fractional Mellin transform
时间: 2023-11-28 20:06:33 浏览: 27
The fractional Mellin transform is a generalization of the classical Mellin transform, which is a mathematical operation that maps a function of a real variable to a function of a complex variable. The fractional Mellin transform extends this concept by allowing the order of the transform to take on fractional values.
The fractional Mellin transform is defined as:
M_{a,b}^\alpha(f(x)) = \frac{1}{\Gamma(\alpha)} \int_0^\infty x^{a-1} (\ln x)^{b-1} f(x) K_\alpha\left(\frac{x}{z}\right) \frac{dz}{z},
where a and b are real constants, f(x) is the function being transformed, and K_\alpha(x) is the modified Bessel function of the second kind.
The order of the transform, denoted by \alpha, determines the degree of smoothness of the transformed function. When \alpha is an integer, the fractional Mellin transform reduces to the classical Mellin transform. When \alpha is a non-integer, the fractional Mellin transform can be used to analyze functions that do not have integer order derivatives, such as fractal functions.
The fractional Mellin transform has applications in a variety of fields, including signal processing, image processing, and fractal analysis. It has been used to analyze the scaling properties of complex systems, to extract features from images, and to filter signals.