Transforms of Some Important Functions
As mentioned above, every constant function is its own Weierstrass transform. The Weierstrass transform of any polynomial is a polynomial of the same degree. Indeed, if Hn denotes the (physicist's) Hermite polynomial of degree n, then the Weierstrass transform of Hn(x/2) is simply xn. This can be shown by exploiting the fact that the generating function for the Hermite polynomials is closely related to the Gaussian kernel used in the definition of the Weierstrass transform.
The Weierstrass transform of the function eax (where a is an arbitrary constant) is ea2 eax. The function eax is thus an eigenvector for the Weierstrass transform. (This is in fact more generally true for all convolution transforms.) By using a=bi where i is the imaginary unit, and using Euler's identity, we see that the Weierstrass transform of the function cos(bx) is e−b2 cos(bx) and the Weierstrass transform of the function sin(bx) is e−b2 sin(bx).
The Weierstrass transform of the function eax2 is if a < 1/4 and undefined if a ≥ 1/4. In particular, by choosing a negative, we see that the Weierstrass transform of a Gaussian function is again a Gaussian function, but a "wider" one.
Read more about this topic: Weierstrass Transform
Famous quotes containing the words transforms, important and/or functions:
“Now, since our condition accommodates things to itself, and transforms them according to itself, we no longer know things in their reality; for nothing comes to us that is not altered and falsified by our Senses. When the compass, the square, and the rule are untrue, all the calculations drawn from them, all the buildings erected by their measure, are of necessity also defective and out of plumb. The uncertainty of our senses renders uncertain everything that they produce.”
—Michel de Montaigne (1533–1592)
“Any important disease whose causality is murky, and for which treatment is ineffectual, tends to be awash in significance.”
—Susan Sontag (b. 1933)
“The English masses are lovable: they are kind, decent, tolerant, practical and not stupid. The tragedy is that there are too many of them, and that they are aimless, having outgrown the servile functions for which they were encouraged to multiply. One day these huge crowds will have to seize power because there will be nothing else for them to do, and yet they neither demand power nor are ready to make use of it; they will learn only to be bored in a new way.”
—Cyril Connolly (1903–1974)