Problem 5

Let

f

and

g

be real-valued functions defined for all real

x,y

, satisfying

f(x+y)+f(x-y)=2f(x)g(y)

for all

x,y

. Prove that if

f

is not identically zero and

|f(x)|\le1

for all

x

, then

|g(y)|\le1

for all

y

Step 2 of 4: Estimate the functional equation

In plain words

The same expression is controlled by the supremum and by $g$ .

2k\ge|f(x+y)+f(x-y)|=2|f(x)||g(y)|

Detailed analysis

The triangle inequality gives $2k\ge|f(x+y)+f(x-y)|$ . The functional equation changes the right side to $2|f(x)||g(y)|$ .