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Extra info for A First Course in Harmonic Analysis (2nd Edition) (Universitext)
F (y)|dy < 4C |y|>T Since a continuous function on a bounded closed interval is uniformly continuous, there is δ > 0 such that |x| ≤ 2T, |x − x | < δ ⇒ |g(x) − g(x )| < ε . 2. CONVOLUTION 45 and |y|>T |f (y)||g(x − y) − g(x0 − y)| dy ≤ 2C |y|>T |f (y)|dy < ε . 2 Together these results imply that for |x − x0 | < δ we have |f ∗ g(x) − f ∗ g(x0 )| < ε, so f ∗ g is continuous at x0 . To see that ||f ∗ g||1 < ∞ we compute ||f ∗ g||1 ∞ = −∞ ∞ |f ∗ g(x)|dx ∞ = ≤ −∞ ∞ −∞ ∞ −∞ ∞ −∞ ∞ −∞ ∞ −∞ = = −∞ f (y)g(x − y)dy dx |f (y)g(x − y)|dy dx |f (y)g(x − y)|dx dy |f (y)|dy ∞ −∞ |g(x)|dx = ||f ||1 ||g||1 .
CHAPTER 2. HILBERT SPACES 32 Examples. • For a ﬁnite dimensional Hilbert space any family that contains a basis is a complete system. • To give an example of an inﬁnite-dimensional separable Hilbert space consider the space 2 (N). For j ∈ N let ψj ∈ 2 (N) be deﬁned by 1 if k = j, ψj (k) = 0 otherwise. Then for every f ∈ 2 (N) we get f, ψj = f (j), which implies that (ψj )j∈N is indeed a complete system. An orthonormal system in a pre-Hilbert space H is a family (hj )j∈J of vectors in H such that for every j, j ∈ J we have hj , hj = δj,j , where δj,j is the Kronecker delta: δj,j = 1, j = j , 0, otherwise.
2 Next, since gn tends to zero uniformly on [−T, T ], there is n0 ∈ N such that for all n ≥ n0 we have T −T |gn (x)|dx < ε . 2 For n ≥ n0 we get ∞ −∞ |gn (x)|dx = ≤ < T −T T −T |gn (x)|dx + |gn (x)|dx + 2 ε ε + = ε. 2 2 This concludes the proof of the lemma. 2. , fn+1 (x) ≥ fn (x) for every n ∈ N and x ∈ R. Then ∞ lim n→∞ −∞ ∞ fn (x)dx = f (x)dx. −∞ ∞ Proof: If −∞ f (x)dx < ∞, then the claim follows from the domi∞ nated convergence theorem, so let us assume that −∞ f (x)dx = ∞. For every C > 0 there is T > 0 such that T f (x)dx > C.