**3. Preimage Problem**

In this section, we describe a form of preimage problem (see Definition 11 below) that we would like to solve in order to describe a similarity between functions. The inverse F-transform of a function *u* provides a vector *F* <sup>ˆ</sup>[*u*] that approximates the vector form of *u*. Therefore, we were motivated to solve the problem of how to find the class of functions that have the same inverse F-transform but decided to solve a more specific problem described in the following paragraph.

Let us have a fixed mapping (based on the F-transform identity in Lemma 3) that maps functions on vectors: *u* → *DF*[*u*] = *Wu*. We would like to characterize the similarity between functions in the sense that two functions are similar if their images (scaled vectors of direct F-transform components) coincide. In that case, their inverse F-transforms coincide as well (*DF*[*u*] = *DF*[*u*] ⇒ *<sup>F</sup>*<sup>ˆ</sup>[*u*] = *<sup>F</sup>*<sup>ˆ</sup>[*u*]). We use the singular value decomposition (formalized, e.g., in [21] and commonly abbreviated as SVD) of the closeness matrix *W* to specify the problem and its solution.

We assume that the universe *X*, its subset *Y* of nodes and the closeness matrix *W* ∈ R*l*×*<sup>L</sup>* given by (8) and (9), where 0 < |*Y*| = *l* ≤ |*X*| = *L* < +<sup>∞</sup>, are fixed. Recall that the node degrees *dtt*'s are given by (7) and form the diagonal of the matrix *D* ∈ R*l*×*l*. In the following text, we will respect the aforegiven assumptions.

Moreover, for simplicity, we will use the language of linear algebra in the following two subsections. That is why the clarification of the correspondence between discrete functions (functional spaces) and vectors (vector spaces) follows.

From the functional viewpoint, let *A* be the space of all functions *u*: *X* → R and let *B* be the space of such functions *f* : *Y* → R that assign to each node *t* ∈ *Y* the value of the corresponding F-transform component multiplied by the corresponding node degree, i.e.,

$$f \in B \quad \Leftrightarrow \quad \exists u \colon X \to \mathbb{R} \,\forall t \in \mathcal{Y} \colon f(t) = d\_{tt}F[u]\_t$$

From the viewpoint of the linear algebra, recalling that |*X*| = *L*, *A* = R*<sup>L</sup>* is a vector space, its elements are later denoted by *u* or *v*. Space *A* contains all vector representations of functions on *X*. As each element of *A* corresponds to one function on the universe *X*, we can further identify *X*, given by (5), with the set of its indices {1, ... , *<sup>L</sup>*}. Hence, the correspondence between the function *u*: *X* → R and the vector *u* ∈ R*<sup>l</sup>* is given by the following equation:

$$\forall \mathbf{x}\_{i} \in X \colon u(\mathbf{x}\_{i}) = u\_{i} \,. \tag{14}$$

.

Following (8) and (9), recall that each row of the closeness matrix *W* ∈ R*l*×*<sup>L</sup>* defines one basic function. Then, based on the F-transform identity, *B* = range *W* (range, or column space, of the matrix *W* ∈ R*l*×*<sup>L</sup>* is a linear subspace of R*<sup>l</sup>* spanned by all columns of *W*) is a vector subspace of R*<sup>l</sup>* containing the scaled images of the operator *F* that are uniquely determined by *W*. In the other words, the set of all vectors (elements) of *B*, *f* ∈ <sup>R</sup>*l*, is defined by

$$f \in B \quad \Leftrightarrow \quad \exists \upharpoonright A \in A \colon f = DF[\mu] = \mathcal{W}\mu \dots$$

Hence, the correspondence between the function *f* : *Y* → R (satisfying ∀*t* ∈ *Y*: *f*(*t*) = *dttF*[*u*]*t*) and the vector *f* ∈ range *W* is given by the following equation:

$$\forall t \in \mathcal{Y} \colon f(t) = f\_t \,. \tag{15}$$

Note that, for convenience, the vectors in R*<sup>l</sup>* are indexed based on the indices of nodes in <sup>R</sup>*L*, so their indices need not form an arithmetic sequence. The same convention will be also respected for the rows of the matrix *W*.

In the following text, we will also respect the notation of the vector spaces *A* and *B*.
