Is there any set of functions $\phi_1(x) , \phi_2(x) , \ldots , \phi_n(x) , \ldots $ defined on $[a,b]$ such that \begin{eqnarray} &&\langle\phi_i, \phi_j\rangle = \int_a^b \phi_i(x)\phi_j(x) \, dx = 0 , & \quad i \neq j\\ \text{and} &&\langle\phi'_i, \phi'_j\rangle = \int_a^b \phi_i'(x)\phi_j'(x) \, dx = 0 , & \quad i \neq j\\ \end{eqnarray}
In other words is it possible to construct a set of orthogonal functions with orthogonal first derivatives.
thanks for any help in advance.
If $\mathbf x_{(n)} = (1,x,...,x^n)^\top$ and $n= 2k$, we can build $k+1$ orthogonal functions as $\{f_1(x) , f_2(x) , \ldots, f_{k+1}(x)\} $ with orthogonal derivatives.
Where $$f_i(x) = \beta_{(i)}^\top \mathbf x_{(n)} $$
for example if $n=4$ we choose $\beta_{(1)} = (1,1,1,1,1) ^\top$, to calculate $\beta_{(2)}$ we obtain a system of $2$ equations with $n+1=5$ unknowns, thus we can choose three of the unknowns arbitrary, in this case we obtained $\beta_{(2)} = (1,1,1, -13863953/737217, 3891559/245739)^\top$
to calculate $\beta_{(3)}$ we obtain a system of $4$ equations with $n+1=5$ unknowns, thus we can choose one of the unknowns arbitrary, i chose $1$ for arbitrary elements and obtained $\beta_{(3)} = (1,-107258998722634059/7523609370275492, 196455394286418897/3761804685137746, -262795627408036863/3761804685137746, 58289000568303981/1880902342568873)^\top$
(to calculate $\beta_{(i)} $ we have $n+3-2i$ choices)
Now I have a question that is important to me: can we choose the elements of $\beta_{(i)}$ (those element which are arbitrary) such that the orthogonal functions $\{f_1(x) , f_2(x) , \ldots, f_{k+1}(x)\} $ generate the space $Span\langle 1,x,x^2,\ldots,x^k\rangle$ ?
You can build a collection of orthogonal polynomials with orthogonal derivatives. Define $\mathbf x_{(n)} = (1,x,...,x^n)^\top$ a polynomial of degre $n$ is just the inner product of $\mathbf x_{(n)}$ with some vector of coefficients $\beta = (\beta_0,...,\beta_n)^\top$. So you want to find a collection of vectors $\beta_{(1)},...,\beta_{(k)}$ in $\mathbb R^n$ such that $$ \forall i \neq j, \ \int \langle \beta_{(j)}, \mathbf x_{(n)} \rangle \langle \beta_{(i)}, \mathbf x_{(n)} \rangle dx = 0 \Longleftrightarrow \beta_{(i)}^\top \left[ \int \mathbf x_{(n)} \mathbf x_{(n)}^\top dx \right] \beta_{(j)} = 0 $$ and $$ \forall i \neq j, \ \int \langle \beta_{(j)}, \mathbf x_{(n)} \rangle' \langle \beta_{(i)}, \mathbf x_{(n)} \rangle' dx = 0 \Longleftrightarrow \beta_{(i)}^\top \left[ \int \mathbf y_{(n)} \mathbf y_{(n)}^\top dx \right] \beta_{(j)} = 0 $$ where $\mathbf y_{(n)} = (0,1,...nx^{n-1})^\top$. So basically, you want to find a collection of vectors that are orthgonal for the two Gram matrices $\int \mathbf x_{(n)} \mathbf x_{(n)}^\top dx$ and $\int \mathbf y_{(n)} \mathbf y_{(n)}^\top dx$. With a sufficiently large $n$, you can find $k$ such vectors (start with any $\beta_{(1)}$, then compute an orthogonal vector $\beta_{(2)}$ for the two Gram matrices and so on...)