• 제목/요약/키워드: Chebyshev's Polynomials

검색결과 19건 처리시간 0.019초

A MODIFIED POLYNOMIAL SEQUENCE OF THE CHEBYSHEV POLYNOMIALS OF THE SECOND KIND

  • Kim, Seon-Hong
    • 대한수학회논문집
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    • 제34권2호
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    • pp.429-437
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    • 2019
  • Dilcher and Stolarsky [1] recently studied a sequence resembling the Chebyshev polynomials of the first kind. In this paper, we follow their some research directions to the Chebyshev polynomials of the second kind. More specifically, we consider a sequence resembling the Chebyshev polynomials of the second kind in two different ways, and investigate its properties including relations between this sequence and the sequence studied in [1], zero distribution and the irreducibility.

Certain Polynomials Related to Chebyshev Polynomials

  • Kim, Seon-Hong
    • 통합자연과학논문집
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    • 제4권3호
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    • pp.227-228
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    • 2011
  • Bae and Kim displayed a sequence of 4th degree self-reciprocal polynomials whose maximal zeros are related in a very nice and far from obvious way. The auxiliary polynomials in their results that parametrize their coefficients are of significant independent interest. In this note we show that such auxiliary polynomials are related to Chebyshev polynomials.

ON THE MAXIMUM AND MINIMUM MODULUS OF POLYNOMIALS ON CIRCLES

  • Chong, Han Kyol;Kim, Seon-Hong
    • 대한수학회논문집
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    • 제33권4호
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    • pp.1303-1308
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    • 2018
  • In this paper, we consider both maximum modulus and minimum modulus on a circle of some polynomials. These give rise to interesting examples that are about moduli of Chebyshev polynomials and certain sums of polynomials on a circle. Moreover, we obtain some root locations of difference quotients of Chebyshev polynomials.

On Two Sequences of Polynomials Satisfying Certain Recurrence

  • Kim, Seon-Hong
    • 통합자연과학논문집
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    • 제5권2호
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    • pp.131-134
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    • 2012
  • Bae and Kim displayed a sequence of 4th degree self-reciprocal polynomials whose maximal zeros are related in a very nice and far from obvious way. Kim showed that the auxiliary polynomials in their results are related to Chebyshev polynomials. In this paper, we study two sequences of polynomials satisfying the recurrence of the auxiliary polynomials with generalized initial conditions. We obtain same results with the auxiliary polynomials from a sequence, and some interesting conjectural properties about resultants and discriminants from another sequence.

Partial Fraction Expansions for Newton's and Halley's Iterations for Square Roots

  • Kouba, Omran
    • Kyungpook Mathematical Journal
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    • 제52권3호
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    • pp.347-357
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    • 2012
  • When Newton's method, or Halley's method is used to approximate the pth root of 1-z, a sequence of rational functions is obtained. In this paper, a beautiful formula for these rational functions is proved in the square root case, using an interesting link to Chebyshev's polynomials. It allows the determination of the sign of the coefficients of the power series expansion of these rational functions. This answers positively the square root case of a proposed conjecture by Guo(2010).

REPRESENTATION OF SOME BINOMIAL COEFFICIENTS BY POLYNOMIALS

  • Kim, Seon-Hong
    • 대한수학회보
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    • 제44권4호
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    • pp.677-682
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    • 2007
  • The unique positive zero of $F_m(z):=z^{2m}-z^{m+1}-z^{m-1}-1$ leads to analogues of $2(\array{2n\\k}\)$(k even) by using hypergeometric functions. The minimal polynomials of these analogues are related to Chebyshev polynomials, and the minimal polynomial of an analogue of $2(\array{2n\\k}\)$(k even>2) can be computed by using an analogue of $2(\array{2n\\k}\)$. In this paper we show that the analogue of $2(\array{2n\\2}\)$. In this paper we show that the analygue $2(\array{2n\\2}\)$ is the only real zero of its minimal polynomial, and has a different representation, by using a polynomial of smaller degree than $F_m$(z).

An Analytical Approach for Design of Nth-band FIR Digital Filters with Equi-Ripple Passband

  • Moon, Dong-Wook;Kim, Lark-Kyo;Lim, Cheng-Chew
    • Journal of Electrical Engineering and Technology
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    • 제4권3호
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    • pp.423-428
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    • 2009
  • In FIR (Finite Impulse Response) filter applications, Nth-band FIR digital filters are known to be important due to their reduced computational requirements. The conventional methods for designing FIR filters use iterative approaches such as the well-known Parks-McClellan algorithm. The Parks-McClellan algorithm is also used to design Nth-band FIR digital filters after Mintzer's research. However, a disadvantage of the Parks-McClellan algorithm is that it needs a large amount of design time. This paper describes a direct design method for Nth-band FIR Filters using Chebyshev polynomials, which provides a reduced design time over indirect methods such as the Parks-McClellan algorithm. The response of the resulting filter is equi-ripple in passband. Our proposed method produces a passband response that is equi-ripple to within a minuscule error, comparable to that of Mintzer's design method which uses the Parks-McClellan algorithm.