Yes, you can. Specialization is a Wizard class feature that affects Wizard spellcasting only. It has no effect on spells from other classes. Complete Arcane, for example, explicitly suggests multiclassing with Sorcerer to get your missing spells.
Complete Arcane pg. 185
One way around at least part of this restriction is for a specialist wizard to take levels in sorcerer, using her sorcerer spellcasting ability to master the spells and magic items she cannot use as a wizard.
The FAQ (which is not a real rules source but may be good for convincing DMs) explicitly answered this question as well.
Specialization would be much clearer if they simply said that the spells were removed from your wizard spell list, and even effects that add spells to your spell list cannot add spells of your banned schools to your wizard spell list. That's what all the specialization rules sum up to, they just chose a long and complicated way to describe it.
However, note that Wizards, like all spellcasters, multiclass poorly. Being behind on spell levels is a very serious blow to your power. The addition of lower-level spells from another class is not nearly enough to justify this, from a purely-mechanical perspective. There are exceptions, but they involve dual-progression PrCs that can be entered with only one level lost, and that have their own class features (like Anima Mage from Tome of Magic or Ultimate Magus from Complete Mage).
Extra Spell is specifically addressed in the FAQ (p. 40):
Can you take spells from spell lists other than your own with the Extra Spell feat (CAr 79)?
The Extra Spell feat allows you to choose a new spell, but it does not remove the restrictions of how you would normally pick your spells—so they must be picked from your own spell list.
This is the general rule -- you have a class ability that lets you cast spells from a particular class list only. For instance, the wizard has the following ability:
Spells: A wizard casts arcane spells which are drawn from the sorcerer/ wizard spell list.
For something to give you spells not on your class list, it has to explicitly call this out as an exception to the more general rule. The Master Specialist ability is worded almost identically to the wizard's regular spell book entry, so it clearly does not allow this:
When you reach 2nd level, you can add one spell of your chosen school to your spellbook. The spell can be of any level that you can cast, and it is in addition to the normal spells gained when increasing your level.
If it isn't on your class list, how would you even determine if it was a level that you can cast?
Best Answer
Dual-school spells appeared only in Player’s Handbook II and Dragon Magic, and so those are the books that explain them (the explanation is the same in both).
(Player’s Handbook II pg. 95, Dragon Magic pg. 60)
So no, your wizard cannot learn or cast Kelgore’s fire bolt as a wizard spell. You could cast it from an item, or using wish or something similar, or take levels in another class that gets access to it, but as a wizard, without something exceptional, it’s not an option for you.