By Anton Deitmar

This primer in harmonic research supplies a lean and stream-lined advent to the vital techniques of this pretty thought. unlike different books at the subject, a primary path in Harmonic research is completely according to the Riemann essential and metric areas rather than the extra difficult Lebesgue critical and summary topology. however, just about all proofs are given in complete and all vital suggestions are awarded sincerely. This booklet introduces Fourier research, major as much as the Poisson Summation formulation, in addition to the recommendations utilized in harmonic research of noncommutative teams.

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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.

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