port readme changes to docs

Author: CosmoMatt

README.md

@@ -84,7 +84,7 @@ from the root directory of the repository. Unit tests can then be
 executed to ensure the installation was successful by running
 
 ``` bash
-pytest tests/         # for pytest
+pytest tests/  
 ```
 
 In the very near future one will be able to install `S2FFT` directly

docs/assets/figures/Wigner_recursion_github_docs.png

@@ -100,12 +100,12 @@
     "header_links_before_dropdown": 5,
     "secondary_sidebar_items": ["page-toc", "edit-this-page", "sourcelink"],
     "icon_links": [
-        # {
-        #     "name": "ArXiv",
-        #     "url": "https://arxiv.org/abs/xxxx.xxxxx",
-        #     "icon": "_static/arxiv-logomark-small.png",
-        #     "type": "local",
-        # },
+        {
+            "name": "ArXiv",
+            "url": "https://arxiv.org/abs/xxxx.xxxxx",
+            "icon": "_static/arxiv-logomark-small.png",
+            "type": "local",
+        },
         # {
         #     "name": "YouTube",
         #     "url": "https://www.youtube.com/channel/UCrCOQsyQOJhOUaIYzmbkKQQ",

docs/assets/figures/sax_schematic_github_docs.png

@@ -18,7 +18,20 @@ distributed, and so map very well onto the architecture of hardware accelerators
 GPUs and TPUs).  In particular, these algorithms are based on new Wigner-d recursions
 that are stable to high angular resolution :math:`L`.  The diagram below illustrates the recursions (for further details see Price & McEwen 2023).
 
-.. image:: ./assets/figures/schematic.png
+.. image:: ./assets/figures/Wigner_recursion_github_docs.png
+
+With this recursion to hand, the spherical harmonic coefficients of an 
+isolatitudinally sampled map may be computed as a two step process. First, 
+a 1D Fourier transform over longitude, for each latitudinal ring. Second, 
+a projection onto the real polar-d functions. One may precompute and store 
+all real polar-d functions for extreme acceleration, however this comes 
+with an equally extreme memory overhead, which is infeasible at L ~ 1024. 
+Alternatively, the real polar-d functions may calculated recursively, 
+computing only a portion of the projection at a time, hence incurring 
+negligible memory overhead at the cost of slightly slower execution. The 
+diagram below illustrates the separable spherical harmonic transform.
+
+.. image:: ./assets/figures/sax_schematic_github_docs.png
 
 Sampling |:earth_africa:|
 -----------------------------------
@@ -68,41 +81,42 @@ article is referenced. A BibTeX entry for this reference may look like:
 
 .. code-block:: 
 
-     @article{price:s2fft, 
-        AUTHOR      = "Matthew A. Price and Jason D. McEwen",
-        TITLE       = "TBA",
-        YEAR        = "2023",
-        EPRINT      = "arXiv:0000.00000"        
-     }
+    @article{price:s2fft, 
+        author      = "Matthew A. Price and Jason D. McEwen and Contributors",
+        title       = "Differentiable and accelerated spherical harmonic and Wigner transforms",
+        journal     = "Journal of Computational Physics",
+        year        = "2023",
+        eprint      = "arXiv:0000.00000"        
+    }
 
 You might also like to consider citing our related papers on which this code builds:
 
 .. code-block:: 
 
     @article{mcewen:fssht,
-        AUTHOR      = "Jason D. McEwen and Yves Wiaux",
-        TITLE       = "A novel sampling theorem on the sphere",
-        JOURNAL     = "IEEE Trans. Sig. Proc.",
-        YEAR        = "2011",
-        VOLUME      = "59",
-        NUMBER      = "12",
-        PAGES       = "5876--5887",        
-        EPRINT      = "arXiv:1110.6298",
-        DOI         = "10.1109/TSP.2011.2166394"
+        author      = "Jason D. McEwen and Yves Wiaux",
+        title       = "A novel sampling theorem on the sphere",
+        journal     = "IEEE Trans. Sig. Proc.",
+        year        = "2011",
+        volume      = "59",
+        number      = "12",
+        pages       = "5876--5887",        
+        eprint      = "arXiv:1110.6298",
+        doi         = "10.1109/TSP.2011.2166394"
     }
 
 .. code-block::
    
     @article{mcewen:so3,
-        AUTHOR      = "Jason D. McEwen and Martin B{\"u}ttner and Boris ~Leistedt and Hiranya V. Peiris and Yves Wiaux",
-        TITLE       = "A novel sampling theorem on the rotation group",
-        JOURNAL     = "IEEE Sig. Proc. Let.",
-        YEAR        = "2015",
-        VOLUME      = "22",
-        NUMBER      = "12",
-        PAGES       = "2425--2429",
-        EPRINT      = "arXiv:1508.03101",
-        DOI         = "10.1109/LSP.2015.2490676"    
+        author      = "Jason D. McEwen and Martin B{\"u}ttner and Boris ~Leistedt and Hiranya V. Peiris and Yves Wiaux",
+        title       = "A novel sampling theorem on the rotation group",
+        journal     = "IEEE Sig. Proc. Let.",
+        year        = "2015",
+        volume      = "22",
+        number      = "12",
+        pages       = "2425--2429",
+        eprint      = "arXiv:1508.03101",
+        doi         = "10.1109/LSP.2015.2490676"     
     }
 
 License |:memo:|

docs/conf.py

@@ -20,7 +20,7 @@ When installing from source we recommend working within an existing conda enviro
 
 .. code-block:: bash
 
-    conda create -n "env_name" python>=3.8
+    conda create -n "env_name" python>=3.9
     conda activate "env_name"
 
 Once within a fresh environment ``S2FFT`` may be installed by cloning the GitHub repository 
@@ -37,8 +37,7 @@ installation was successful by running
 
 .. code-block:: bash 
 
-    pytest tests/         # for pytest
-    tox -e py38           # for tox 
+    pytest tests/ 
 
 In the very near future one will be able to install ``S2FFT`` directly from `PyPi` by ``pip install s2fft`` but this is not yet supported.