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Results tagged with harmonic-analysis
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user 14414
Harmonic analysis is a generalisation of Fourier analysis that studies the properties of functions. Check out this tag for abstract harmonic analysis (on abelian locally compact groups), or Euclidean harmonic analysis (eg, Littlewood-Paley theory, singular integrals). It also covers harmonic analysis on tube domains, as well as the study of eigenvalues and eigenvectors of the Laplacian on domains, manifolds and graphs.
17
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Is this statement which relates the Fourier transform of a function to its singularities cor...
I am working on a problem, which would possibly relate the Fourier transform/series with the jump singularities of the function where the function itself or one of its derivatives jump. ((some kind of …
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5
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3
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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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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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Is flatness of Wigner Ville Distribution of error function in Fourier Approximation possible...
For a real valued function $f(t)$ I want to check the information left, after taking a Fourier partial sum/integral. Let $\hat{f}$ be its Fourier transform and let $$e_{\omega}(t) = f(t) - \int\limits …
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2
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request for any expository works in pointwise convergence of double Fourier series and espec...
Quart. J. Math. Volume 37, Issue 1, Pages 53-79
On double Fourier series, and especially those which represent the double zeta-function with real and incommensurable parameters.
Hardy, G.H.
I am not …
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2
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0
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194
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Shattering with sinusoids
Let $d \geq 2$ and $K$ some positive integer. Consider distinct points $\theta_1, \ldots, \theta_K\in \mathbb{T}^d$ and (not necessarily distinct) $z_1, \ldots, z_K \in \{-1,1\}$ such that $\sum\limit …
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Does the Total variation of the Fourier partial sum of a bv function with jumps converge to ...
Does the total variation of the Fourier partial sum of a piecewise continuous bv function converge to the total variation of the function as $N\to\infty$. To explain briefly,
Let $f$ be a periodic B …
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Is this statement which relates the Fourier transform of a function to its singularities cor...
I would like to prove for the case of jump discontinuity of the function itself. (rather than that of one of its derivatives).
Let $t_0>0$ be a point where $f$ jumps. The curve $$(X_{t_0}(s),Y_{t_0}( …
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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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What is the growth of sum of absolute values of Fourier coefficients
For a periodic BV function $f$ which has jump discontinuties, is there any theorem in Fourier analysis which gives like $$\sum_{k=0}^n\left|c_k\right|\sim C\log\left(n\right)$$ where $C$ is a constant …
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A question about pointwise convergence of Fourier transform in $N$-dimensions
I am retreating back on this statement, after some explorations and calculation
Bow to Willie and others who were skeptical on this. Main difficulty can be seen in this reference. But I must mention t …
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