How to calculate this sum using orthonormal closed system?

Question

Let $\left\{\phi_{n}\right\}_{n=1}^{\infty}$ be a closed orthonormal system of real valued functions in the inner product space V of continuous piecewise continuously differentiable functions on [0, 1] with the inner product $$ \langle f, g\rangle=f(0) \overline{g(0)}+\int_{0}^{1} f^{\prime}(t) \overline{g^{\prime}(t)} d t $$ Calculate : $$ \sum_{n=1}^{\infty}\left(\alpha \phi_{n}(0)+\int_{0}^{a} \phi_{n}^{\prime}(t) d t\right)\left(\beta \phi_{n}(0)+\int_{0}^{b} \phi_{n}^{\prime}(t) d t\right) $$ while $\alpha , \beta$ are complex numbers and $0\leq a,b \leq1$

My idea was to open the integrals values in the sum and get $$\int_{0}^{a} \phi_{n}^{\prime}(t) d t = \phi_{n}(a) - \phi_{n}(0)$$ $$\int_{0}^{b} \phi_{n}^{\prime}(t) d t = \phi_{n}(b) - \phi_{n}(0)$$ But it doesn't go well and I am losing the similarity for the inner product. I thought on another idea define a piece wise smooth function which get the value 1 in some interval and $0$ otherwise but I am not sure how to perform this idea (if it is true)

Answer 1

Hint: Consider the coordinates of the following functions: $$f(x)=\left\{\matrix{ \alpha+x& \text{if }x<a\\ \alpha+a& \text{if }x\ge a}\right.\\ g(x)=\left\{\matrix{ \beta+x& \text{if }x<b\\ \beta+b& \text{if }x\ge b}\right. $$

Answer 2

Hint:

For functions $f,g \in V$ we have $$\sum_{n=1}^\infty \langle f, \phi_n\rangle \langle \phi_n,g\rangle = \langle f,g\rangle.$$