任意ε>0,1.有K>0,使∫_{(K,+∞)}|f(x)|dx+∫_{(0,1/K)}|f(x)|dx≤ε2.又有
g(x)=∑_{1≤s≤S}c(s)E_{[d(s-1),d(s))(x),其中E_{[d(s-1),d(s))(x)為[d(s-1),d(s))的特徵函式,1/K=d(0)<...<k=d(s),滿足: ∫_{[1/K,K]}|f(x)-g(x)|dx≤ε3.==>|∫_{[a(λ),b(λ)]}f(x)cos(λx)dx--∫_{[1/K,K]∩[a(λ),b(λ)]}g(x)cos(λx)dx|≤≤∫_{(K,+∞)∩[a(λ),b(λ)]}|f(x)cos(λx)|dx++∫_{(0,1/K)∩[a(λ),b(λ)]}|f(x)cos(λx)|dx++∫_{[1/K,K]∩[a(λ),b(λ)]}|[f(x)-g(x)]cos(λx)|dx≤2ε4.∫_{[1/K,K]∩[a(λ),b(λ)]}g(x)cos(λx)dx==∑_{1≤s≤S}c(s)∫_{[d(s-1),d(s)∩[a(λ),b(λ)]}cos(λx)dx==>M>0,當λ>M|∫_{[1/K,K]∩[a(λ),b(λ)]}g(x)cos(λx)dx|≤ε==>|∫_{[a(λ),b(λ)]}f(x)cos(λx)dx|≤3ε==>Lim_{λ→+∞}∫_{[a(λ),b(λ)]}f(x)cos(λx)dx=0.
任意ε>0,1.有K>0,使∫_{(K,+∞)}|f(x)|dx+∫_{(0,1/K)}|f(x)|dx≤ε2.又有
函式g(x)=∑_{1≤s≤S}c(s)E_{[d(s-1),d(s))(x),其中E_{[d(s-1),d(s))(x)為[d(s-1),d(s))的特徵函式,1/K=d(0)<...<k=d(s),滿足: ∫_{[1/K,K]}|f(x)-g(x)|dx≤ε3.==>|∫_{[a(λ),b(λ)]}f(x)cos(λx)dx--∫_{[1/K,K]∩[a(λ),b(λ)]}g(x)cos(λx)dx|≤≤∫_{(K,+∞)∩[a(λ),b(λ)]}|f(x)cos(λx)|dx++∫_{(0,1/K)∩[a(λ),b(λ)]}|f(x)cos(λx)|dx++∫_{[1/K,K]∩[a(λ),b(λ)]}|[f(x)-g(x)]cos(λx)|dx≤2ε4.∫_{[1/K,K]∩[a(λ),b(λ)]}g(x)cos(λx)dx==∑_{1≤s≤S}c(s)∫_{[d(s-1),d(s)∩[a(λ),b(λ)]}cos(λx)dx==>M>0,當λ>M|∫_{[1/K,K]∩[a(λ),b(λ)]}g(x)cos(λx)dx|≤ε==>|∫_{[a(λ),b(λ)]}f(x)cos(λx)dx|≤3ε==>Lim_{λ→+∞}∫_{[a(λ),b(λ)]}f(x)cos(λx)dx=0.