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Topic: Trigonometric polynomial

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	Reference.com/Encyclopedia/Trigonometric polynomial
		Hence the term trigonometric polynomial as the sin(nx)s and cos(nx)s are used similar to the monomial basis for a polynomial.
		The trigonometric polynomials are used in trigonometric interpolation to interpolate periodic functions.
		A basic result is that the trigonometric polynomials are dense in the space of continuous functions on the unit circle, with the uniform norm.
	www.reference.com /browse/wiki/Trigonometric_polynomial (317 words)

	Trigonometric polynomial - Wikipedia, the free encyclopedia
		Hence the term trigonometric polynomial as the sin( nx)s and cos( nx)s are used similar to the monomial basis for a polynomial.
		The trigonometric polynomials are used in trigonometric interpolation to interpolate periodic functions.
		A basic result is that the trigonometric polynomials are dense in the space of continuous functions on the unit circle, with the uniform norm.
	en.wikipedia.org /wiki/Trigonometric_polynomial (252 words)

	Trigonometric function - Wikipedia, the free encyclopedia
		The computation of trigonometric functions is a complicated subject, which can today be avoided by most people because of the widespread availability of computers and scientific calculators that provide built-in trigonometric functions for any angle.
		The six trigonometric functions can also be defined in terms of the unit circle, the circle of radius one centered at the origin.
		The earliest systematic study of trigonometric functions and tabulation of their values was performed by Hipparchus of Nicaea ( 180 - 125 BC), who tabulated the lengths of circle arcs (angle A times radius r) with the lengths of the subtending chords (2 r sin( A /2)).
	en.wikipedia.org /wiki/Trigonometric_function (3060 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
		econd is the indeterminate form of the trigonometric form as
	www.unc.edu /~wjt/ChebyshevPolynomials2.htm (342 words)

	Generating trigonometric tables (Site not responding. Last check: 2007-10-17)
		Tables of trigonometric function s are useful in a number of areas.
		Interpolation of simple look-up tables of trigonometric functions are still used in computer graphics, where accurate calculations are either not needed, or cannot be made fast enough.
		Another important application of trigonometric tables and generation schemes is for fast Fourier transform (FFT) algorithms, where the same trigonometric function values (called twiddle factors) must be evaluated many times in a given transform, especially in the common case where many transforms of the same size are computed.
	www.serebella.com /encyclopedia/article-Generating_trigonometric_tables.html (862 words)

	Polynomial (Site not responding. Last check: 2007-10-17)
		The culmination of these efforts is Taylor's theorem, which roughly states that every differentiable function locally looks like a polynomial, and the Stone-Weierstrass theorem, which states that every continuous function defined on a compact interval of the real axis can be approximated on the whole interval as closely as desired by a polynomial.
		Depending on the degree of the polynomial to be considered, simply checking if the polynomial has linear factors can eliminate several cases, and then resorting to checking divisibility of some other irreducible polynomials, however Eisenstein's criterion can be used to more efficiently determine irreducibility.
		Polynomial regression analysis applet - It calculate a calibration for a MALDI TOF (and possibly other) mass spectrometers using any kind of (semi-)repetitive peptide as the calibrant.
	www.nebulasearch.com /encyclopedia/article/Polynomial.html (2001 words)

	Polynomial interpolation - Wikipedia, the free encyclopedia
		In the mathematical subfield of numerical analysis, polynomial interpolation is the interpolation of a given data set by a polynomial.
		One method is to write the interpolation polynomial in the Newton form and use the method of divided differences to construct the coefficients.
		This can be seen as a form of polynomial interpolation with harmonic base functions, see trigonometric interpolation and trigonometric polynomial.
	en.wikipedia.org /wiki/Polynomial_interpolation (1293 words)

	[No title]
		Classic problems connected with trigonometric polynomial approximation of periodic functions, extremal properties of algebraic and trigonometric polynomials, orthogonal polynomials and their asymptotic, approximating and extremal properties.
		The fundamental research was done to study extremal and asymptotic properties of orthogonal (algebraic on the interval or on the unit circle and trigonometric) polynomials and their derivatives in the case when the weight function has singularities.
		Existence theorems for the interpolating and interpolating in the mean polynomial splines with uniform nodal points of splines and uniform points of interpolation in the case of their arbitrary mutual location were proved, the bounds of approximation errors were obtained for various classes of functions (Yu.N.Subbotin).
	www.imm.uran.ru /DEPARTMT/OTP.HTM (2160 words)

	Chris Fenwick's mathematics education (self-organised reading-to-learn) research and teaching page
		Polynomial equations are equations containing a sum of terms (i.e.
		A polynomial is therefore generally expressed as where are the coefficients.
		This specific case is the Factor Theorem and occurs when the remainder of a polynomial division is 0.
	www.ucl.ac.uk /~uczlcfe/documents/teaching/allgroups.html (1713 words)

	Polynomial/Trig Regression Data Fit
		For the trigonometric data fit, the data should represent a periodic function, and the x data should be scaled so that values between 0 and 1 are proportional to 2 Pi radians (or 360 degrees).
		For both types of regression, the choice of degree and the evaluation of the fit's quality depend on judgments that are left up to the user.
		The optimal fit is that in which the degree of the equation is equal to the number of data pairs minus one.
	vps.arachnoid.com /polysolve (343 words)

	Lab for Trigonometric Polynomial Approximation (FFT) (Site not responding. Last check: 2007-10-17)
		Trigonometric curve fitting at discrete points is equivalent to finding the Fast Fourier Transform (FFT) for a discrete data set.
		The coefficients of the trigonometric polynomial can be obtained using Mathematica 's built in "Fourier" procedure.
		Since there are 13 data points, the trigonometric polynomial of degree m = 6 will fit the data "exactly".
	www.ecs.fullerton.edu /~mathews/numerical/at.htm (418 words)

	Definition of polynomial
		In [[graph theory]] the [[chromatic polynomial ]] of a [[graph (mathematics)graph]] encodes the...
		Thus a trigonometric polynomial can be considered a periodic function on the [[re...
		14: The Ehrhart polynomial of the [[interior (topology)interior]] of a clos...
	www.wordiq.com /search/polynomial.html (1099 words)

	Trigonometry and Basic Functions - Numericana
		Polynomial functions: The value y is obtained from the variable x using only a finite number of additions and/or multiplications involving given constants.
		Specific adjectives are virtually unused for polynomials beyond degree 6: degree 7 is either septic [sic!] or heptic, degree 8 is octic, 9 is nonic, 10 is decic, and we are told that some have called degree 100 "hectic"...
		Modern usage is to consider only the 3 preferred trigonometric functions (sine, cosine and tangent) whereas their 3 reciprocals (cosecant, secant and cotangent) are being deprecated.
	home.att.net /~numericana/answer/functions.htm (3957 words)

	Lab for Trigonometric Polynomial Approximation (FFT)
		Trigonometric curve fitting at discrete points is equivalent to finding the Fast Fourier Transform (FFT) for a discrete data set.
		The coefficients of the trigonometric polynomial can be obtained using Mathematica's built in "Fourier" procedure.
		Since there are 13 data points, the trigonometric polynomial of degree m = 6 will fit the data "exactly".
	math.fullerton.edu /mathews/numerical/at.htm (416 words)

	A Levinson--Galerkin Algorithm for Regularized Trigonometric Approximation
		Trigonometric polynomials are widely used for the approximation of a smooth function from a set of nonuniformly spaced samples.
		If the samples are perturbed by noise, a good choice for the polynomial degree of the trigonometric approximation becomes an essential issue to avoid overfitting and underfitting of the data.
		Standard methods for trigonometric least squares approximation assume that the degree for the approximating polynomial is known a priori, which is usually not the case in practice.
	epubs.siam.org /sam-bin/dbq/article/32925 (263 words)

	Quantifier Elimination for Trigonometric Polynomials
		We have chosen to adapt to the trigonometric case the method of George Collins, cylindrical algebraic decomposition.
		We have adapted this method to the trigonometric case; a description of the algorithm, together with the background theory, can be found in RISC technical report 99-07.
		the coefficient is an algebraic polynomial with integer coefficients:
	www.risc.uni-linz.ac.at /projects/basic/SFB/reals/themes/tpolqe (616 words)

	Nyquist-Shannon sampling theorem - Wikipedia, the free encyclopedia
		Equivalently, the entire spectrum of the bandlimited signal should be expressible in terms of the finitely many time-domain coefficients obtained from sampling the signal.
		Mathematically this is equivalent to requiring that a (trigonometric) polynomial can have infinitely many zeros since the bandlimited signal must be zero on an interval beyond a critical frequency which has infinitely many points.
		However, it is well-known that polynomials do not have more zeros than their orders due to the fundamental theorem of algebra.
	en.wikipedia.org /wiki/Nyquist-Shannon_sampling_theorem (1619 words)

	Harmonic Knots
		(t) be trigonometric polynomials of degree less than or equal to N. We say T
		in R 3 an Nth degree trigonometric polynomial curve.
		Definition 1.3 An Nth degree harmonic knot is a knot with a regular Nth degree trigonometric polynomial parametrization.
	www.carthage.edu /~trautwn/hknots.html (1156 words)

	Math Problems, Questions and Online Self Tests
		Polynomial and Rational Inequalities, questions and their answers: Questions on solving Polynomial and Rational Inequalities are presented.
		Graphs of Polynomial Functions - Questions with answers: Questions and their answers on graphs of polynomial functions are presented.
		Graphs Trigonometric Functions - Questions with answers: A set of questions, with their answers, on identifying the graphs of trigonometric functions sin(x), cos(x), tan(x), sec(x), csc(x), cot(x) are presented.
	www.analyzemath.com /precalculustests.html (592 words)

	downldabs
		We construct bounded polynomial operators, similar to the classical de la Valle\'e Poussin operators in Fourier series, which preserve polynomials of a certain degree, but are defined in terms of the values of the function rather than its Fourier coefficients.
		Applications to the mean boundedness of a quasi-interpolatory operator in the case of trigonometric polynomials, Jacobi polynomials, and Freud polynomials are presented.
		Our analysis is based on orthogonal polynomials with respect to the weight in question, but the frame bounds are independent of the system of orthogonal polynomials used.
	www.calstatela.edu /faculty/hmhaska/downldabs.htm (1539 words)

	Example: Symbolic Algebra
		as a polynomial in x with coefficients that are trigonometric functions of polynomials in y whose coefficients are integers.
		We will define a polynomial to be a sum of terms, each of which is either a coefficient, a power of the indeterminate, or a product of a coefficient and a power of the indeterminate.
		If the coefficients of the coefficients were themselves polynomials (as might be used to represent polynomials in three variables), the data direction would ensure that the system would follow through another level of recursive calls, and so on through as many levels as the structure of the data dictates.
	mitpress.mit.edu /sicp/full-text/sicp/book/node49.html (3488 words)

	Approximation and Interpolation
		Download Mixed Polynomial and Trigonometric Approximations (Tabulated) Approximates tabulated functions in 1 to 4 independent variables by finite multivariable power and/or trigonometric series.
		The program determines the coefficients of the polynomial, the generalized correlation coefficient and the standard error of estimate.
		Note that, even though there are 5 values in each direction, only 4 of the function's values are independent because its value at the interval's end point must be the same as at the initial point because of the required periodicity.
	www.numericalmathematics.com /approximation_and_interpolation.htm (711 words)

	Course Syllabus - Lee College (Site not responding. Last check: 2007-10-17)
		Catalog Description: This course covers the applications of algebra and trigonometry to the study of elementary functions and their graphs including polynomial, rational, exponential, logarithmic, and trigonometric functions and may include topics from analytical geometry.
		Find the rational zeros of a polynomial by use of the rational zeros theorem, Descartes' rule of signs, and the upper/lower bound theorem.
		Define the trigonometric functions, their even-odd properties, their signs in different quadrants, and the domains and ranges of each.
	www.lee.edu /syllabus_descr.asp?CRSE_ID=001841 (1136 words)

	Citations: Class of global minimum bounds of polynomial functions - Shor (ResearchIndex) (Site not responding. Last check: 2007-10-17)
		It is shown that the new scheme has certain theoretical advantages over the classical one: in particular, a trigonometric polynomial is positive if and only if it can be represented as a sum of squares of a....
		The method is based on rewriting a given polynomial as a sum of squares of polynomials.
		For an in depth analysis of this particular problem, including numerous examples and a comparison with traditional algebraic techniques, we refer the reader to [PS01] The condition F (x) is a sum of squares is affine in, and therefore it is possible to efficiently compute the maximum....
	citeseer.lcs.mit.edu /context/896295/0 (2273 words)

	Learn more about Pre-Calculus Math (Site not responding. Last check: 2007-10-17)
		This course includes set concepts, real and complex number systems, regression, function concepts, graphical representations, polynomial functions and theory of equations, exponential and logarithmic functions, trigonometric functions and their applications, systems and equations, matrices and determinants.
		This course is a sequel to intermediate algebra and trigonometry, and you should already be comfortable (perhaps with a little review) working with algebraic expressions, solving linear and quadratic equations, understanding and using integer and fractional exponents as well as radicals and roots, and working with the basics of exponentials and logarithms.
		In this course we concentrate on understanding the properties, graphs, and applications of elementary functions (polynomial, exponential and logarithmic), solving and understanding the properties of solutions for polynomial equations, using trigonometric functions and their graphs to solve triangles and application problems involving triangles, and understanding trigonometric identities and equations and their uses.
	www.stclair.cc.mi.us /preview/mth113.htm (313 words)

	ACM TOMS: Bibliographic record for `Schweikard:1992:RZI' (Site not responding. Last check: 2007-10-17)
		An efficient method for the calculation of the coefficients of a corresponding algebraic polynomial is stated.
		The complexity of trigonometric zero isolation depending on the degree and the coefficient size of the given trigonometric polynomial is analyzed.
		In an experimental evaluation, the performance of the method is compared to the performance of recently developed numeric techniques for the approximate determination of all roots of trigonometric polynomials.
	math.nist.gov /toms/cgi-bin/TOMSbibget?Schweikard:1992:RZI (141 words)

	[No title]
		Topics covered include trigonometry, polar coordinates, polynomial functions, linear and polynomial inequalities, conic sections, sequences and series, and an introduction to calculus -- all with an emphasis on graphing.
		Analyze the relationships among the Unit 2: Polynomial and 2nd Qtr.
		Determine roots of polynomial Unit 2: Polynomial and 2nd Qtr.
	jj014.k12.sd.us /Classes/Precalculus/PrecCourse.doc (1083 words)

	SOA Honors Pre-Calculus Course Outline
		The course includes a study of polynomial, rational, exponential, logarithmic, and trigonometric functions.
		These functions include those that are linear, polynomial, rational, exponential, logarithmic, trigonometric, inverse trigonometric, and piecewise defined.
		In particular, before studying calculus, students must be familiar with the properties of functions, the algebra of functions, and the graphs of functions.
	www.soamath.net /psmyth/precalculus/SOAHPrecalculusCourseOutline.htm (735 words)

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