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The Chebyshev polynomials form a complete orthogonal system. The Chebyshev series converges to f(x) if the function is piecewise smooth and continuous. The smoothness requirement can be relaxed in most cases – as long as there are a finite number of discontinuities in f(x) and its derivatives. At a discontinuity, the series will converge to ...
Clenshaw–Curtis quadrature and Fejér quadrature are methods for numerical integration, or "quadrature", that are based on an expansion of the integrand in terms of Chebyshev polynomials. Equivalently, they employ a change of variables x = cos θ {\displaystyle x=\cos \theta } and use a discrete cosine transform (DCT) approximation for ...
The Chebyshev pseudospectral method for optimal control problems is based on Chebyshev polynomials of the first kind. It is part of the larger theory of pseudospectral optimal control , a term coined by Ross . [ 1 ]
It was picked up by Shrapnel Games in 2000 and spawned the follow-on products, BCT Expansion Pack 1 & 2 and BCT Construction Kit. Just before the roll out of ProSIM's second generation engine, Armored Task Force , the game and all of its expansions were repackaged as a single product and, along with a number of upgrades, renamed BCT Commander.
There are a very large number of ideas that fall under the general banner of pseudospectral optimal control. [7] Examples of these are the Legendre pseudospectral method, the Chebyshev pseudospectral method, the Gauss pseudospectral method, the Ross-Fahroo pseudospectral method, the Bellman pseudospectral method, the flat pseudospectral method and many others.
In applied mathematics, a discrete Chebyshev transform (DCT) is an analog of the discrete Fourier transform for a function of a real interval, converting in either direction between function values at a set of Chebyshev nodes and coefficients of a function in Chebyshev polynomial basis. Like the Chebyshev polynomials, it is named after Pafnuty ...
As a result of its generality it may not (and usually does not) provide as sharp a bound as alternative methods that can be used if the distribution of the random variable is known. To improve the sharpness of the bounds provided by Chebyshev's inequality a number of methods have been developed; for a review see eg. [12] [37]
The Chebyshev functions, especially the second one ψ (x), are often used in proofs related to prime numbers, because it is typically simpler to work with them than with the prime-counting function, π (x) (see the exact formula below.) Both Chebyshev functions are asymptotic to x, a statement equivalent to the prime number theorem.