关于函数方程sum(α_ix+β_iy)from i=1 to s=sum(p_k(x)q_h(y)),Ⅱ

On the Functional Equations sum(α_ix+β_iy)from i=1 to s=sum(p_k(x)q_h(y)),Ⅱ

本文主要考虑函数方程f(x+y)+F(x-y)=f_1(x)+f_1(y)+sum(X_i(x)Y_i(y) from i=1 to n设f, F分别在〔A, B〕+〔C, D〕和〔A, B〕-〔C,D〕上Lebesgue可积,又设X_1, X_2, …, X_n, 1在〔A, B〕上,和Y_1, Y_2, …, Y_0, 1在〔C, D〕上几乎处处线性无关,我们得到方程(1)的一般解.我们也考虑函数方程?,?在一定条件下,分别给出它们的一般解.

The followiqng euation is concerned in this paperf(x+y)+F(x-y)=f_1(x)+f_1(y)+sum(X_i(x)Y_i(y) from i=1 to n The general solution of Eg. (1) can be obtained when it is assumed that the functions f and F are Lebesgue-integrable on [A, B]-[C,D] or [A, B]-[C,D], and the functions X_1, X_2…, X_n, 1 and Y_1, Y_2, …, Y_n, 1 are almost everywhere linearly independent on [A, B] or [C, D]。The general integrable solutions of the following equations (2) and (3) are also given:f(x+y)+F(x-y)=f_1(x)+f_1(y)+sum(X_i(x)Y_i(y) from i=1 to n and f(x+y)+F(x-y)=sum(X_i(x)Y_i(y) from i=1 to n