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In mathematics, trigonometric interpolation is interpolation with trigonometric polynomials. Interpolation is the process of finding a function which goes through some given data points . For trigonometric interpolation, this function has to be a trigonometric polynomial, that is, a sum of sines and cosines of given periods.
A trigonometric polynomial can be considered a periodic function on the real line, with period some divisor of , or as a function on the unit circle.. Trigonometric polynomials are dense in the space of continuous functions on the unit circle, with the uniform norm; [4] this is a special case of the Stone–Weierstrass theorem.
If we consider as a variable in a topological space, and the function () mapping to a Banach space, then the problem is treated as "interpolation of operators". [11] The classical results about interpolation of operators are the Riesz–Thorin theorem and the Marcinkiewicz theorem. There are also many other subsequent results.
The fractional part function has Fourier series expansion [19] {} = = for x not an integer. At points of discontinuity, a Fourier series converges to a value that is the average of its limits on the left and the right, unlike the floor, ceiling and fractional part functions: for y fixed and x a multiple of y the Fourier series given ...
One of the applications of the Mittag-Leffler function is in modeling fractional order viscoelastic materials. Experimental investigations into the time-dependent relaxation behavior of viscoelastic materials are characterized by a very fast decrease of the stress at the beginning of the relaxation process and an extremely slow decay for large ...
We fix the interpolation nodes x 0, ..., x n and an interval [a, b] containing all the interpolation nodes. The process of interpolation maps the function f to a polynomial p. This defines a mapping X from the space C([a, b]) of all continuous functions on [a, b] to itself.
When = + /, +, the Matérn covariance can be written as a product of an exponential and a polynomial of degree . [5] [6] The modified Bessel function of a fractional order is given by Equations 10.1.9 and 10.2.15 [7] as
The process of interpolation maps the function to a polynomial . This defines a mapping X {\displaystyle X} from the space C ([ a , b ]) of all continuous functions on [ a , b ] to itself. The map X is linear and it is a projection on the subspace Π n of polynomials of degree n or less.