Schwartz functions with values in Hilbert space

Let $mathfrak{s}$ denote the rapidly decaying sequence space $(x_n)_{nge 0}$ with seminorms $||x||_k=sup_n (1+n)^k|x_n|$. The aim is to create a theory for both “${rm Hom}(mathfrak{s},l^2)$” / “$mathfrak{s}otimes l^2$” and “${rm Hom}(mathfrak{s}’,l^2)$” / “$mathfrak{s}’otimes l^2$”. However, the sources found so far, such as the book “Interpolation Spaces” (section 6.1) by Bergh and Löfström, only briefly discuss these spaces without developing the basic theory required.


Question:

Assuming $H$ is a separable complex Hilbert space, we can define
schwartz function
s $fcolonmathbb R^nto H$ as smooth functions that satisfy the following condition: for all $minmathbb N$ and all multi-indices $alpha$, $sup_{xinmathbb R^n}|(1+|x|^2)^mD^alpha f(x)|_H<infty$. Instead of constructing the theory from the beginning, I prefer to use $S(mathbb R^n,H)$, which is the space of $H$-valued Scwhartz functions and its dual $S'(mathbb R^n,H)$.

Could you recommend a useful resource, such as a book or reference material, that covers the topics I’m interested in? Despite my efforts, I couldn’t come across anything myself. These are the areas that I’d like the resource to address:

  • Assuming $fin S(mathbb R^n,H)$ and ${e_k;kinmathbb N}$ is an orthonormal basis for $H$, the individual functions $f_k(x)=langle f(x),e_krangle_H$ are considered as
    Schwartz function
    functions in the conventional sense. Additionally, the series $sum_kf_k(x)e_k$ converges to $f$ in the given space. Can these outcomes be extended to $fin S’$?
  • Are the expected outcomes of Fourier transforms in both $S$ and $S’$ consistent with each other?
  • How can I extend the definition of things from $S$ to $S’$ using duality?
  • In what way do the Schwartz structure and continuous linear functions interplay in Hilbert spaces?

The questions provided are solely for example purposes and should not be answered here. If there are gaps in the material, I may inquire about more specific topics separately.

Regarding the sources I have gathered, I have come across a brief discussion of these spaces in “Interpolation Spaces” by Bergh and Löfström in section 6.1, but it doesn’t cover the basic theory I need. Additionally, I found a treatise on vector-
valued distributions
, but it fails to answer my questions or develop Fourier theory. Although a MathOverflow post is related, it’s not what I’m looking for. One of the answers suggests “Vector-Valued Distributions And Fourier Multipliers” by Herbert Amann, which seems to be the most promising source so far. If you believe that there might be a better source, please let me know. Also, feel free to correct me if I have misunderstood any of my sources.


Solution:

Regrettably, the concept of vector-valued distributions lacks significant advancement. Therefore, if Amann does not meet your requirements, you have limited alternatives.

  1. Create the necessary theory from the beginning, which is the opposite of what you’re attempting to do.
  2. Acquire French language skills and peruse Schwartz’s lengthy AIF pieces.
  3. Do a toy model first

The third option available to you could be of practical use.

Consider the space of rapidly decaying sequences $(x_n)_{nge 0}$ denoted by $mathfrak{s}$ with the defining norm $||x||_k=sup_n (1+n)^k|x_n|$. The goal is to develop a theory for ” ${rm Hom}(mathfrak{s},l^2)$ ” / ” $mathfrak{s}otimes l^2$ ” as well as ” ${rm Hom}(mathfrak{s}’,l^2)$ ” / ” $mathfrak{s}’otimes l^2$ “.

The task at hand is to analyze matrices with an infinite number of rows and columns, otherwise known as doubly infinite matrices. An example of such matrices are those that meet the condition where the supremum of the product between the $(1+m)^k$ and the sum of the squared absolute values of the entries in each row are finite.

You should be capable of rapidly creating this toy hypothesis from the ground up.

Ultimately, leverage the isomorphism between $S(mathbb{R}^d)$ and $mathfrak{s}$ utilizing
Hermite functions
as a basis, to convert the principles of the simplified theory into principles of the desired theory. This approach could potentially be applied to the Fourier transform since
hermite functions
serve as eigenvectors for this transformation.

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