the convolute function of both f and g is marked f∗g
And define by (f∗g)(x)=∫_0 ^x f(x−t)g(t)dt
Let noted E=the set of function define on R_+
0) Prove that there exist a function f_0 ∈E such as for all x>0 , ∫_0 ^∞ f_0 (t)e^(−xt) dt=1
1)Prove that (E,∗) is a semigroup
2)Prove that (E,+,∗) is an integrity domain.Is it a hull?
3) Prove that the sub−set J(x)={f∈E , ∫_0 ^∞ f(t)e^(−xt) dt=0} is a maximal ideal of E for all x>0.
4) let U(E) be the set of units of E
Prove that H(x)={f∈U(E), f≡f_0 modJ(x)} is an invariant sub−group of U(E).
5) let be I_n the ideal formed by g_n : x→x^n witb n≥1, let mark I_n =(g_n )=g_n E
Prove that , I_n ={∫∫..∫fdx_1 ...dx_n , f∈E}
6) By using the fondamental analysis theorem , can we say that E is a principal ring??
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