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The first Chebyshev function ϑ (x) or θ (x) is given by = where denotes the natural logarithm, with the sum extending over all prime numbers p that are less than or equal to x. The second Chebyshev function ψ (x) is defined similarly, with the sum extending over all prime powers not exceeding x
Plot of the Chebyshev polynomial of the first kind () with = in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D. The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as () and ().
Chebyshev's inequality is important because of its applicability to any distribution. 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.
The type-generic macros that correspond to a function that is defined for only real numbers encapsulates a total of 3 different functions: float, double and long double variants of the function. The C++ language includes native support for function overloading and thus does not provide the <tgmath.h> header even as a compatibility feature.
A typical example of a Chebyshev space is the subspace of Chebyshev polynomials of order n in the space of real continuous functions on an interval, C[a, b]. The polynomial of best approximation within a given subspace is defined to be the one that minimizes the maximum absolute difference between the polynomial and the function.
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 ...
One can obtain polynomials very close to the optimal one by expanding the given function in terms of Chebyshev polynomials and then cutting off the expansion at the desired degree. This is similar to the Fourier analysis of the function, using the Chebyshev polynomials instead of the usual trigonometric functions.
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 ...