# Accurate estimate/lower bound for the l2 approximation error of trigonometric polynomial approximation

This is a restated version of my original very broad question.

Let $$P$$ be probability a measure on an interval $$[a,b]$$ ($$-\infty) that's dominated by Lebesgue measure. Let $$\langle f, g \rangle_P=\int_{[a,b]} f \cdot g \, dP$$ be an inner product for integrable functions associated with $$P$$ and $$\| \cdot \|_{L_2(P)}$$ denote the induced $$L_2$$-norm. Let $$f: [a,b] \rightarrow \mathbb{R}$$ be integrable and $$f \in C^r[a,b]$$ for some $$r \leq \infty$$. Consider approximating $$f$$ by a trigonometric polynomial $$a_0+\sum_{k=1}^K [a_k \cos(kx) + b_k \sin(kx)]$$. Let $$\pi_{K,P}(f)$$ denote the projection of $$f$$ onto the space of $$K$$-th order trigonometric polynomials with respect to $$\langle \cdot, \cdot \rangle_P$$. Consider the approximation error $$\| f-\pi_{K,P}(f) \|_{L_2(P)}$$. Assume $$f$$ is not a trigonometric polynomial itself.

I've seen an upper bound of this error that looks like $$C K^{-r}$$ when $$r < \infty$$ and is derived from the $$L_\infty$$ error (https://www.springer.com/us/book/9783540506270). Is there a lower bound for the $$L_2(P)$$ error, ideally with the same structure as the upper bound (maybe under mild conditions)? Also, what would be the upper bound and lower bound if $$r=\infty$$? I'm particularly interested in the case where $$f(x)=x$$.

You may take $$[a,b]$$ to be any interval you want for convenience.

I would be very happy if you know of any results of this kind, perhaps under similar settings (e.g. a different notion of smoothness for $$f$$ in terms of Soblev spaces).

Original question:

Let $$P$$ be probability a measure on an interval $$[a,b]$$. Let $$\langle f, g \rangle_P=\int_{[a,b]} f \cdot g \, dP$$ be an inner product for integrable functions associated with $$P$$ and $$\| \cdot \|_{L_2(P)}$$ denote the induced $$L_2$$-norm. Let $$f: [a,b] \rightarrow \mathbb{R}$$ be integrable. Let $$\phi_1,\phi_2,\ldots: [a,b] \rightarrow \mathbb{R}$$ be a basis and I want to use $$\sum_{k=1}^K \beta_k \phi_k$$ to approximate $$f$$ for some $$\beta_k \in \mathbb{R}$$. In particular, let $$\pi_{K,P}(f)$$ denote the projection of $$f$$ onto $$Span\{\phi_1,\ldots,\phi_K\}$$ with respect to $$\langle \cdot, \cdot \rangle_P$$. Consider the approximation error of $$\pi_{K,P}(f)$$: $$\| f-\pi_{K,P}(f) \|_{L_2(P)}$$.

I know there are results on the upper bound of this error. But is there a more accurate estimate (not just an upper bound)? Or is there a lower bound? If so, can you provide a reference?

Here I always assume $$f$$ is not a linear combination of $$\phi_1,\phi_2,\ldots$$, i.e. $$\pi_{K,P}(f) \neq f$$ for any $$K<\infty$$ and any $$P$$, so there might be a nontrivial lower bound.

You may consider simplified/restricted versions of this problem. For example, you may take $$\phi_k=x^{k-1}$$ or $$\phi_{2k-1}=\cos((k-1)x), \phi_{2k}(x)=\sin(kx)$$ (trigonometric polynomial); you may take $$[a,b]=[0,1]$$ or $$[a,b]=[0,2\pi]$$.

You may add other assumptions on $$f$$ or $$P$$ as long as they are not too restrictive. For example, assume $$f$$ is continuously differentiable up to some order or infinitely differentiable; or, assume $$P$$ is dominated by Lebesgue measure.

Edit 1: Let me try to be more specific. Is there a lower bound of the approximation error when $$f(x)=x$$ and I use a trigonometric polynomial (with both $$\sin$$ and $$\cos$$ series) to approximate? You may assume $$[a,b]=[0,2\pi],[0,\pi],[-\pi,\pi]$$ or whatever finite interval as you want. The probability measure $$P$$ has no special property. If necessary, you may assume $$P$$ is dominated by Lebesgue measure.

Edit 2: Just want to emphasize that I only care the behavior on a finite interval $$[a,b]$$ instead of the whole real line, so if I use trigonometric polynomial for approximation, I feel $$f(x)=x$$ not being periodic should not make the approximation behave super badly.

This is very broad, and largely depend on some details you have not provided. Usually the questions are -

1. How smooth is $$f$$?
2. Is $$P$$ a measure with any special properties, e.g., has an associated Sturm Liouville operator?
3. What is the choice of the basis $$\phi _1, \ldots , \phi _N, \ldots$$.

All these questions change the answer considerably. For example, let $$P=m$$ the Lebesgue measure, then for a function $$f\in C^r([0,1])$$, using the right polynomial expansion, $$\| f - \pi _{K,m} \| _2 \leq CK^{-r}$$

Hence, if $$f$$ is only integrable, convergence might be agonizingly small. That also means that if $$f$$ is analytic, but not periodic, its expansion by cosines will be very slow, since it will be discontinuous when considered as a function on $$\mathbb{T}$$.

Two good general reference books are "Interpolation and Approximation" by the late Philip Davis, and "Approximation Theory and Approximation Practice" by Nick Trefethen. I would advise you to be more specific in your question, because as it is, it's just too broad.

• I've just edited my question trying to be more specific. Can you see whether you know any relevant reference? Thanks. – David Hongxiang QIU Oct 7 '18 at 17:28
• If $P$ has no general property whatsoever, I suspect you may run into serious issues. $f(x)=x$ is not periodic, and therefore its $L^2$ error will be pretty bad for every $P$ absolutely continuous with respect to Lebesgue measure. But without any limitation, if $P= \delta(x)$, the point mass at $x=0$, a sine-only basis can give zero $L^2$ norm. – Amir Sagiv Oct 7 '18 at 17:55
• The assumption of Edit 2 should not change the result, in my opinion - if $f(a) \neq f(b)$, then we should expect a very slow convergence rate using Fourier with any scaling I can think of. – Amir Sagiv Oct 7 '18 at 19:48
• But you can always somehow extend the definition of f to the whole real line to make it periodic and at least continuous. Also, if we take $[a,b]=[−\pi,\pi]$, then $f(-x)=-f(x)$, so there seems to be a sine series that well approximates it. – David Hongxiang QIU Oct 7 '18 at 20:25
• I am quite lost with your question, and the added edits don't help - restate it from the top, and state your assumptions (not what they can be, but what they are concretely). – Amir Sagiv Oct 7 '18 at 21:33