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7. 140). 57. In case of Banach spaces there is a huge literature about unconditional convergence of series and related properties of bases and unconditional bases. A discussion in connection with wavelets and functions, especially in the spaces Lp (Rn ) with 1 < p < ∞ may be found in [Woj97], Section 7. 17 or subspaces of them. 140), J PJ b = λj (b) bj , J ∈ N. 58. 54. (i) Then Hn is an unconditional basis in Lp (Rn ) with 1 < p < ∞. s (Rn ) if (ii) Then Hn is a basis in Bpq either 1 < p < ∞, or n n+1 < p ≤ 1, 1 p − 1 < s < p1 , 0 < q < ∞, n( p1 − 1) < s < 1, 0 < q < ∞.

23/24, 27. 8 and in detail in Chapter 3. 44. We add some comments. 118) is again a frame. 36, replacing there ψ ν+N,m and xν+N,m by ψ ν,m and xν,m , respectively, ν ∈ N0 . 114) the extra factor 2−N |β| on the right-hand side. 39 that ≥ 0 can be chosen arbitrarily large (at the expense of equivalence constants which depend on ). 13, p. 24. 39 as given in [Tri ], pp. 15–21, which will also be of some use later on. 17. 45. 16), and let s ∈ R, 0 < p ≤ ∞, 0 < q ≤ ∞. Let J ∈ N0 and for f ∈ S (Rn ), λνm = 2ν(s−n/p) (ϕν f )∨ (2−ν−J m), ν ∈ N0 , m ∈ Zn .

This justifies the restriction of the tvariable to 0 < t < ε < 1, where ε < 1 is immaterial, but convenient in connection with | log t|. 211). In other words, for any t > 0 there is a measurable set Mt with |Mt | ≤ t such that if x ∈ Rn \Mt . 195) are equivalent to a decreasing function, for example to its rearrangement ω ∗ . This is not obvious but one finds corresponding remarks in [Tri ], p. 196, with a reference to [DeL93], Chapter 2, §6, pp. 40–44. 219) are of the same nature. 73 we assume |supp f | ≤ ε < 1.

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A New Equation for the Distribution of Radiant Energy by Lewis G.N.


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