This is a question posted in MSE before-<https://math.stackexchange.com/questions/3169269/references-on-equivalent-characterization-for-sobolev-spaces-of-functions-of-one>:


I cited a result which characterizes Sobolev spaces of functions of one variable as



> 
> $ H^p(a,b):= \{ x \in C^{p-1} [a,b]:
>  x^{(p-1)}(t) = \alpha + \int^t\_a \Psi ds, \ \alpha \in \mathbb{R}, \Psi \in L^2 \}$,
> 
> 
> where $ p \in \mathbb{N} $.
> 
> 
> 


from page 14 of



> 
> A. Kirsch: An Introduction to the Mathematical Theory of Inverse
>  Problems. Springer, New York, 1996.
> 
> 
> 


However, the result in this monograph lacks details for proof. Could any researcher help with references with details?


Thanks!



References on equivalent characterization for Sobolev spaces of functions of one variable
In your notation $H^k=W^{k,2}$, where $W^{k,p}$ represents the Sobolev space of functions whose derivatives of orders $\leq k$ are in $L^p$. 


A function $f\in W^{1,p}(a,b)$ if and only if there is $g\in L^p(a,b)$ and a constant $\alpha$ such that $f(x)=\alpha+\int\_a^x g(t)\, dt$ for almost all $x\in (a,b)$. Since $f\in W^{k,p}(a,b)$ if and only if $f'\in W^{k-1,p}(a,b)$ your characterization follows. 


For the proof of the above fact see Theorem 1 on p. 163 (in the first edition of the book) in the section Sobolev functions of one variable. You can find this book online.


**L. C. Evans, R. F. Gariepy**, *Measure theory and fine properties of functions.* Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. 


