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Question Number 78400 by ~blr237~ last updated on 17/Jan/20

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??

$$\mathrm{the}\:\mathrm{convolute}\:\mathrm{function}\:\mathrm{of}\:\mathrm{both}\:\mathrm{f}\:\mathrm{and}\:\mathrm{g}\:\mathrm{is}\:\mathrm{marked}\:\mathrm{f}\ast\mathrm{g} \\ $$$$\mathrm{And}\:\mathrm{define}\:\mathrm{by}\:\:\left(\mathrm{f}\ast\mathrm{g}\right)\left(\mathrm{x}\right)=\int_{\mathrm{0}} ^{\mathrm{x}} \mathrm{f}\left(\mathrm{x}−\mathrm{t}\right)\mathrm{g}\left(\mathrm{t}\right)\mathrm{dt} \\ $$$$\mathrm{Let}\:\mathrm{noted}\:\mathrm{E}=\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{function}\:\mathrm{define}\:\mathrm{on}\:\mathbb{R}_{+} \\ $$$$\left.\mathrm{0}\right)\:\mathrm{Prove}\:\mathrm{that}\:\mathrm{there}\:\mathrm{exist}\:\mathrm{a}\:\mathrm{function}\:\mathrm{f}_{\mathrm{0}} \in\mathrm{E}\:\mathrm{such}\:\mathrm{as}\:\mathrm{for}\:\mathrm{all}\:\mathrm{x}>\mathrm{0}\:,\:\int_{\mathrm{0}} ^{\infty} \mathrm{f}_{\mathrm{0}} \left(\mathrm{t}\right)\mathrm{e}^{−\mathrm{xt}} \mathrm{dt}=\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\mathrm{Prove}\:\mathrm{that}\:\left(\mathrm{E},\ast\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{semigroup} \\ $$$$\left.\mathrm{2}\right)\mathrm{Prove}\:\mathrm{that}\:\left(\mathrm{E},+,\ast\right)\:\mathrm{is}\:\mathrm{an}\:\mathrm{integrity}\:\mathrm{domain}.\mathrm{Is}\:\mathrm{it}\:\mathrm{a}\:\mathrm{hull}?\: \\ $$$$\left.\mathrm{3}\right)\:\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{sub}−\mathrm{set}\:\mathrm{J}\left(\mathrm{x}\right)=\left\{\mathrm{f}\in\mathrm{E}\:,\:\int_{\mathrm{0}} ^{\infty} \mathrm{f}\left(\mathrm{t}\right)\mathrm{e}^{−\mathrm{xt}} \mathrm{dt}=\mathrm{0}\right\}\:\mathrm{is}\:\mathrm{a}\:\mathrm{maximal}\:\mathrm{ideal}\:\mathrm{of}\:\mathrm{E}\:\:\:\mathrm{for}\:\mathrm{all}\:\mathrm{x}>\mathrm{0}. \\ $$$$\left.\mathrm{4}\right)\:\mathrm{let}\:\mathrm{U}\left(\mathrm{E}\right)\:\mathrm{be}\:\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{units}\:\mathrm{of}\:\mathrm{E} \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\:\mathrm{H}\left(\mathrm{x}\right)=\left\{\mathrm{f}\in\mathrm{U}\left(\mathrm{E}\right),\:\mathrm{f}\equiv\mathrm{f}_{\mathrm{0}} \mathrm{modJ}\left(\mathrm{x}\right)\right\}\:\mathrm{is}\:\mathrm{an}\:\mathrm{invariant}\:\mathrm{sub}−\mathrm{group}\:\mathrm{of}\:\mathrm{U}\left(\mathrm{E}\right). \\ $$$$\left.\mathrm{5}\right)\:\mathrm{let}\:\mathrm{be}\:\mathrm{I}_{\mathrm{n}} \:\mathrm{the}\:\mathrm{ideal}\:\mathrm{formed}\:\mathrm{by}\:\mathrm{g}_{\mathrm{n}} :\:\mathrm{x}\rightarrow\mathrm{x}^{\mathrm{n}} \:\mathrm{witb}\:\mathrm{n}\geqslant\mathrm{1},\:\mathrm{let}\:\mathrm{mark}\:\:\mathrm{I}_{\mathrm{n}} =\left(\mathrm{g}_{\mathrm{n}} \:\right)=\mathrm{g}_{\mathrm{n}} \mathrm{E} \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\:,\:\mathrm{I}_{\mathrm{n}} =\left\{\int\int..\int\mathrm{fdx}_{\mathrm{1}} ...\mathrm{dx}_{\mathrm{n}} \:,\:\:\:\:\:\mathrm{f}\in\mathrm{E}\right\} \\ $$$$\left.\mathrm{6}\right)\:\mathrm{By}\:\mathrm{using}\:\mathrm{the}\:\mathrm{fondamental}\:\mathrm{analysis}\:\mathrm{theorem}\:,\:\mathrm{can}\:\mathrm{we}\:\mathrm{say}\:\mathrm{that}\:\mathrm{E}\:\mathrm{is}\:\mathrm{a}\:\mathrm{principal}\:\mathrm{ring}?? \\ $$

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