Note 1 to entry: For a function f(t) and a wavelet ψ(t):
Cf(a,b)=∫+∞−∞f(t) ψ∗a,b(t)dt
where a is the scale parameter, b is the position parameter, and * denotes the complex conjugate.
Note 2 to entry: A discrete wavelet transform is obtained by choosing a finite number of values of the two parameters. The inverse transform expresses the function of time as a superposition of wavelets.
Note 1 to entry: For a function f(t) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaiqaciWacmGadaGadeaabaGaaqaaaOqaaiaadAgacaGGOaGaam iDaiaacMcaaaa@388F@ and a wavelet ψ(t) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaiqaciWacmGadaGadeaabaGaaqaaaOqaaiabeI8a5jaacIcaca WG0bGaaiykaaaa@3972@ :
C f (a,b)= ∫ −∞ +∞ f(t) ψ a,b ∗ (t)dt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaiqaciWacmGadaGadeaabaGaaqaaaOqaaiaadoeadaWgaaWcba GaamOzaaqabaGccaGGOaGaamyyaiaacYcacaWGIbGaaiykaiabg2da 9maapedabaGaamOzaiaacIcacaWG0bGaaiykaaWcbaGaeyOeI0Iaey OhIukabaGaey4kaSIaeyOhIukaniabgUIiYdGccaaMc8UaeqiYdK3a a0baaSqaaiaadggacaGGSaGaamOyaaqaaiabgEHiQaaakiaacIcaca WG0bGaaiykaKqzaeGaciizaOGaamiDaaaa@51E2@