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This sum is called a Chebyshev series or a Chebyshev expansion. Since a Chebyshev series is related to a Fourier cosine series through a change of variables, all of the theorems, identities, etc. that apply to Fourier series have a Chebyshev counterpart. [16] These attributes include: The Chebyshev polynomials form a complete orthogonal system.
Ocean mixed layer carbon, c m, is the only explicitly modelled ocean stock of carbon; though to estimate carbon cycle feedbacks the total ocean carbon is also calculated. [ 107 ] Current trends in climate change lead to higher ocean temperatures and acidity , thus modifying marine ecosystems. [ 108 ]
Atmospheric carbon is exchanged quickly between the oceans and the terrestrial biosphere. This means that at times the atmosphere acts as a sink, and at other times as a source of carbon. [2] The following section introduces exchanges between the atmospheric and other components of the global carbon cycle.
The use of Chebyshev polynomials enables highly precise, efficient calculations for any given point in time. DE405 calculation for the inner planets "recovers" accuracy of about 0.001 seconds of arc (arcseconds) (equivalent to about 1 km at the distance of Mars); for the outer planets it is generally about 0.1 arcseconds.
In numerical analysis, the Clenshaw algorithm, also called Clenshaw summation, is a recursive method to evaluate a linear combination of Chebyshev polynomials. [1] [2] The method was published by Charles William Clenshaw in 1955.
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.
Many applications for Chebyshev nodes, such as the design of equally terminated passive Chebyshev filters, cannot use Chebyshev nodes directly, due to the lack of a root at 0. However, the Chebyshev nodes may be modified into a usable form by translating the roots down such that the lowest roots are moved to zero, thereby creating two roots at ...
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.