Boolean function denotes the map from $\mathbb{F}_2^n$ to $\mathbb{F}_2$. All the $n$-variable Boolean function consist the set $\mathfrak{B}_{n}$. Any elements $\alpha=(\alpha_0,\alpha_1,...,\alpha_{n-1})$ of $\mathbb{F}_2^n$ can be identified with an integer modulo $2^n$ by $\tilde{\alpha} =\sum_{u=0}^{n-1}\alpha_i2^{n-i-1}$.
The vector $[(−1)^{f(\tilde{0})},(−1)^{f(\tilde{1})}, ··· ,(−1)^{f(\widetilde{2^n-1}))}] $is called the polarity truth table of $f$.
The Walsh-Hadamard of an n-variable Boolean function f is defined as \begin{equation} W_{f}(\alpha)=\sum_{x \in \mathbb{F}_{2}^{n}}(-1)^{f(x) \oplus \alpha \cdot x}, \alpha \in \mathbb{F}_{2}^{n} \end{equation} The vector $\hat{f}=[W_f (\tilde{0}), W_f (\tilde{1}), ..., W_f (\widetilde{2^n-1})] $is called the Walsh spectrum of a Boolean function $f$,
There is a one-to-one correspondence between any Boolean function and its complete walsh spectrum $\hat{f}$: \begin{equation} (-1)^f=\frac{1}{2^n}H_n\hat{f}(\text{Inversion formula}) \end{equation} Here $H_n$ is Hadamard matrix,i.e. $H_1=\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} $ ,$H_{n+1}=\begin{pmatrix} H_n & H_n \\ H_n & -H_n \end{pmatrix} $.
From the inversion formula, Boolean function $f$ is uniquely determined if we know the complete Walsh spectrum $\hat{f}$. Now the problem is how to find all the Boolean functions when the partial Walsh spectrum is known.
More specifically, give a set consisting of $\alpha$ and corresponding Walsh spectral values pairs $\{(\alpha_j,\gamma_{\alpha_j}):0\leq\alpha_{j}\leq2^n-1\}_{0\leq j\leq m},0<m<2^n$, how to find as many elements of the set $F$ as possible. Here : \begin{equation} F=\left\{f \in \mathfrak{B}_{n} \mid \hat{f}\left(\alpha_{j}\right)=\gamma_{i_{j}}, 0 \leq j \leq m-1\right\} \end{equation}
Informally this is difficult since specifying more and more transform coefficients will make the function more complex.
To brute force this, you can take advantage of the fact that the Walsh transform is essentially self-inverse, and choose all possible unknown Walsh coefficients and multiply by the transform matrix to obtain the boolean function.
A good reference for all this is the chapter by Carlet on boolean functions available at his homepage https://www.math.univ-paris13.fr/~carlet/chap-fcts-Bool-corr.pdf.
This computation will sometimes yield non $\pm 1,$ valued functions which will need to be discarded. Note that some constraints exists on the Walsh coefficients resulting from the fact that the function is $\pm 1-$valued. What makes this problem difficult in general is that for example every boolean function of odd weight has a full Walsh transform support, all the transform coefficients are nonzero.
Even computing the partial Walsh transform (if you are not interested in all the coefficients) from the anf (algebraic normal form) is a tricky problem. This was addressed in this paper. You want to go in the reverse direction and obtain all compatible boolean functions. The way to go between the anf and the Walsh spectrum is via the Mobius transform.
You will have an undetermined linear system, plus the extra information that you know the unknown function is $\pm 1$ valued. Being able to shortcut this computation may be quite hard if you want all compatible functions.
If you add more constraints such as degree constraints, sometimes a unique function can be determined.
Let the support set of the known spectral values be $W_0:=\{\alpha_j:\hat{f}(\alpha_j)~is~known\}.$ By linearity of the transform you can determine the functions $f$ only up to componentwise addition modulo 2 by all functions $g$ such that $$ \hat{g}(\alpha_j)=0,\quad \forall \alpha_j \in W_0. $$ If the set $W_0$ has a simple structure, say it is the set of all frequencies with bounded Hamming weight, i.e., $W_0=\{\alpha:w_H(\alpha)\leq \ell\}$, then we know that after we solve for one function $f_0$ satisfying the partial Walsh spectrum, for example by setting the Walsh coefficients outside $W_0$ to zero, then you have the set of functions $$ f(x)=f_0(x)+g(x),\quad \hat{g}(\alpha_j)=0,\forall \alpha_j\in W_0. $$ Under the above example for $W_0,$ the set of functions $g(x)$ is simply the set of boolean functions with resilience $\ell.$
Finally, if the set $W_0$ on which the transform values are known is a subspace of $\mathbb{F}_2^n,$ then there is a moment relationship with the transform values on the dual subspace $W_0^\top.$