So I was reading the two papers by Zeev Ditzian: https://arxiv.org/abs/1409.5397 http://ac.els-cdn.com/S0021904514000021/1-s2.0-S0021904514000021-main.pdf?_tid=5e49c95a-c978-11e6-a788-00000aab0f6b&acdnat=1482543129_c57ca9a6ce1cd665ac9933e2cc4e5803
For the moment I am looking at the second paper above, on page 4 they argue that by using the reference of Stein's and Weiss' called "Introduction to Fourier Analysis on Euclidean Spaces" on page 144, 1987th edition, they argue in the article that: $$\sum_{\ell =1}^{d_{n-2j}}S_{n-2j,\ell}(x)^2 = \| x \|^{2n-4j}d_{2n-j}|S^{d-1}|$$
Where in the notation in the article the $S_{n-2j,\ell}(x)=\| x\|^{n-2j} Y_{n-2j,\ell}$ where $Y_{k,\ell}$ are the spherical harmonic polynomials of degree $k$ on $S^{d-1}$ which the $d-1$-dimensional unit sphere.
Now in the book of Stein's and co, in my edition of 1971 (first edition), on page 144 in corolloary 2.9b they write that:
$$\sum_{k=1}^{a_k} |Y_m(x')|^2 = a_k\omega_{n-1}^{-1}$$ where $x'\in S^{d-1}$ and $a_k$ is the dimension of the basis of spherical harmonic polynomials above.; $\omega_{n-1}$ is the surface are of $S^{d-1}$.
Now what I don't understand is why in the paper of Ditzian the measure of $|S^{d-1}|$ doesn't appear with to the power of $-1$ like in the book, unless there's a mistake in the paper or the edition of the book that I have.
I still haven't read the proof of Corollary 2.9, anyone care to enlighten me this simple matter?
Thanks.
