Core Audio and the Accelerate platform are built into iOS. No need for any third-party libraries. A lot if the documentation is on the Apple Developer website. For real-time low-latency audio input, try the RemoteIO Audio Unit, also built into iOS Core Audio.
There is also a book about learning Core Audio.
The Accelerate structure contains many basic DSP blocks, such as FFT, matrix math, and biquad filters.
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