You can't new up an instance. new instanceofAudioContext() isn't a valid thing to do.
Your interface doesn't match the API you're trying to use. In the browser, window.AudioContext isn't a instance of AudioContext. It's the class definition method, typeof AudioContext. You'll also have the same problem with the Function.
If you'd still like to keep your interface. Try this
interface Window {
AudioContext: typeof AudioContext;
webkitAudioContext: typeof AudioContext
}
declare var window: Window;
var AudioContextDecl = window.AudioContext || window.webkitAudioContext;
var audioCtx = new AudioContext();
Or if you don't want the interface,
var AudioContextDecl = <typeof AudioContext>(window.AudioContext || window.webkitAudioContext);
var audioCtx = new AudioContext();
Or even just use any. It all transpiles to the same thing.