• Title/Summary/Keyword: Series expansion

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NEUMANN SERIES EXPANSION OF THE INVERSE OF A FRAME OPERATOR

  • Lim, Jae-Kun
    • Communications of the Korean Mathematical Society
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    • v.13 no.4
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    • pp.791-800
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    • 1998
  • We present a proof that, among all complex numbers, Duffin-Schaeffer's choice in the Neumann series expansion of the inverse of a frame operator has the best possible convergence rate.

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Identification Using Orthonormal Functions

  • Bae, Chul-Min;Wada, Kiyoshi;Imai, Jun
    • 제어로봇시스템학회:학술대회논문집
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    • 1998.10a
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    • pp.285-288
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    • 1998
  • A least-squares identification method is studied that estimates a finite number of coefficients in the series expansion of a transfer function, where the expansion is in terms of recently introduced generalized basis functions, We will expand and generalize the orthogonal functions as basis functions for dynamical system representations. To this end, use is made of balanced realizations as inner transfer functions. The orthogonal functions can be considered as generalizations of, for example, the pulse functions, Laguerre functions, and Kautz functions, and give rise to an alternative series expansion of rational transfer functions. We show that the Laplace transform of the expansion for some sets$\Psi_{\kappa}(Z)$ is equivalent to a series expansion . Techniques based on this result are presented for obtaining the coefficients $c_{n}$ as those of a series. One of their important properties is that, if chosen properly, they can substantially increase the speed of convergence of the series expansion. This leads to accurate approximate models with only a few coefficients to be estimated. The set of Kautz functions is discussed in detail and, using the power-series equivalence, the truncation error is obtained.

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ON THE PERIOD OF β-EXPANSION OF PISOT OR SALEM SERIES OVER 𝔽q((x-1))

  • RIM, GHORBEL;SOUROUR, ZOUARI
    • Bulletin of the Korean Mathematical Society
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    • v.52 no.4
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    • pp.1047-1057
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    • 2015
  • In [6], it is proved that the lengths of periods occurring in the ${\beta}$-expansion of a rational series r noted by $Per_{\beta}(r)$ depend only on the denominator of the reduced form of r for quadratic Pisot unit series. In this paper, we will show first that every rational r in the unit disk has strictly periodic ${\beta}$-expansion for Pisot or Salem unit basis under some condition. Second, for this basis, if $r=\frac{P}{Q}$ is written in reduced form with |P| < |Q|, we will generalize the curious property "$Per_{\beta}(\frac{P}{Q})=Per_{\beta}(\frac{1}{Q})$".

Reliability Calculation of Power Generation Systems Using Generalized Expansion

  • Kim, Jin-O
    • Journal of Electrical Engineering and information Science
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    • v.2 no.6
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    • pp.123-130
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    • 1997
  • This paper presents a generalized expansion method for calculating reliability index in power generation systems. This generalized expansion with a gamma distribution is a very useful tool for the approximation of capacity outage probability distribution of generation system. The well-known Gram-Charlier expansion and Legendre series are also studied in this paper to be compared with this generalized expansion using a sample system IEEE-RTS(Reliability Test System). The results show that the generalized expansion with a composite of gamma distributions is more accurate and stable than Gram-Charlier expansion and Legendre series as addition of the terms to be expanded.

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REMARKS ON KERNEL FOR WAVELET EXPANSIONS IN MULTIDIMENSIONS

  • Shim, Hong-Tae;Kwon, Joong-Sung
    • Journal of applied mathematics & informatics
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    • v.27 no.1_2
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    • pp.419-426
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    • 2009
  • In expansion of function by special basis functions, properties of expansion kernel are very important. In the Fourier series, the series are expressed by the convolution with Dirichlet kernel. We investigate some of properties of kernel in wavelet expansions both in one and higher dimensions.

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BETA-EXPANSIONS WITH PISOT BASES OVER Fq((x-1))

  • Hbaib, Mohamed
    • Bulletin of the Korean Mathematical Society
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    • v.49 no.1
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    • pp.127-133
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    • 2012
  • It is well known that if the ${\beta}$-expansion of any nonnegative integer is finite, then ${\beta}$ is a Pisot or Salem number. We prove here that $\mathbb{F}_q((x^{-1}))$, the ${\beta}$-expansion of the polynomial part of ${\beta}$ is finite if and only if ${\beta}$ is a Pisot series. Consequently we give an other proof of Scheiche theorem about finiteness property in $\mathbb{F}_q((x^{-1}))$. Finally we show that if the base ${\beta}$ is a Pisot series, then there is a bound of the length of the fractional part of ${\beta}$-expansion of any polynomial P in $\mathbb{F}_q[x]$.

FOURIER SERIES OF A DEVIL'S STAIRCASE

  • Kwon, DoYong
    • Honam Mathematical Journal
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    • v.43 no.2
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    • pp.259-267
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    • 2021
  • Given 𝛽 > 1, we consider real numbers whose 𝛽-expansions are Sturmian words. When the slope of Sturmian words varies, their behaviors have been well studied from analytical point of view. The regularity enables us to find the Fourier series expansion, while the singularity at rational slopes yields a new kind of trigonometric series representing 𝜋.

Error Estimation for the Semi-Analytic Design Sensitivity Using the Geometric Series Expansion Method (기하급수 전개법을 이용한 준해석 민감도의 오차 분석)

  • Dan, Ho-Jin;Lee, Byung-Chai
    • Transactions of the Korean Society of Mechanical Engineers A
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    • v.27 no.2
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    • pp.262-267
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    • 2003
  • Error of the geometric series expansion method for the structural sensitivity analysis is estimated. Although the semi-analytic method has several advantages, accuracy of the method prevents it from practical application. One of the promising remedies is the use of geometric series formula for the matrix inversion. Its result of the sensitivity analysis converges that of the global difference method which is known as reliable one. To reduce computational efforts and to obtain reliable results, it is important to know how many terms need to expand. In this paper, the error formula is presented and Its usefulness is illustrated through numerical experiments.