There are two questions.

1. For the specific functional equation considered in this question On equation $f(z+1)-f(z)=f'(z)$, the formula I gave covers all entire solutions. I added the references there.

2. On the general question about "Fourier transform" of entire functions or functions on the real line which are not in $L^p$. One usually replaces Fourier transform with various versions of Laplace transform. There are many versions, for various problems. I recommend Hormander, Analysis of differential operators..., Chap. 9, or the paper MR0199747 Ljubič, Ju. I.; Tkačenko, V. A. Theory and certain applications of the local Laplace transform. (Russian) Mat. Sb. (N.S.) 70 (112) 1966 416–437. There is an English translation in Math USSR Sbornik. There is also a nice little book by Carleman of Fourier transform (in French).

Edit. See also On linear independence of exponentials for an example how Laplace transform of entire functions is used.

There are two questions.

1. For the specific functional equation considered in this question On equation $f(z+1)-f(z)=f'(z)$, the formula I gave covers all entire solutions. I added the references there.

2. On the general question about "Fourier transform" of entire functions or functions on the real line which are not in $L^p$. One usually replaces Fourier transform with various versions of Laplace transform. There are many versions, for various problems. I recommend Hormander, Analysis of differential operators..., Chap. 9, or the paper MR0199747 Ljubič, Ju. I.; Tkačenko, V. A. Theory and certain applications of the local Laplace transform. (Russian) Mat. Sb. (N.S.) 70 (112) 1966 416–437. There is an English translation in Math USSR Sbornik. There is also a nice little book by Carleman of Fourier transform (in French).

Edit. See also On linear independence of exponentials for an example how Laplace transform of entire functions is used.

There are two questions.

1. For the specific functional equation considered in this question On equation $f(z+1)-f(z)=f'(z)$, the formula I gave covers all entire solutions. I added the reference there.

2. On the general question about "Fourier transform" of entire functions or functions on the real line which are not in $L^p$. One usually replaces Fourier transform with various versions of Laplace transform. There are many versions, for various problems. I recommend Hormander, Analysis of differential operators..., Chap. 9, or the paper MR0199747 Ljubič, Ju. I.; Tkačenko, V. A. Theory and certain applications of the local Laplace transform. (Russian) Mat. Sb. (N.S.) 70 (112) 1966 416–437. There is an English translation in Math USSR Sbornik. There is also a nice little book by Carleman of Fourier transform (in French).

There are two questions.

1. For the specific functional equation considered in this question On equation $f(z+1)-f(z)=f'(z)$, the formula I gave covers all entire solutions. I added the references there.

2. On the general question about "Fourier transform" of entire functions or functions on the real line which are not in $L^p$. One usually replaces Fourier transform with various versions of Laplace transform. There are many versions, for various problems. I recommend Hormander, Analysis of differential operators..., Chap. 9, or the paper MR0199747 Ljubič, Ju. I.; Tkačenko, V. A. Theory and certain applications of the local Laplace transform. (Russian) Mat. Sb. (N.S.) 70 (112) 1966 416–437. There is an English translation in Math USSR Sbornik. There is also a nice little book by Carleman of Fourier transform (in French).

Edit. See also On linear independence of exponentials for an example how Laplace transform of entire functions is used.