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Results tagged with fourier-analysis
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user 14414
The representation of functions (or objects which are in some generalize the notion of function) as constant linear combinations of sines and cosines at integer multiples of a given frequency, as Fourier transforms or as Fourier integrals.
1
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What's the relationship between the roots of a function and that of a filtered Fourier serie...
If the cut off for the low pass filtering (partial sum) is sufficiently high, the number of zeros of $f(M)$ is equal to the number of sign changes of the binary wave $M$. Although the Gibb's phenomeno …
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0
answers
211
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Computing a closed form representation for a Fourier series summation
I want to compute a closed form representation for the below given summation expession.
$$g_{\lambda}(\boldsymbol{x}) = \sum\limits_{\boldsymbol{l}\in\mathbb{Z}^m} \frac{1}{1+\lambda\|\boldsymbol{l}\ …
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3
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2
answers
440
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A problem on real valued functions in $\mathbb{R}^2$ with least variation
Let $\alpha(s) = (x(s),y(s))$ be the arc length parametrization of a plane, smooth, closed, convex curve, of length $L$. Let $J:(0,L)\to\mathbb{R}$ be a smooth and Bounded variation (BV introduced aft …
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A problem on real valued functions in $\mathbb{R}^2$ with least variation
I roughly sketch a solution without a proof, and also a possible minimum possible value for total variation of $f$.
Let $f$ assume values of $J$ on the boundary curve $\alpha$. We construct infinite …
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-4
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1
answer
203
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A Question in Fourier Analysis proposing a conjecture
Let $f$ be a $2\pi$ periodic BV function whose derivative is also BV.Let the amount of jump at a point $x$ is denoted as $\lfloor f \rfloor (x) = f(x+0)-f(x-0)$ Define function $J:\mathbb{R} \to\mathb …
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A Question in Fourier Analysis proposing a conjecture
These are proofs for first and second conjectures.
This is a proof to the second conjecture.
Let $$V_n = V_a^b(Y_n) = \int_a^b\left|\frac{1}{\log(n) G(n)}\sum_{k=1}^{n}\tilde{S}'_k(x;f)\left|c_k\righ …
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0
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Does the Total variation of the Fourier partial sum of a bv function with jumps converge to ...
Does the total variation of the Fourier partial sum of a piecewise continuous bv function converge to the total variation of the function as $N\to\infty$. To explain briefly,
Let $f$ be a periodic B …
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4
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1
answer
486
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What is the importance of convergence of variation of Fourier reconstruction to that of vari...
Let $f$ be a periodic function of bounded variation which jumps at a point $x_0\in\mathbb{R}$. Let $S_{N}[f]$ denote the partial Fourier sum of $f$ and let $C_{N}[f]$ denote the Cesaro partial sum. It …
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0
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1
answer
227
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Can we construct a sequence of trigonometric polynomials that converges pointwise to a given...
Consider any continuous function $f$ on an $m$-dimensional Torus $\mathbb{T}^m$. Can we construct a sequence of band limited functions (trigonometric polynomials), with the band width (degree of the t …
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2
votes
0
answers
99
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Closed form expression for this Fourier summation?
Consider the function $f:\mathbb{T}^m\to\mathbb{R}$
$$f(\boldsymbol{x}) = \sum\limits_{{\pmb{\eta}\in\mathbb{Z}^m}}\frac{1}{1+\lambda\|\pmb{\eta}\|_{2k}^{2k} } \cos({2\pi \pmb{\eta}\cdot\pmb{x}})$$
…
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5
votes
0
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219
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Can we construct a computable sequence of trigonometric polynomials that converges pointwise...
Consider any continuous function $f$ on an $m$-dimensional torus $\mathbb{T}^m$. Can we construct a sequence of band limited functions (trigonometric plynomials), with the band width (degree of the tr …
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1
vote
1
answer
194
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A metric on the set of BV functions, is it mentioned/studied in literature?
I'd like to propose the following metric which operates on the set $M$ of all square integrable functions that are also of bounded variation, of the form $f : (0,1) \to \mathbb{R}$.
Given any $x,y \in …
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5
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3
answers
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Property/Relations using Fourier series/transform, which give complete information about all...
Consider a function which has only jump singularities of the form of the function itself or one of its derivatives jumping. Now let $\hat{f}(k)$ be its Fourier transform/series. We know the decay of t …
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-1
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1
answer
222
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separable BV space for PDE's, Whats stopping us? [closed]
Consider the metric space BV(0,1) with the following metric
$$ d(u,v) = \|u-v\|_{L^1} + |TV(u)-TV(v)| $$. It is separable, compact, uniformly bounded and complete. So What is the really obvious thing …
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2
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0
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290
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A question on convergence rates of Fourier series and strict convergence
Consider BV functions on a torus. The Fourier partial sum using the first $n$ coefficients will converge to the function at every point of continuity, as $n\to\infty$. The convergence rate is $O(1/n)$ …
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