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From Wikipedia

An integral transform is any transform $T$ of the following form: $$ (Tf)(u) = \int \limits_{t_1}^{t_2} K(t, u)\, f(t)\, dt $$ $K: \mathbb{R}^2 \to \mathbb{C}$ is called the kernel function or nucleus of the transform.

Is $\{K(t,), \forall t \in \mathbb{R}\}$ considered as some generalized "basis" of the space for $Tf$?

Some kernels have an associated inverse kernel $K^{-1}(u, t)$ which (roughly speaking) yields an inverse transform: $$ f(t) = \int \limits_{u_1}^{u_2} K^{-1}( u,t )\, (Tf(u))\, du $$

Is $\{K^{-1}(,t), \forall t \in \mathbb{R}\}$ considered as some generalized "basis" of the space for $f$?

By "generalized basis", I realize that:

  • in a vector space $V$, a basis is defined to be a minimal set of vectors such that every vector in $V$ can be written as linear combination of finitely many vectors in the basis, and

  • in a topological vector space $V$, the concept of basis is generalized to be a minimal set of vectors such that every vector in $V$ can be written as convergent series of countably many vectors in the basis.

  • here for an integral transform, finite or countable sum is replaced by integral, and each function $f$ and its coordinate function may come from different functional space.

For example, consider Fourier transform FT on $L^p(\mathbb{R}), p \in [1,2]$. Then we have $K(t,u)= e^{-2\pi i tu}$ and $K^{-1}(u,t)= e^{2\pi i tu}$. Is $\{K(t,), \forall t \in \mathbb{R}\}$ considered as some generalized "basis" of FT($L^p(\mathbb{R})$) or some of its subsets? Is $\{K^{-1}(,t), \forall t \in \mathbb{R}\}$ considered as some generalized "basis" of $L^p(\mathbb{R})$ or some of its subsets? What I have realized is that $\{K(t,), \forall t \in \mathbb{R}\}$ forms an orthonormal set of functions, which is something very similar to a basis.

My questions may sound pointless, but they come from the unnaturalness or discomfort that I have been feeling about integral transforms, and the attempt to relate them to concepts that I feel more natural, such as a basis of a vector space or a TVS, in the hope of deepening my understanding.

Thanks and regards!

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First of all I suggest you restrict yourself to the Hilbertian case $p=2$, otherwise things will soon become too messy. In this setting there is a way to formalize your intuitions and it is called "theory of generalized eigenfunctions". You can find more information on Berezin-Shubin, The Schrödinger Equation. –  Giuseppe Negro Dec 30 '12 at 21:58
    
Thanks,@GiuseppeNegro! –  Tim Dec 30 '12 at 21:59
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