A Wizard can:
To learn, prepare, or cast a spell, the wizard must have an Intelligence score equal to at least 10 + the spell level. The Difficulty Class for a saving throw against a wizard's spell is 10 + the spell level + the wizard's Intelligence modifier.
Unlike a bard or sorcerer, a wizard may know any number of spells.
According to another question, a Wizard does not need to be able to cast a spell to copy it to their spellbook:
There are no restrictions on spell level aside from the usual checks required to copy a spell into the spellbook.
A Beguiler has the ability to:
A beguiler casts arcane spells, which are drawn from the beguiler spell list. When you gain access to a new level of spells, you automatically know all the spells for that level on the beguiler's spell list. You can cast any spell you know without preparing it ahead of time. Essentially, your spell list is the same as your spells known list. You also have the option of adding to your existing spell list through your advanced learning class feature (see below) as you increase in level.
This could be read to only allow the Beguiler spell list to be expanded with the Advanced Learning class feature, but there are numerous other recognized ways to add spells to spell lists - e.g. the Arcane Disciple feat or the Rainbow Servant prestige class - so Beguilers being able to cast other spells than those on the Beguiler spell list or selected with Advanced Learning seems clear.
And finally, the rules for multiclass characters (PHB, p59) state:
Spells: The character gains spells from all of his or her spellcasting classes. Thus, an experienced ranger/druid may have access to the spell protection from elements both as a ranger and as a druid. Since the spell’s effect is based on the class level of the caster, the player must keep track of whether the character is preparing and casting protection from elements as a ranger or as a druid.
So, can a Beguiler with a 1-level dip in Wizard who has learned Wizard/Sorcerer spells from scrolls cast them using Beguiler spell slots?
Or in other words, does a character have a single pool of arcane spells known, or one for each base casting class?