Multiple Fourier Series and Fourier Integrals

Classical Fourier Analysis

“…3) holds for all f , g in L p = L p (X, μ), whenever 1 ≤ p ≤ ∞. Since in addition f L p (X,μ) = 0 implies that f = 0 (μ-a.e.…”

“…For 0 < p < 1, inequality (1.1. 3) is reversed when f , g ≥ 0. However, the following substitute of (1.1.3) holds:…”

Invariants of Isospectral Deformations and Spectral Rigidity

“…Let A be a -PDO on n−1 with a symbol A I = a 0 I + −1 a 0 I , and let W 0 be a -FIO operator of the form (3.31) with a symbol W 0 I = p 0 I + −1 p 0 I . Suppose that a 0 I = p 0 I = 1 in a neighborhood D 0 of I 0 , and that the symbols satisfy(2) and(3)in Proposition 3.9 with l ≥ M + 2 + 2n and > l + n − 1 /2. R 1 and R 0 are -FIOs of order 0 of the form (3.31) and such that (1) the symbol of R 0 satisfies (3.40), Downloaded by [The UC Irvine Libraries] at 02:obtain from (A.51) the homological equations (3.33) for any with ≤ N − 1, which we solve by recurrence with respect to .…”

On the Pinsky Phenomenon

“…We need condition (i) here to use (2.2) k + 1 times, and condition (ii) guarantees the convergence of partial integrals E λ (φf )(0), φ ∈ C ∞ 0 ( ) (see [1]). So we have Lemma 3 Let n 0 = [ n− 3 2 ] and f (x) ∈ W P n 0 ( ).…”

“…If A is the self-adjoint extension in L 2 (R n ) of the elliptic differential operator A(D) with the domain of definition C ∞ 0 (R n ) and {E λ }-be a decomposition of the identity of A, then the corresponding eigenfunction expansion of any function f ∈ L 2 (R n ) will coincide with (1.1) (see [1]). …”