Talks given at NuHAG events

Convergence of Greedy Approximation for the Trigonometric System

  Vladimir Temlyakov

  given at  strobl07 (20.06.07 11:00)
  id:  671
  length:  50min
  status:  invited
  type:  talk

We study the following nonlinear method of approximation by trigonometric
polynomials. For a periodic function $f$ we take as an approximant a trigonometric polynomial of the form $G_m(f) := \sum_{k \in \Lambda} \hat f(k) e^{i(k,x)} $, where $\Lambda \subset \Bbb Z^d$ is a set of cardinality $m$ containing the indices of the $m$ biggest (in absolute value) Fourier coefficients $\hat f(k)$ of function $f$. Note that $G_m(f)$ gives the best $m$-term approximant in the $L_2$-norm and, therefore, for each $f\in L_2$, $\|f-G_m(f)\|_2 \to 0$ as $m\to \infty$.
It is known from previous results that in the case of $p\neq 2$ the condition $f\in L_p$ does not guarantee the convergence $\|f-G_m(f)\|_p \to 0$ as $m\to \infty$. We study the following question. What conditions (in addition to $f\in L_p$) provide the
convergence $\|f-G_m(f)\|_p \to 0$ as $m\to \infty$?

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