Group posts by weekly or monthly

Sorry, but what’s the use scope of JSON API plugin here?

The only thing you need is ad the week (or wahtever information you need) in the markup of the page.

Assuming your loop is somethin like this:

<ul id="postlist">

  <?php while( have_posts() ) : the_post(); ?>

  <li id="post_<?php the_ID(); ?>" class="post">
    <h2><?php the_title(); ?></h2>
    <p><?php the_content(); ?></p>
  </li>

  <?php endwhile; ?>

</ul>

You should only add the week information:

<ul id="postlist">

  <?php
  while( have_posts() ) : the_post();
  $week = get_the_time('W');
  ?>

  <li id="post_<?php the_ID(); ?>" data-week="<?php echo $week; ?>" class="post">
    <h2><?php the_title(); ?></h2>
    <p><?php the_content(); ?></p>
  </li>

  <?php endwhile; ?>

</ul>

After that your js code can be (please proper enqueue it on the page):

$( "#postlist" ).listview({
  autodividers: true,
  autodividersSelector: function ( li ) {
    var out="Week " + $( li ).data('week');
    return out;
  }
});

I’m not an expert of jQuery UI Mobile, but this should work.