This covers all transformations of functions by integrals, including but not limited to the Radon, X-Ray, Hilbert, Mellin transforms. Use (wavelet-transform), (laplace-transform) for those respective transforms, and use (fourier-analysis) for questions about the Fourier and Fourier-sine/cosine ...

learn more… | top users | synonyms

47
votes
4answers
8k views

Connection between Fourier transform and Taylor series

Both Fourier transform and Taylor series are means to represent functions in a different form. My question: What is the connection between these two? Is there a way to get from one to the other (and ...
43
votes
3answers
974 views

Laplace, Legendre, Fourier, Hankel, Mellin, Hilbert, Borel, Z…: unified treatment of transforms?

I understand "transform methods" as recipes, but beyond this they are a big mystery to me. There are two aspects of them I find bewildering. One is the sheer multiplicity of such transforms. Is ...
40
votes
1answer
1k views

To evaluate $\int_0^{+\infty} \frac{\;\mathrm dx}{\sqrt[3]{x^3+a^3}\sqrt[3]{x^3+b^3}\sqrt[3]{x^3+c^3}}$

$$f(a,b)=\int_0^{+\infty} \frac{\;\mathrm dx}{\sqrt{x^2+a^2}\sqrt{x^2+b^2}}$$ To use Landen's transformation $$f(a,b)=\int_0^{+\infty} \frac{\;\mathrm ...
26
votes
2answers
3k views

Why do engineers use the Z-transform and mathematicians use generating functions?

For a (complex valued) sequence $(a_n)_{n\in\mathbb{N}}$ there is the associated generating function $$ f(z) = \sum_{n=0}^\infty a_nz^n$$ and the $z$-Transform $$ Z(a)(z) = \sum_{n=0}^\infty ...
20
votes
7answers
13k views

Laplace transformations for dummies

Is there a simple explanation of what the Laplace transformation do exactly and how they work? Reading my math book has left me in a foggy haze of proofs that I don't completely understand. I'm ...
18
votes
1answer
2k views

Does a Fourier transformation on a (pseudo-)Riemannian manifold make sense?

the Fourier transformation of a scalar function with respect to one variable might be defined as $\mathcal{F}\left[w\right](\omega )\equiv ...
15
votes
1answer
458 views

Is Fourier transform characterized by its diagonalization properties?

Let us fix the following convention for the Fourier transform in $L^1(\mathbb{R})$ space: $$\hat{f}(\xi)=\int_{-\infty}^\infty f(x)\, e^{-2\pi i x\xi}\, dx.$$ We then have the following properties: ...
10
votes
1answer
183 views

A particular case of Truesdell's unified theory of special functions

I'm reading through Clifford Truesdell's "An essay toward a unified theory of special functions", Princeton Univ. Press, 1948. All his exposition is based on the functional equation ...
10
votes
4answers
363 views

Image of closed ball under degenerate integral operator is a closed set

I will be putting a bounty on this problem as soon as it lets me. For those who want to understand where the problem came from I encourage reading the edits, as I cut out several failed attempts and ...
10
votes
1answer
507 views

Are Laplace Transforms a Special Case of Fourier Transforms?

A Laplace Transform is based on the integral: $F(\xi) = \int_0^{\infty} f(x) e^ {-\xi x}\,dx.$ In a roundabout way, a Fourier transform can get to $\hat{f}(\xi) = \int_{-\infty}^{\infty} f(x)\ e^{- ...
9
votes
1answer
62 views

What are the “right” spaces for the Laplace transform

There are for example several canonical spaces to define the Fourier transform (i.e. Schwartz's space). Is there also a particularly suitable space to define the Laplace transform, so that the Laplace ...
8
votes
2answers
481 views

Fourier series is to Fourier transform what Laurent series is to …?

Since the coefficients $$a_k = \frac1{2\pi i}\oint_C\frac{f(z)}{(z-c)^{k+1}}\,dz$$ for the Laurent series $$f(z)\Big|_{r\le|z|\le R} = \sum_{k=-\infty}^{\infty}a_k\cdot(z-c)^k $$ of a function ...
7
votes
1answer
2k views

What's the connection between the Laplace transform and the Fourier transform?

Both the Laplace transform and the Fourier transform in some sense decode the "spectrum" of a function. The Laplace transform gives a power-series decomposition whereas the Fourier transform gives a ...
7
votes
1answer
224 views

Mellin transform of digamma function

what is the Mellin trasnform of the Digamma function ?? from Ramanujan master theorem http://mathworld.wolfram.com/RamanujansMasterTheorem.html y believe it should be equal to $$ ...
7
votes
1answer
146 views

Fourier transform using principal value

Can anyone help me compute the Fourier transform of $ 1/|x|^{n-\alpha} $ in $\mathbb{R}^n $ where $ 0 < \alpha < n $ ? Somehow it becomes the principal value of $ 1/|x|^\alpha $ which I can't ...
7
votes
1answer
711 views

Meaning of polynomially bounded

I am currently reading through some lecture notes on Fourier Transform and Distributions on my own, and came upon the notion of a polynomially bounded function. I am not sure I understand this ...
7
votes
1answer
279 views

Evaluate $\int\limits_{\mathbb{R}^2} \frac{e^{i \langle \xi, x \rangle} d\xi}{ \langle\xi,\theta\rangle}$

The task is to evalute $$ \int\limits_{\mathbb{R}^2} \frac{e^{i \langle \xi, x \rangle} d\xi}{ \langle\xi,\theta\rangle}, \;\;\; \theta \in \mathbb{C}^2 \setminus ( \mathbb{S}^1 \cup \left\{ 0 ...
7
votes
1answer
74 views

Where to submit a small work of applied math written by a physicist?

Working in physics, I recently discovered a mathematical identity useful to solve a particular partial differential equation. Using the same idea, I found several other identities but I do not know ...
6
votes
1answer
1k views

Fourier transform of $\text{sinc}^3 {\pi t}$

$$f(t)=\frac{\sin^3(\pi t)}{(\pi t)^3}$$ I want to calculate the Fourier transform. I can't calculate this integral: $$\int_0^\infty\frac{\sin^3(\pi t)}{(\pi t)^3}\cos(ut)\,\mathrm{d}t$$
6
votes
4answers
141 views

Change of variables Double integral

I have $$\iint_A \frac{1}{(x^2+y^2)^2}\,dx\,dy.$$ $A$ is bounded by the conditions $x^2 + y^2 \leq 1$ and $x+y \geq 1$. I initially thought to make the switch the polar coordinates, but the line ...
6
votes
2answers
280 views

Usage of inverse Laplace transform

At my current study level in college, use of inverse Laplace transform is not mentioned well - textbooks say "use tables." So, can anyone show me how to use inverse Lapalce transform? And also proof? ...
6
votes
2answers
82 views

Mellin Transform of the floor function

I have from integral transform tables that: $$\operatorname{Mellin}(\lfloor x\rfloor) = -\dfrac{\zeta(-z)}{z},\quad \operatorname{Re}(z)<-1$$ How can this be proved?
6
votes
1answer
967 views

Bessel function integral and Mellin transform

Gradshteyn&Ryzhik 6.635.3 provides the following integral, with the usual constraints on $\nu,\alpha,\beta$, $$\int\limits_0^\infty \exp\left(-\frac{\alpha}{x}-\beta x\right)J_\nu(\gamma ...
6
votes
1answer
237 views

Stieltjes Transform

Stieltjes transform for a distribution $F(x)$ is defined as $$m(z)=\int \frac{dF(x)}{x−z}$$ where z is complex with positive imaginary parts and $F(x)$ is a distribution function. Basically, I am ...
6
votes
1answer
43 views

Integral transforms and uncertainty products

Heisenberg's uncertainty principle is well-studied and has become a bit of a pop science phenomenon due to its widespread implications in quantum mechanics. (Though interpretations are often ...
6
votes
1answer
157 views

How fast does the function $\displaystyle f(x)=\lim_{\epsilon\to0}\int_\epsilon^{\infty} \dfrac{e^{xt}}{t^t} \, dt $ grow?

Let $x$ be a positive real number and $f(x):=\lim_{\epsilon\to0}\int_\epsilon^{\infty} \dfrac{e^{xt}}{t^t} \, dt $. How fast does this function grow ? In other words can we find a good asymptote for ...
6
votes
1answer
356 views

Qualitative interpretation of Hilbert transform

the well-known Kramers-Kronig relations state that for a function satisfying certain conditions, its imaginary part is the Hilbert transform of its real part. This often comes up in physics, where ...
5
votes
2answers
515 views

integral transforms: why do roots in frequency domain correspond to eigenvalues in time domain (and how does it help solve differential equations)?

In Wikipedia you can read about integral transforms, esp. the Laplace transform which maps a differential equation in the time domain into a polynomial equation in the complex frequency domain: ...
5
votes
1answer
193 views

A Parseval-like theorem for Mellin transforms

A particular case of Parseval's theorem for Fourier transforms says that if $f$ is square integrable on $\mathbb{R}$, then $$ \int_{-\infty}^{\infty} |f(t)|^{2} \ dt = \int_{-\infty}^{\infty} ...
5
votes
1answer
255 views

Convolution square root of $\delta $

I want to somehow classify the distributional solutions of the equation $$ f \ast f = \delta $$ where $\delta = \delta _0$ is the Dirac delta distribution. Clearly, by Fourier transformation, we ...
5
votes
1answer
235 views

Multiple Fourier Integrals involving Heaviside Theta Function

I want to evaluate the integral: $$I=\int_{-\infty}^{\infty}dx_1 \int_{-\infty}^{\infty}dx_2 \ \Theta(x_1-x_2) \ e^{i(ax_1+bx_2)}$$ where $\Theta(x)$ is the Heaviside function. What I was doing now ...
5
votes
1answer
1k views

Hilbert transform and Fourier transform

Assume the following relationship between the Hilbert and Fourier transforms: $$ \mathcal{H}(f) = {\mathcal{F}^{-1}}(-i ~ \text{sgn}(\cdot) \cdot \mathcal{F}(f)), $$ where $ \displaystyle ...
4
votes
2answers
187 views

Why does it seem I can't apply the Radon transform to the Helmholtz equation?

Say we a function $u$ and a bounded region $\Omega \subset \mathbb{R}^2$, such that $(\Delta+\lambda)u = 0$ everywhere, and $u=0$ on the boundary. We extend it to the entire plane by defining $u=0$ ...
4
votes
1answer
4k views

What exactly is the Probability Integral Transform?

I've been going back over my notes from Stats class and came across the Probability Integral Transform. From my limited understanding, the basic idea is that from a cdf in terms of one variable, can ...
4
votes
2answers
189 views

Laplace transform of $[f''[x]]^n$

Can anyone help me get this Laplace transform, $$ L[(f''(x))^n] $$ where $f'(0)=0$ and $f''(0)=0$ and $n$ is power of $$f''(x)$$?
4
votes
1answer
89 views

Continuous version of the Möbius inversion theorem

Is there a continuous version of Möbius Inversion. Essentially, using integrals instead of sums.
4
votes
1answer
437 views

Is the Mellin transform useful to solve differential equations?

The Mellin transform is defined as: $$F(\mu)=\int_0^\infty f(x)x^{\mu-1}dx$$ The derivative of the Mellin transform is: $$F'(\mu)=-(\mu-1)F(\mu-1)$$ Applying this property, for example to the Bessel ...
4
votes
1answer
2k views

Solving an initial value ODE problem using fourier transform

I am a physics undergrad and studying some transform methods. The question is as follows: $y^{\prime \prime} - 2 y^{\prime}+y=\cos{x}\,\,\,\,y(0)=y^{\prime}(0)=0\,\,\, x>0$ I am having some ...
4
votes
3answers
208 views

A convolution like integral equation

I would like to solve the following integral equation for $g(z)$. $$\int_z^\infty g(\zeta)(\zeta-z)^{\alpha-1} d\zeta = e^{-bz}, \tag{1}$$ where $\alpha$ and $b$ are constants. I would also like to ...
4
votes
1answer
120 views

An inequality involving arctan of complex argument

I have the following conjecture: \begin{equation} \text{Re}\left[(1+\text{i}y)\arctan\left(\frac{t}{1+\text{i}y}\right)\right] \ge \arctan(t), \qquad \forall y,t\ge0. \end{equation} Which seems to be ...
4
votes
2answers
139 views

Fourier, Laplace, … and other Integral-transformations

I know Laplace, Fourier and Mellin-Transformation. Is there a general theory of transformations? My main interest is about classification of transformations satisfying specified properties like ...
4
votes
2answers
65 views

Computation of the fourier transformation of a function with a matrix

I want to compute the Fourier transformation of the following function: \begin{align} f:& \mathbb R^n \rightarrow \mathbb R \\ & x \mapsto \exp(-\left<Ax,x\right>) \end{align} where ...
4
votes
2answers
291 views

Mellin Transform of $\sin$

Can one show that the following integral converges on $-1<\Re s < 1$ and define holomorphic function of $s$? $$\int_0^\infty \sin(y) y^{s-1} dy$$ I've googled for a while, but I could not ...
4
votes
0answers
29 views

Estimating the (double) Riesz transform.

I'm trying to verify the following estimate, which appears in a paper I'm reading. It seems I'm missing something easy, I just can't figure this out. $\textbf{Background}:$ For a function $f \in ...
4
votes
1answer
86 views

How to develop the Fourier Transform in my mind now that I know the Fourier Seires?

I know that we can represent some function $f$ in this way: $$f(t) = \frac{a_0}{2} + \sum_{n=1}^\infty a_n\cos\left(\frac{n\pi t}{L}\right) + \sum_{n=1}^\infty b_n\sin\left(\frac{n\pi t}{L}\right)$$ ...
4
votes
0answers
104 views

Involutive fourier transform

The writer here states I am introducing a viewpoint (the involutive convention) which makes the Fourier transform its own inverse (i.e., the Fourier transform so defined is an involution). ...
4
votes
0answers
62 views

Some properties of an analogue of the integral Fourier operator

Let $\phi(x,\theta)$ be an infinitely differentiable function in $X^2 = X \times X$, where $X = \mathop{\mathsf{int}}\mathbb{R}^n_{+}$, and let$\phi(\lambda x, \theta) = \phi(x,\lambda \theta) = ...
4
votes
0answers
580 views

Do kernel functions of integral transforms have any special properties?

From the Wikipedia page on integral transforms, it states that: ...an integral transform is any transform $T$ of the following form: $$ (Tf)(u)=\int^{t_2}_{t_1}K(t,u)f(t)dt $$ ...There are ...
4
votes
0answers
8k views

Relationship Between The Z-Transform And The Laplace Transform

Below I've quoted Wikipedia's entry that relates the Z-Transform to the Laplace Transform. The part I don't understand is $z \ \stackrel{\mathrm{def}}{=}\ e^{s T}$; I thought $z$ was actually an ...
3
votes
2answers
422 views

Series of nested integrals

I'm trying to calculate the following series of nested integrals with $\varepsilon(t)$ being a real function. $$\sigma = 1 + \int_{t_0}^t\mathrm dt_1 \, \varepsilon(t_1) + \int_{t_0}^t\mathrm dt_1 ...