Tagged Questions

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 ...

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Properties of Eigenfunctions of a Kernel

I'm a newbie and may be this question is bit simple for you but pardon me if it's too simple and provide me some references. I've and Kernel function $K(x,y)$ $f(x)=(Kg)(x)=\int_{\Omega}K(x,y)g(y)dy$...
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calculate $\int_{-\infty}^{+\infty} \cos(at) e^{-bt^2} dt$

Could someone please help me to calculate the integral of: $$\int_{-\infty}^{+\infty} \cos (at) e^{-bt^2} dt.$$ a and b both real, b>0. I have tried integration by parts, but I can't seem to ...
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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 ...
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fancy about inverse discrete Fourier sine and cosine transform (i.e. Fourier sine and cosine series)

In order to find $f(x)$ so that $F(u)=\sum\limits_{x=0}^\infty f(x)\sin\dfrac{\pi ux}{L}$ and $F(u)=\sum\limits_{x=0}^\infty f(x)\cos\dfrac{\pi ux}{L}$ , we can borrow the idea from Fourier sine ...
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Help me to understand the Gaussian blurring

Here is an unknown luminosity function $f(x,y)$ and its integration results: $$p_{i,j}= \frac{\iint\limits_{D_{i,j}} \! f(x,y) \, dx \, dy}{\iint\limits_{D_{i,j}} \,dx\,dy}$$ I need to express the ...
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Can we solve for $a$ in $b(x) =\int_{-\infty}^\infty b(s)a(x,s)ds$

For all $x \in \mathbb{R}$, $$b(x) =\int_{-\infty}^\infty b(s)a(x,s)ds.$$ If it helps, we can assume that $a, b$ are continuous, nonnegative, and $\int_{-\infty}^\infty$ of $a$ or $b$ are both ...
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Domain rotations in the Mellin integral transform

Let's consider the following form of the Mellin integral transform: $$m_{pq} =\iint\limits_{D_R} \! x^p y^q f(x,y) \, dx\; dy, \, D_R={\{(x,y)\,|\,x^2 + y^2 \le R^2\}}$$ If we scale the domain of the ...
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Commutativity of Convolution in higher dimensions

I have a basic question about how to show that convolution in dimension $n$ is commutative - or maybe it is rather a question about change of variables .. So on $\mathbb{R}$ I know how to show ...
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What does it mean to carry out calculations modulo $S^{- \infty}$

I have trouble making sense of the following remark that is given in a set of lecture notes I am currently going through on my own: " Oscillatory integrals are used for the study of singularities of ...
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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 concept,...
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How to apply the solution of $y(n) = (0.85)y(n-1) + x(n)$ to data

I learned how to solve difference equation $y(n) = (0.85)y(n-1) + x(n)$ using z Transform, and inverse z Transform, I get $h(n) = 0.85^n u(n)$ where $u(n)$ is unit step sequence. Now my ...
Recently I was studying the Mellin convolution technique to solve definite integrals. I am just wondering is the technique valid only for any definite integrals (with any range $\int^b_a$)? Or only ...