**4. Inverse F-Transform**

In this section, we omit the assumption (6) stating every point of the universe *X* must be covered by at least one basic function. Admitting this degenerated case allows us to express a useful characterization of the case when the inverse F-transform, given by Definition 10, provides a solution to the preimage problem.

**Lemma 10.** *Let the assumptions of Lemma 3 be satisfied. If*

$$\mathcal{W}\mathcal{W}^{\top} = D \,, \tag{17}$$

*then the inverse F-transform v* = *<sup>F</sup>*<sup>ˆ</sup>[*u*] *given by Definition 10 is a solution to the preimage problem according to Definition 11, DF*[*v*] = *Wv* = *Wu* = *DF*[*u*] = *b for any vector u* ∈ *A.*

**Proof.** We see that the inverse F-transform is a solution to the preimage problem if and only if for any *u* ∈ *A*, it holds:

$$\mathcal{W}u = DF[u] = b \quad = \quad \mathcal{W}v = \mathcal{W}\mathcal{F}[u] = \mathcal{W}\mathcal{W}^\top F[u] = \mathcal{W}\mathcal{W}^{-1}D^{-1}\mathcal{W}u,$$

which holds if

$$WW^\top D^{-1} = I\_{\prime\prime}$$

or equivalently if (17) holds true.

**Corollary 3.** *Condition* (17) *holds true if both of the following equations are fulfilled:*

*i*=1

*1.* ∀*t* ∈ *Y*: *dtt* = *L* ∑ *i*=1 *wti* = *L* ∑ *i*=1 *w*2*ti , 2.* ∀*t*,*<sup>s</sup>* ∈ *Y*, *t* = *s*: *dts* = 0 = *L* ∑ *wtiwsi*

Recalling that *wti* = *<sup>W</sup>*(*<sup>t</sup>*, *xi*) = *At*(*xi*) ∈ [0, 1], we infer that condition 1 is equivalent to stating that ∀*t* ∈ *Y*, *xi* ∈ *X* : *wti* ∈ {0, <sup>1</sup>}, i.e., only simple-minded assignment of closeness is possible to be used. Condition 2 is equivalent to stating that every column of the matrix *W* contains at most one non-zero value—let us denote the system of all such matrices as W. Putting these two conditions together, we obtain the following corollary.

 *.*

**Corollary 4.** *If the matrix W* ∈ W *contains only values 0 and 1, then the inverse F-transform v* = *F* <sup>ˆ</sup>[*u*] *given by Definition 10 is a solution to the preimage problem.*

This property is demonstrated in Examples A2 and A3 in the Appendix A.

#### **5. New Set of Basic Functions**

This section was inspired by the paper [17] where one of the main goals consists in finding conditions under which a noisy signal must be sampled so that it can be reconstructed from its F-transform components. The authors showed that a fuzzy partition adjoint to the partition (with the same nodes) that determined the F-transform components, ensures that the associated inverse F-transform provides the best approximation of the original, continuous function in a certain space.

We analyzed a possible connection between discrete fuzzy partitions on the closeness space and decided to address the following question: can we find a solution to the preimage problem *Wv* = *b* by creating a new set of basic functions, s.t. they determine a linear transformation of the vector *b* that would solve the problem? If so, what are the conditions that this new set of basic functions must satisfy?

Throughout this section, assume that the sets *X* = {1, ... , *<sup>L</sup>*}, *Y* and {*At* : *X* → [0, 1] | *t* ∈ *<sup>Y</sup>*}, the matrices *W* and *D* and the vector *b* are related as in Lemma 3 and fixed.

We are looking for a new set of basic functions (a new fuzzy partition) {*Bt* : *X* → R | *t* ∈ *<sup>Y</sup>*}, written in rows of a newly created matrix *M* ∈ <sup>R</sup>*l*×*L*, i.e.,

$$\forall t \in \mathcal{Y}, \mathbf{x} \in \mathcal{X} \colon m\_{t\mathbf{x}} = B\_t(\mathbf{x}) \,, \tag{18}$$

s.t. the vector *v* = *Mb* solves the preimage problem.

Therefore, we require that *WMb* = *b* and from this, we deduce properties of the matrix *M* and express them in terms of basic functions *At*'s and *Bt*'s. Let for an arbitrary matrix *N*, *N<sup>t</sup>* denote its *t*-th row, *Nt* denote its *t*-th column and *Nts* denote its entry *nts*.

Hence, we require that

$$\forall t \in \mathcal{Y} \colon (\mathcal{W}\mathcal{M}^{\top})^{t} b = \sum\_{s \in \mathcal{Y}} (\mathcal{W}\mathcal{M}^{\top})\_{ts} b\_{s} = \sum\_{s \in \mathcal{Y}} \mathcal{W}^{t} (\mathcal{M}^{\top})\_{s} b\_{s} = b\_{t} \dots$$

Recalling that *Wtx* = *At*(*x*) and (*M*)*xs* = *Msx* = *Bs*(*x*), for *t*,*<sup>s</sup>* ∈ *Y*, *x* ∈ *X*, we have

$$\forall t \in \mathcal{Y} \colon \sum\_{s \in \mathcal{Y}} \left( b\_s \sum\_{x \in X} A\_t(x) B\_s(x) \right) = b\_t \,. \tag{19}$$

Substituting the constants

$$\mathfrak{c}\_{\mathfrak{ts}} = \sum\_{\mathfrak{x} \in X} \mathcal{A}\_{\mathfrak{t}}(\mathfrak{x}) B\_{\mathfrak{s}}(\mathfrak{x}), \quad (\mathfrak{t}, \mathfrak{s}) \in \mathcal{Y} \times \mathcal{Y}\_{\mathfrak{s}}.$$

in (19), we obtain that

=

$$\forall t \in \mathcal{Y} \colon \sum\_{s \in \mathcal{Y}} b\_s c\_{ts} = b\_t \dots$$

In the other words, by creating the matrix *C* = *WM* ∈ R*l*×*<sup>l</sup>* from the constants *cts*, (*<sup>t</sup>*,*<sup>s</sup>*) ∈ *Y* × *Y*, we solve *Cb* = *b*. As *B* can be the whole space <sup>R</sup>*l*, we generally demand that *C* is an identity matrix *I* (in the special case of rank *W* < *l*, this requirement would be unnecessarily strong but leads again to a good choice of new basic functions *Bt*'s). demand

$$\begin{aligned} \text{This means that for all } (t, s) \in \mathcal{Y} \times \mathcal{Y}\_\prime \text{ we denote} \\ 1. \qquad \forall t = s \text{; } \mathfrak{c}\_{t\star} = \ \Gamma \quad A\_t(\mathfrak{x}) B\_t(\mathfrak{x}) = 1. \text{ and} \end{aligned}$$

$$\begin{aligned} 1. \quad \forall t = s \colon c\_{ts} &= \sum\_{\mathbf{x} \in X} A\_{t}(\mathbf{x}) B\_{s}(\mathbf{x}) = 1, \mathbf{a},\\ 2. \quad \forall t \neq s \colon c\_{ts} &= \sum\_{\mathbf{x} \in X} A\_{t}(\mathbf{x}) B\_{s}(\mathbf{x}) = 0. \end{aligned}$$

=

This observation motivates the following definition.

=

**Definition 13** (Compatible Set of Basic Functions)**.** *Let the assumptions of Lemma 3 be satisfied. We say that the set* {*Bt* : *X* → R | *t* ∈ *Y*} *is a compatible set of basic functions with respect to* {*At* : *X* → R | *t* ∈ *Y*} *in the closeness space* (*<sup>X</sup>*, *<sup>w</sup>*)*, if*

$$\forall t \in \mathcal{Y} \colon \sum\_{\mathbf{x} \in \mathcal{X}} A\_t(\mathbf{x}) B\_t(\mathbf{x}) = 1\_{\mathcal{X}}$$

*and*

$$\forall t, \mathbf{s} \in \mathcal{Y}, \ t \neq \mathbf{s} \colon \sum\_{\mathbf{x} \in \mathcal{X}} A\_t(\mathbf{x}) B\_{\mathbf{s}}(\mathbf{x}) = \mathbf{0} \dots$$

**Theorem 2** (Solution Determined by a Compatible Set of Basic Functions)**.** *Let the assumptions of Lemma 3 be satisfied and let* {*Bt* : *X* → R | *t* ∈ *Y*} *be a compatible set of basic functions with respect to* {*At* : *X* → R | *t* ∈ *Y*} *in the closeness space* (*<sup>X</sup>*, *<sup>w</sup>*)*. Let b* ∈ *B and a matrix M* ∈ R*l*×*<sup>L</sup> be defined by Mtx* = *Bt*(*x*)*, t* ∈ *Y, x* ∈ *X, then the vector v* = *Mb is a solution to the preimage problem Wv* = *b.*

**Proof.** *Wv* = *WMb* ∈ R*<sup>l</sup>* and for each of its components, we have

$$\begin{aligned} \forall t \in Y \colon \mathcal{W}^t \boldsymbol{\upsilon} &= (\mathcal{W}\mathcal{M}^\top)^t \boldsymbol{b} = \sum\_{s \in Y} (\mathcal{W}\mathcal{M}^\top)\_{ts} \boldsymbol{b}\_s = \sum\_{s \in Y} \mathcal{W}^t (\mathcal{M}^\top)\_s \boldsymbol{b}\_s \\ &= \sum\_{s \in Y} \sum\_{\mathbf{x} \in \mathcal{X}} \mathcal{W}\_{tx} \mathcal{M}\_{s\mathbf{x}} \boldsymbol{b}\_s = \sum\_{s \in Y} \sum\_{\mathbf{x} \in \mathcal{X}} A\_t(\mathbf{x}) B\_s(\mathbf{x}) \boldsymbol{b}\_s = \boldsymbol{b}\_t \dots \boldsymbol{b}\_s \end{aligned}$$

This proves that *Wv* = *b*.

If we can find a set of new basic functions *Bt*'s compatible with *At*'s, then the vector *v* = *Mb* can be described as an inverse F-transform of an unknown vector *u* ∈ *A*, s.t. *b* = *DF*[*u*], with respect to {*Bt* : *X* → R | *t* ∈ *Y*} written in the rows of *M*.

In other words, the direct mapping between the vector spaces *A* and *B* is given by *W* (*W* : *A* → *B*) and any *M* gives an element of the induced inverse relation, i.e., (*b*, *Mb*) ∈ *W*−<sup>1</sup> where *W*−<sup>1</sup> ⊆ *B* × *A* is the inverse relation to the mapping *W*. This leads to multiple solutions forming a subset of the affine subspace described in Lemma 9.

**Lemma 11.** *Let the assumptions of Lemma 3 be satisfied. If* rank *W* = *l, then there always exists a compatible set of basic functions with respect to* {*At* : *X* → R | *t* ∈ *Y*} *in the closeness space* (*<sup>X</sup>*, *<sup>w</sup>*)*.*

**Proof.** Following Lemma 8, *W*−<sup>1</sup> = *<sup>W</sup>*(*WW*)−<sup>1</sup> is a right inverse to *W*. By setting *M* = *<sup>W</sup>*−1, we found a compatible set of basic functions {*Bt* : *X* → R | *t* ∈ *Y*} given by (18).

To conclude, we found the inverse-F-transform-like procedure associated with the system of new basic functions that, applied on *b*, gives a solution to the preimage problem. Examples A4 and A5 in the Appendix A illustrate the proposition of this section.
