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Topic: Chebyshev form

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	Chebyshev biography
		Chebyshev always acknowledged the great influence Brashman had been on him while studying at university, and credited him as the main influence in directing his research interests, referring to their "precious personal talks".
		Chebyshev continued to aim at international recognition with his second paper, written again in French, appearing in 1844 published by Crelle in his journal.
		In the summer of 1846 Chebyshev was examined on his Master's thesis and in the same year published a paper based on that thesis, again in Crelle's journal.
	www-history.mcs.st-andrews.ac.uk /Biographies/Chebyshev.html (3026 words)

	Chebyshev
		In 1847 Chebyshev was appointed to the University of St Petersburg.
		Chebyshev also came close to proving the prime number theorem, proving that if pi (n)log n)/n had a limit as n-> infinity then that limit is 1.
		Chebyshev was also interested in mechanics and studied the problems involved in converting rotary motion into rectilinear motion by mechanical coupling.
	library.wolfram.com /examples/quintic/people/Chebyshev.html (234 words)

	Polynomial
		This form is sometimes taken as the definition of a polynomial in one variable.
		A polynomial equation is an equation in the form of a polynomial equal to zero or equal to another polynomial.
		The latter form is usually converted to the former by subtracting the second polynomial from both sides of the equation.
	www.brainyencyclopedia.com /encyclopedia/p/po/polynomial.html (2753 words)

	Chebyshev polynomials - Wikipedia Light! (Site not responding. Last check: 2007-10-17)
		Chebyshev polynomials are important in approximation theory because the roots of the Chebyshev polynomials of the first kind, which are also called Chebyshev nodes, are used as nodes in polynomial interpolation.
		The Chebyshev polynomials are a special case of the ultraspherical or Gegenbauer polynomials, which themselves are a special case of the Jacobi polynomials.
		A polynomial of degree N in Chebyshev form is a polynomial p(x) of the form
	godseye.com /wiki/index.php?title=Chebyshev_polynomials (1003 words)

	NationMaster - Encyclopedia: Butterworth filter (Site not responding. Last check: 2007-10-17)
		The Bode plot for a first-order Butterworth filter A Bode plot, named after Hendrik Wade Bode, is usually a combination of a Bode magnitude plot and Bode phase plot: A Bode magnitude plot is a graph of log magnitude against log frequency often used in signal processing to show...
		Compared with a Chebyshev Type I/Type II filter or an elliptic filter, the Butterworth filter has a slower roll-off, and thus will require a higher order to implement a particular stopband specification.
		The group delay is defined as the derivative of the phase with respect to angular frequency and is a measure of the distortion in the signal introduced by phase differences for different frequencies.
	www.nationmaster.com /encyclopedia/Butterworth-filter (2351 words)

	Chebyshev polynomials
		Chebyshev's polynomial of the second kind can be defined in terms of the differential equation
		A plot of the Chebyshev polynomials of the second kind as functions of
		x near ±1 this form becomes indeterminate, therefore a suitable approximation—which requires accuracy parameter eps—is needed.
	www.unc.edu /~wjt/ChebyshevPolynomials2.htm (342 words)

	clenshaw algorithm - Article and Reference from OnPedia.com
		In the mathematical subfield of numerical analysis the Clenshaw algorithm is a recursive method to evaluate polynomials in Chebyshev form.
		A polynomial of degree N in Chebyshev form is a polynomial p(x) of the form
		The Clenshaw algorithm can be used to evaluate a polynomial in the Chebyshev form.
	www.onpedia.com /encyclopedia/Clenshaw-algorithm (120 words)

	cheby2 (Signal Processing Toolbox)
		Chebyshev Type II filters are monotonic in the passband and equiripple in the stopband.
		For high order filters, the state-space form is the most numerically accurate, followed by the zero-pole-gain form.
		The transfer function coefficient form is the least accurate; numerical problems can arise for filter orders as low as 15.
	www.technion.ac.il /guides/matlab/toolbox/signal/cheby2.html (463 words)

	Chebyshev Interpolation
		Chebyshev pseudospectral methods, which are based on the interpolating Chebyshev approximation (12), are well established as powerful methods for the numerical solution of PDEs with sufficiently smooth solutions.
		In these cases, the Chebyshev pseudospectral method produces approximations that are contaminated with Gibbs oscillations and suffer from the corresponding loss of spectral accuracy, just like the Chebyshev interpolation methods that the pseudospectral methods are based on.
		Chebyshev approximation and its relation to polynomial interpolation at equidistant nodes has been discussed.
	www.joma.org /images/upload_library/4/vol6/Sarra/Chebyshev.html (2857 words)

	Chebyshev polynomials - Wikipedia, the free encyclopedia
		are polynomials of degree n and the sequence of Chebyshev polynomials of either kind composes a polynomial sequence.
		Polynomials in Chebyshev form can be evaluated using the Clenshaw algorithm.
		The spread polynomials are in a sense equivalent to the Chebyshev polynomials of the first kind, but enable one to avoid square roots and conventional trigonometric functions in certain contexts, notably in rational trigonometry.
	en.wikipedia.org /wiki/Chebyshev_polynomials (898 words)

	Determination of optimal Chebyshev-expanded hydrophobic discrimination function for globular proteins
		The scoring function is expanded using the Chebyshev polynomials, for which the coefficients are determined by minimizing the Z-score of native structures in the ensembles of alternate conformations.
		First, it assumes nothing of the form of the interaction other than the fact that the energy depends on the amino acid type and its degree of burial.
		In fact, the Chebyshev approximating polynomial is very nearly the same as the minimax polynomial.
	www.research.ibm.com /journal/rd/453/fain.html (3467 words)

	opolyls5.html
		The discrete Chebyshev polynomials have been generalized to the case of arbitrary positive weights and arbitrary independent variable values.
		A necessary condition for a minimum of this sum is that all the first partial derivatives with respect to the parameters must be set equal to zero.
		to bidiagonal form, then a modified eigenvalue QR algorithm can be used to reduce the size of the off diagonal elements so that they are all less than the given off diagonal tolerance.
	www.math.sfu.ca /~gfee/opolyls/opolyls51.html (1838 words)

	Chebyshev Polynomials
		The signal property of Chebyshev polynomials is the trigonometric representation on [-1,1].
		In the construction, vector operations are used to assist the computations, since, similar terms occur in the summation for each of the coefficients.
		Form several Chebyshev polynomials of degree n = 1,2, 3, 4, and 5 for the function
	math.fullerton.edu /mathews/n2003/ChebyshevPolyMod.html (435 words)

	NAG C Library, Mark 7 : e01aec
		The Newton form of the polynomial is then converted to Chebyshev-series form as described in Section 8.3.
		Though by no means the worst example, interpolation by a single polynomial to a large number of function values given at points equally spaced across the range is notoriously ill-conditioned and the polynomial interpolating such a data set is prone to exhibit enormous oscillations between the data points, especially near the ends of the range.
		The second condition takes two different forms according to whether or not the performance indices (see Section 7) of the current best polynomial are all less than 8 times the machine precision.
	www.nag.co.uk /numeric/CL/manual/xhtml/E01/e01aec.xml (1364 words)

	ellip (Signal Processing Toolbox)
		The transfer function form is the least accurate; numerical problems can arise for filter orders as low as 15.
		The design of elliptic filters is the most difficult and computationally intensive of the Butterworth, Chebyshev type I and II, and elliptic designs.
		It transforms the lowpass filter to a bandpass, highpass, or bandstop filter with the desired cutoff frequencies using a state-space transformation.
	www.tau.ac.il /cc/pages/docs/matlab/help/toolbox/signal/ellip.html (475 words)

	Prime numbers
		He proved a speculation of Albert Girard that every prime number of the form 4 n + 1 can be written in a unique way as the sum of two squares and was able to show how any number could be written as a sum of four squares.
		formed by summing the reciprocals of the prime numbers, is also divergent.
		Century with notable progress being made by Chebyshev and Riemann who was able to relate the problem to something called the Riemann Hypothesis: a still unproved result about the zeros in the Complex plane of something called the Riemann zeta-function.
	www-gap.dcs.st-and.ac.uk /~history/HistTopics/Prime_numbers.html (1600 words)

	Chebyshev (Site not responding. Last check: 2007-10-17)
		In 1847 Chebyshev was appointed to the University of St Petersburg.
		Chebyshev was also interested in mechanics and studied the problems involved in converting rotary motion into rectilinear motion by mechanical coupling.
		He wrote about many subjects, including probability theory, quadratic forms, orthogonal functions, the theory of integrals, the construction of maps, and the calculation of geometric volumes.
	www.mathsoc.spb.ru /pantheon/chebyshe/index.html (265 words)

	Analog Filter Design Demystified - Maxim/Dallas
		Poles usually occur as pairs, in the form of a complex number (a + jb) and its complex conjugate (a - jb).
		Similar to the Chebyshev response, it has ripple in the passband and severe rolloff at the expense of ripple in the stopband.
		A 5th-order, 1dB-ripple Chebyshev lowpass filter is constructed from two non-identical 2nd-order sections and an output RC network.
	www.maxim-ic.com /appnotes.cfm/appnote_number/1795 (2295 words)

	Chebyshev's bias
		Chebyshev’s Bias deserves to be much better known than it is, though, so to get the word out, I’m going to blog it, right here.
		The situation resembles those “first past the post” election systems, where a nationwide majority of 51 percent can give your party a landslide in terms of parliamentary seats; or a foot race with very well-matched runners, in which one runner manages to stay slightly ahead for most of the race, and gets all the glory.
		The wonderful and amazing result got by Rubinstein and Sarnak is this: Take a Chebyshev bias; for illustration, I'll take the bias to remainder 3 when you divide a prime by 4.
	olimu.com /Notes/ChebyshevsBias.htm (1564 words)

	Cubic equation Summary
		A cubic equation is one of the form ax
		In fact, all cubic equations can be reduced to this form if we allow m and n to be negative, but negative numbers were not known at that time.
		Therefore the Chebyshev cube root is in fact an analytic function on the whole of the domain D. An alternative construction of the Chebyshev cube root in terms of hypergeometric functions is sketched in the next subsection.
	www.bookrags.com /Cubic_equation (2260 words)

	Polynomials From Pascal's Triangle
		These functions (also known as linear fractional transformations) have the general form ax + b f(x) = -------- cx + d where a,b,c,d are constants.
		However, if N is a prime, it IS possible to factor the polynomial P_N[-x^2], which is the polynomial formed by substituting -x^2 in place of x.
		Of course, if we construct a polynomial whose coefficients are all the numbers on the Nth horizontal row of Pascal's triangle, then the roots are all -1's, because the polynomial is just (1+x)^N. However, if you take every OTHER number from a horizontal row, you get an interesting result.
	www.mathpages.com /home/kmath304.htm (507 words)

	Wei Qi, Cellular Automata, Ising Model, Feynman Checkerboard
		M_n is a 1-dimensional lattice with structure of the Chebyshev measures on B2.
		As M goes to infinity, the Chebyshev measure goes to the arccosine measure.
		From this Octonion Chebyshev Bernoulli Shift point of view, only 3 {i,j,E} of the 7 {i,j,k,E,I,J,K} imaginary octonions are algebraically independent, so physical spacetime should have 3 spatial dimensions.
	www.valdostamuseum.org /hamsmith/ficw.html (5783 words)

	Chebyshev's Theorem -- from Wolfram MathWorld
		There are at least two theorems known as Chebyshev's theorem.
		The first is Bertrand's postulate, proposed by Bertrand in 1845 and proved by Chebyshev using elementary methods in 1850 (Derbyshire 2004, p.
		The second is a weak form of the prime number theorem stating that the order of magnitude of the prime counting function
	mathworld.wolfram.com /ChebyshevsTheorem.html (136 words)

	Even Order Passband and Stopband Synthesis by Nuhertz Technologies
		The even order filters of Chebyshev I, Chebyshev II, Hourglass, and Elliptic create a special problem in that their standard forms are not realizable with passive RCL circuits when their source and load resistances are comparable.
		The result is no longer a true Chebyshev I filter, but is rather a modified Chebyshev I filter that is realizable with RCL circuits of comparable source and load resistances.
		To realize the 6th order Chebyshev II circuit, Filter Solution modifies the frequency response such that the highest frequency transmission zero is mapped to infinity.
	www.filter-solutions.com /even.html (557 words)

	Monomial basis - Wikipedia, the free encyclopedia
		This description, the monomial form of a polynomial, is often used because of the simple structure of the monomial basis.
		Polynomials in monomial form can be evaluated efficiently using the Horner algorithm.
		A polynomial can always be converted into monomial form by calculating its Taylor expansion around 0.
	en.wikipedia.org /wiki/Monomial_basis (126 words)

	Filter Design and Implementation (Signal Processing Toolbox)
		Impulse invariance works for some lowpass and bandpass filters, but is not appropriate for highpass and bandstop filters.
		Design a Chebyshev Type I filter and plot its frequency and phase response using FVTool:
		Note that the band edges are in rad/s for analog filters, whereas for the digital case, frequency is normalized:
	www.mathworks.com /access/helpdesk_r13/help/toolbox/signal/filter15.html (527 words)

	cheby1 :: Functions (Signal Processing Toolbox)
		Chebyshev Type I filters are equiripple in the passband and monotonic in the stopband.
		To obtain zero-pole-gain form, use three output arguments as shown below.
		For data sampled at 1000 Hz, design a 9th-order lowpass Chebyshev Type I filter with 0.5 dB of ripple in the passband and a passband edge frequency of 300 Hz, which corresponds to a normalized value of 0.6:
	www.mathworks.com /access/helpdesk/help/toolbox/signal/cheby1.html (505 words)

	math lessons - Polynomial
		non-zero, a polynomial of degree n is a function of the form
		In this article polynomials are written using a monomial basis (i.e.
		Which algorithm is used for a given polynomial depends on the form of the polynomial and the chosen x.
	www.mathdaily.com /lessons/Polynomial (1657 words)

	Orthogonal Polynomials
		Series of Chebyshev polynomials are often used in making numerical approximations to functions.
		The name "Chebyshev" is a transliteration from the Cyrillic alphabet; several other spellings, such as "Tschebyscheff", are sometimes used.
		Legendre, Gegenbauer and Chebyshev polynomials can all be viewed as special cases of Jacobi polynomials.
	documents.wolfram.com /v5/TheMathematicaBook/AdvancedMathematicsInMathematica/MathematicalFunctions/3.2.9.html (232 words)

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