious at first sight but follows by rewriting and using de Moivre's formula or by using the angle sum formulas for and repeatedly. For example, the double angle formulas, which follow directly from the angle sum formulas, may be used to obtain and , which are respectively a polynomial in and a polynomial in multiplied by Hence and
An important and convenient property of the Tn(x) is that they are orthogonal with respect to the inner product:
and Un(x) are orthogonal with respect to another, analogous inner product, given below. The Chebyshev polynomials Tn are polynomials with the largest possible leading coefficient whose absolute value on the interval is bounded by 1. They are also the "extremal" polynomials for many other properties. In 1952, Cornelius Lanczos showed that the Chebyshev polynomials are important in approximation theory for the solution of linear systems; the roots of Tn(x), which are also called Chebyshev nodes, are used as matching points for optimizing polynomial interpolation. The resulting interpolation polynomial minimizes the problem of Runge's phenomenon and provides an approximation that is close to the best polynomial approximation to a continuous function under the maximum norm, also called the "minimax" criterion. This approximation leads directly to the method of Clenshaw–Curtis quadrature. These polynomials were named after Pafnuty Chebyshev. The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebycheff, Tchebyshev (French) or Tschebyschow (German).
The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as and They can be defined in several equivalent ways, one of which starts with trigonometric functions:
The Chebyshev polynomials of the first kind are defined by:
Similarly, the Chebyshev polynomials of the second kind are defined by:
That these expressions define polynomials in may not be obvious at first sight but follows by rewriting and using de Moivre's formula or by using the angle sum formulas for and repeatedly. For example, the double angle formulas, which follow directly from the angle sum formulas, may be used to obtain and , which are respectively a polynomial in and a polynomial in multiplied by Hence and
An important and convenient property of the Tn(x) is that they are orthogonal with respect to the inner product:
and Un(x) are orthogonal with respect to another, analogous inner product, given below. The Chebyshev polynomials Tn are polynomials with the largest possible leading coefficient whose absolute value on the interval is bounded by 1. They are also the "extremal" polynomials for many other properties. In 1952, Cornelius Lanczos showed that the Chebyshev polynomials are important in approximation theory for the solution of linear systems; the roots of Tn(x), which are also called Chebyshev nodes, are used as matching points for optimizing polynomial interpolation. The resulting interpolation polynomial minimizes the problem of Runge's phenomenon and provides an approximation that is close to the best polynomial approximation to a continuous function under the maximum norm, also called the "minimax" criterion. This approximation leads directly to the method of Clenshaw–Curtis quadrature. These polynomials were named after Pafnuty Chebyshev. The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebycheff, Tchebyshev (French) or Tschebyschow (German).
және Un(x) басқа, ұқсас ішкі көбейтуге қатысты ортогоналды. Чебышев көптамалары Tn – аралықта абсолюттік мәні 1-ден аспайтын ең үлкен жетекші коэффициенті бар полиномдар. Олар сонымен қатар көптеген басқа қасиеттер үшін "шекті" полиномдар болып табылады. 1952 жылы Корнелиус Ланчос Чебышев көптамаларының сызықтық жүйелерді шешу үшін жуықтау теориясында маңызды екенін көрсетті; Tn(x) түбірлері, сонымен қатар Чебышев түйіндері деп аталады, полиномиалдық интерполяцияны оңтайландыру үшін сәйкес келетін нүктелер ретінде қолданылады. Нәтижесінде алынған интерполяциялық полином Рундж құбылысының мәселесін азайтады және максималды нормадағы үздіксіз функцияға ең жақсы полиномиалдық жуықтамаға жақын жуықтама береді, бұл "минимакс" критерийі деп те аталады. Бұл жуықтау тікелей Кленшоу-Куртис квадратурасы әдісіне әкеледі. Бұл көптамалар Пафнутий Чебышевтің құрметіне аталған. Әріп T, Чебышев есімінің Чебичефф, Чебышев (француз тілінде) немесе Чебышов (неміс тілінде) деген баламалы транслитерацияларына байланысты қолданылады.
Polynomial sequence
The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as and They can be defined in several equivalent ways, one of which starts with trigonometric functions:
The Chebyshev polynomials of the first kind are defined by:
Similarly, the Chebyshev polynomials of the second kind are defined by:
That these expressions define polynomials in may not be obvious at first sight but follows by rewriting and using de Moivre's formula or by using the angle sum formulas for and repeatedly. For example, the double angle formulas, which follow directly from the angle sum formulas, may be used to obtain and , which are respectively a polynomial in and a polynomial in multiplied by Hence and
An important and convenient property of the Tn(x) is that they are orthogonal with respect to the inner product:
and Un(x) are orthogonal with respect to another, analogous inner product, given below. The Chebyshev polynomials Tn are polynomials with the largest possible leading coefficient whose absolute value on the interval is bounded by 1. They are also the "extremal" polynomials for many other properties. In 1952, Cornelius Lanczos showed that the Chebyshev polynomials are important in approximation theory for the solution of linear systems; the roots of Tn(x), which are also called Chebyshev nodes, are used as matching points for optimizing polynomial interpolation. The resulting interpolation polynomial minimizes the problem of Runge's phenomenon and provides an approximation that is close to the best polynomial approximation to a continuous function under the maximum norm, also called the "minimax" criterion. This approximation leads directly to the method of Clenshaw–Curtis quadrature. These polynomials were named after Pafnuty Chebyshev. The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebycheff, Tchebyshev (French) or Tschebyschow (German).
The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as and They can be defined in several equivalent ways, one of which starts with trigonometric functions:
The Chebyshev polynomials of the first kind are defined by:
Similarly, the Chebyshev polynomials of the second kind are defined by:
That these expressions define polynomials in may not be obvious at first sight but follows by rewriting and using de Moivre's formula or by using the angle sum formulas for and repeatedly. For example, the double angle formulas, which follow directly from the angle sum formulas, may be used to obtain and , which are respectively a polynomial in and a polynomial in multiplied by Hence and
An important and convenient property of the Tn(x) is that they are orthogonal with respect to the inner product:
and Un(x) are orthogonal with respect to another, analogous inner product, given below. The Chebyshev polynomials Tn are polynomials with the largest possible leading coefficient whose absolute value on the interval is bounded by 1. They are also the "extremal" polynomials for many other properties. In 1952, Cornelius Lanczos showed that the Chebyshev polynomials are important in approximation theory for the solution of linear systems; the roots of Tn(x), which are also called Chebyshev nodes, are used as matching points for optimizing polynomial interpolation. The resulting interpolation polynomial minimizes the problem of Runge's phenomenon and provides an approximation that is close to the best polynomial approximation to a continuous function under the maximum norm, also called the "minimax" criterion. This approximation leads directly to the method of Clenshaw–Curtis quadrature. These polynomials were named after Pafnuty Chebyshev. The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebycheff, Tchebyshev (French) or Tschebyschow (German).
Симметрия
Яғни, Чебышев полиномдарының жұп реттік саны жұп симметрияға ие және сондықтан x-тің тек жұп дәрежелерін ғана қамтиды. Чебышев полиномдарының тақ реттік саны тақ симметрияға ие және сондықтан x-тің тек тақ дәрежелерін ғана қамтиды.
That is, Chebyshev polynomials of even order have even symmetry and therefore contain only even powers of x. Chebyshev polynomials of odd order have odd symmetry and therefore contain only odd powers of x.
Ескертпе
Эквиосцилляция теоремасы бойынша, ≤ n дәрежелі барлық полиномиалдардың арасында, f полиномиалы егер және тек қана егер -1 ≤ x0 < x1 < ⋯ < xn + 1 ≤ 1 болатын n + 2 нүкте болса, онда минималданады. Әрине, [интервал] аралығындағы нөлдік полиномиалды өзімен-өзі жуықтауға болады және ол ∞ нормасын минималдайды. Бірақ жоғарыда көрсетілгенде, оның максимумы тек n + 1 рет жетеді, себебі біз n ≥ 1 дәрежелі ең жақсы полиномиалды іздеп отырмыз (сондықтан бұрын айтылған теорема қолданылмайды).
By the equioscillation theorem, among all the polynomials of degree ≤ n, the polynomial f minimizes on if and only if there are n + 2 points −1 ≤ x0 < x1 < ⋯ < xn + 1 ≤ 1 such that
Of course, the null polynomial on the interval can be approximated by itself and minimizes the ∞ norm. Above, however, reaches its maximum only n + 1 times because we are searching for the best polynomial of degree n ≥ 1 (therefore the theorem evoked previously cannot be used).
Of course, the null polynomial on the interval can be approximated by itself and minimizes the ∞ norm. Above, however, reaches its maximum only n + 1 times because we are searching for the best polynomial of degree n ≥ 1 (therefore the theorem evoked previously cannot be used).
Чебышев түріндегі көптік
N дәрежелі кез келген көпмүшелікті бірінші тектес Чебышев көпмүшеліктері арқылы өрнектеуге болады. Мұндай көпмүшелік p(x) мына түрде жазылады:
An arbitrary polynomial of degree N can be written in terms of the Chebyshev polynomials of the first kind. Such a polynomial p(x) is of the form:
Чебышев пішіндес көпмүшеліктерді Кленшоу алгоритмімен есептеуге болады.
Polynomials in Chebyshev form can be evaluated using the Clenshaw algorithm.