RECOVERING PROBABILITY FUNCTIONS WITH FOURIER SERIES

Functions on circles : Fourier series

1. Provocative example 2. Natural function spaces on the circle S = R/2πZ 3. Topology on C∞(S1) 4. Pointwise convergence of Fourier series 5. Distributions: generalized functions 6. Invariant integration, periodicization 7. Hilbert space theory of Fourier series 8. Levi-Sobolev inequality, Levi-Sobolev imbedding 9. C∞(S1) = limC(S) = limH(S) 10. Distributions, generalized functions, again 11. T...

Recovering piecewise smooth functions from nonuniform Fourier measurements

In this paper, we consider the problem of reconstructing piecewise smooth functions to high accuracy from nonuniform samples of their Fourier transform. We use the framework of nonuniform generalized sampling (NUGS) to do this, and to ensure high accuracy we employ reconstruction spaces consisting of splines or (piecewise) polynomials. We analyze the relation between the dimension of the recons...

Lp Computable Functions and Fourier Series

This paper studies how well computable functions can be approximated by their Fourier series. To this end, we equip the space of L-computable functions (computable Lebesgue integrable functions) with a size notion, by introducing L-computable Baire categories. We show that L-computable Baire categories satisfy the following three basic properties. Singleton sets {f} (where f is L-computable) ar...

L-computable Functions and Fourier Series

This paper studies the notion of Lebesgue integrable computable functions (denoted L-computable where p is an integer), that naturally extends the classical model of bitcomputable functions. We introduce L-computable Baire categories. We observe that L-computability is incomparable to the recently introduced notion of graph-computable functions. We study the convergence of Fourier series for L-...

Nonanalytic Functions of Absolutely Convergent Fourier Series.