Geodesic
On Linear Functional Equations with Nonpolynomial C ∞ Solutions
Canadian mathematical bulletin, Tome 14 (1971) no. 2, pp. 239-244

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It is known (cf. M. A. McKiernan [6]) that the only measurably bounded solutions ƒ of the equations 1 where x ∊ Rn , t ∊ R, αi(i= 1, ..., m) span the space , and Σi ∊ 1 μi ≠0 for any I ⊂ {1, ..., m}, are polynomials. The degree of these polynomials and the dimension of the solution space can be estimated by numbers depending on m and n. (For estimates and other details concerning equations (1) see see [1], [2], [3], [4], [5], [6].) 
Światak, Halina. On Linear Functional Equations with Nonpolynomial C ∞ Solutions. Canadian mathematical bulletin, Tome 14 (1971) no. 2, pp. 239-244. doi: 10.4153/CMB-1971-041-x
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     author = {\'Swiatak, Halina},
     title = {On {Linear} {Functional} {Equations} with {Nonpolynomial} {C} \ensuremath{\infty} {Solutions}},
     journal = {Canadian mathematical bulletin},
     pages = {239--244},
     year = {1971},
     volume = {14},
     number = {2},
     doi = {10.4153/CMB-1971-041-x},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1971-041-x/}
}
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