Real Analysis: Modern Techniques and Their Applications

By G. Folland. 1984

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In Pure Mathematics

Citations in the corpus

Minimax Estimation of Kernel Mean Embeddings (2016, neutral): “j Z∞ −∞ e−δ 2 w2 ℓe−iaℓwℓdw ℓ Z∞ −∞ e− δ 2 w2 jw2 je −iajwjdw j = (ez)2 j Y ℓ6= j r 2π δ e− a2 ℓ 2δ · Z∞ −∞ e− δ 2 w2 jw2 je −iajwjdw j , 38 where we used Lemma C.1. It follows from (Folland, 1999, Theorem 8.22(d)) that if g= x2f∈ L1(R), then f∧ is twice diﬀerentiable and g∧(y) = − ∂2f∧(y) ∂2”
Density Estimation in Infinite Dimensional Exponential Families (2013, neutral): “∗ p∧ 0 (ω) 2 (·) θ 2 = Xd j=1 iω jg ∧(ω) ∗ p∧ 0 (ω) 2 θ (∗) ≤ Pd j=1 kiωjg ∧(ω)k2 1 kp ∧ 0 k 2 θ (∗∗) ≤ kp0k 2 r Pd j=1 kiωjg ∧(ω)k2 1 (‡) <∞, where we have invoked generalized Young’s inequality (Folland, 1999, Proposition 8.9) in (∗), Hausdorﬀ-Young inequality (Folland, 1999, p. 253) in (∗∗), a”
On the optimal estimation of probability measures in weak and strong topologies (2013, methodology): “X K Following the analysis as above, it is easy to show that U n ( ; [12, Theorem 2.27(b)], deﬁne and so K satisﬁes ( iv ) in Theorem 4.”
Universality, Characteristic Kernels and RKHS Embedding of Measures (2010, methodology): “By the Riesz representation theorem (Folland, 1999, Theorem 7.17), C ′ 0 ( X ) = M b ( X ) in the sense that there is a bijective linear isometry µ 7→ T µ from M b ( X ) onto C ′ 0 ( X ), given by the natural mapping, Therefore, by Theorem 5, H is dense in C 0 ( X ) if and only if ( ⇐ ) Suppose…”
Hilbert Space Embeddings and Metrics on Probability Measures (2009, background): “These distributions are constructed so as to differ at a sufficiently high frequency, which is then penalized by the RKHS norm when computing γk.”
Convolution Theorems for Quaternion Fourier Transform: Properties and Applications (2013, background): “(24) In the following, we summarize the elementary properties of quaternion convolution as shown in Table 1 (compare to Folland [15]).”
The Geometric Foundations of Hamiltonian Monte Carlo (2014, background): “tion between two generic spaces, F : R→ S, to deﬁne measures on Rfor some subsets of S along with an abundance of technicalities. It is only when Sis endowed with the quotient topology relative to F (Folland, 1999; Lee, 2011) that we can deﬁne regular conditional probability measures that shed many ”
VECTOR GRÜNWALD FORMULA FOR FRACTIONAL DERIVATIVES (2004, methodology): “[12] G.B. Folland (1999) Real Analysis: modern techniques and their applications, 2nd Ed., WileyInterscience, New York.”
A new application of random matrices: Ext(C^*_{red}(F_2)) is not a group (2002, background): “satisfying that F and all its rst order partial derivatives”
Theory and Use of the EM Algorithm (2011, unknown): “27 of [13]), so we also have”
SIMPLE PROOFS OF SOME RESULTS OF RESHETNYAK (2011, background): “9 in [8]), the result is demonstrated.”