Search results
Results from the WOW.Com Content Network
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
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.
The abundance of the theorems and identities inherited from Fourier series make the Chebyshev polynomials important tools in numeric analysis; for example they are the most popular general purpose basis functions used in the spectral method, [16] often in favor of trigonometric series due to generally faster convergence for continuous functions ...
For better Chebyshev nodes, however, such an example is much harder to find due to the following result: Theorem — For every absolutely continuous function on [−1, 1] the sequence of interpolating polynomials constructed on Chebyshev nodes converges to f ( x ) uniformly.
Even order Chebyshev filters implemented with passive elements, typically inductors, capacitors, and transmission lines, with terminations of equal value on each side cannot be implemented with the traditional Chebyshev transfer function without the use of coupled coils, which may not be desirable or feasible, particularly at the higher ...
The algorithm is most useful when () are functions that are complicated to compute directly, but () and () are particularly simple. In the most common applications, α ( x ) {\displaystyle \alpha (x)} does not depend on k {\displaystyle k} , and β {\displaystyle \beta } is a constant that depends on neither x {\displaystyle x} nor k ...
Since the function is antisymmetric, it is seen there are three zeroes and three poles. Between the zeroes, the function rises to a value of 1 and, between the poles, the function drops to the value of the discrimination factor L n Plot of the absolute value of the fourth order elliptic rational function with ξ=1.4. Since the function is ...
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.