I have two real valued functions $f_1$ and $f_2$ such that

- $\int_0^Tf_1=\int_0^Tf_2=a_1$
- $\int_0^Tf_1^2=\int_0^Tf_2^2=a_2$
- $\forall \\ t, f_1(t),f_2(t)\in[0,1]$

Now,I want to construct a family of functions using $f_1$ and $f_2$ which also have all the above mentioned properties. For example $cf_1+(1-c)f_2$ have the first and third property and $\sqrt{cf_1^2+(1-c)f_2^2}$ have the second and third property.

But I want the functions to satisfy all three. Is there any way to combine the two functions in the required manner. Necessary smoothness conditions could be assumed.