The Cardinal Function pp(λ)

In this paper we consider various properties of Jensen's □ principles and use them to construct several examples concerning the so-called Novák number of partially ordered sets.In §1 we give the relevant definitions and review some facts about □ principles. Apart from some simple observations most of the results in this section are known.In §2 we consider the Novák number of partially ordered sets and, using □ principles, give counterexamples to the productivity of this cardinal function. We also formulate a principle, show by forcing that it is consistent and use it to construct an ℵ2-Suslin tree T such that forcing with T × T collapses ℵ1.In §3 we briefly consider games played on partially ordered sets and relate them to the problems of the previous section. Using a version of □ we give an example of a proper partial order such that the game of length ω played on is undetermined.In §4 we raise the question of whether the Novák number of a homogenous partial order can be singular, and show that in some cases the answer is no.We assume familiarity with the basic techniques of forcing. In §1 some facts about large cardinals (e.g. weakly compact cardinals are -indescribable) and elementary properties of the constructible hierarchy are used. For this and all undefined terms we refer the reader to Jech [10].

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Interpolated Derivatives

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500014061 ◽

1958 ◽

Vol 9 (4) ◽

pp. 166-167

Author(s):

B. Spain

Keyword(s):

Fractional Derivative ◽

Analytical Continuation ◽

Cardinal Function ◽

Complementary Function ◽

Familiar Expression ◽

Image Position

In a previous paper [Spain, Proc. Roy. Soc. Edinburgh, Vol. LX (1940), 134], I have shown that the application of the cardinal function to the problem of interpolating the derivatives yields the resultThis formula is valid for x > a (the constant of integration), and R(n) < 0. The analytical continuation for R(n) ≥ 0. is indicated in the paper just quoted. The first term is the familiar expression for a fractional derivative, but the second term is not Riemann's complementary function. Furthermore, this result is unsatisfactory because it is impossible to perform the repeated operation of a fractional derivative of a fractional derivative.

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Fourier's Integral

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500029527 ◽

1915 ◽

Vol 34 ◽

pp. 3-10 ◽

Cited By ~ 4

Author(s):

T. A. Brown

Keyword(s):

Integral Formula ◽

Fourier Integral ◽

Cardinal Function ◽

Repeated Integral

The purposes of the following note are these:—(1) To show the relation between Whittaker's Cardinal Function and Fourier's Repeated Integral; (2) to give a new derivation of Fourier's Integral Formula; and (3) to extend the notion of the Fourier Integral to the case in which the variables involved are complex.

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X.—Interpolated Derivatives

Proceedings of the Royal Society of Edinburgh ◽

10.1017/s0370164600020125 ◽

1940 ◽

Vol 60 (2) ◽

pp. 134-140

Author(s):

B. Spain

Keyword(s):

Infinite Number ◽

Cardinal Function ◽

Positive Integers ◽

Generalised Derivatives

I. Various writers (Ferrar, 1927) have started out with different definitions of generalised derivatives. Essentially, the problem of the generalised derivative is a problem in interpolation. The values of the derivatives are known for all integer values of n; for all positive integers, being the ordinary derivatives; for zero, being the function itself; for negative integers, being repeated integrals. Any function of n which has the above values at the integers (i.e. any cotabular function) is a solution of the problem. Out of the infinite number of cotabular functions, there exists one discovered by E. T. Whittaker (Whittaker, 1915; Ferrar, 1925; Whittaker, 1935), called the cardinal function, possessing rather remarkable properties. In particular, if the cardinal series defining the cardinal function is convergent, then it is equivalent to the Newton-Gauss formula of interpolation.

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On the Cardinal Function of Interpolation Theory

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500007318 ◽

1927 ◽

Vol 1 (1) ◽

pp. 41-46 ◽

Cited By ~ 23

Author(s):

J. M. Whittaker

Keyword(s):

Interpolation Function ◽

Interpolation Theory ◽

Integral Theorem ◽

Cardinal Function ◽

Image Position

1. The cardinal function is the interpolation functionwhich takes the values αr at the points α + rw. Its principal properties were discovered by Professor Whittaker, amongst others that(A) When C (x) is analysed into periodic constituents by Fourier's integral-theorem, all constituents of period less than 2w are absent.

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Cardinal Functions of Purely Atomic Measures

Results in Mathematics ◽

10.1007/s00025-020-01260-x ◽

2020 ◽

Vol 75 (4) ◽

Author(s):

Szymon Gła̧b ◽

Jacek Marchwicki

Keyword(s):

Finite Measure ◽

Topological Properties ◽

Cardinal Function ◽

Cardinal Functions

AbstractLet $$\mu $$ μ be a purely atomic finite measure. Without loss of generality we may assume that $$\mu $$ μ is defined on $${\mathbb {N}}$$ N , and the atoms with smaller indexes have larger masses, that is $$\mu (\{k\})\ge \mu (\{k+1\})$$ μ ( { k } ) ≥ μ ( { k + 1 } ) for $$k\in {\mathbb {N}}$$ k ∈ N . By $$f_\mu :[0,\infty )\rightarrow \{0,1,2,\dots ,\omega ,{\mathfrak {c}}\}$$ f μ : [ 0 , ∞ ) → { 0 , 1 , 2 , ⋯ , ω , c } we denote its cardinal function $$f_{\mu }(t)=\vert \{A\subset {\mathbb {N}}:\mu (A)=t\}\vert $$ f μ ( t ) = | { A ⊂ N : μ ( A ) = t } | . We study the problem for which sets $$R\subset \{0,1,2,\dots ,\omega ,{\mathfrak {c}}\}$$ R ⊂ { 0 , 1 , 2 , ⋯ , ω , c } there is a measure $$\mu $$ μ such that $$R=\text {rng}(f_\mu )$$ R = rng ( f μ ) . We are also interested in the set-theoretic and topological properties of the set of $$\mu $$ μ -values which are obtained uniquely.

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A generalization of some cardinal function inequalities

Topology and its Applications ◽

10.1016/j.topol.2019.02.009 ◽

2019 ◽

Vol 256 ◽

pp. 228-234

Author(s):

Alejandro Ramírez-Páramo

Keyword(s):

Cardinal Function

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XVIII.—On the Consistency of Cardinal Function Interpolation. Parts I, II

Proceedings of the Royal Society of Edinburgh ◽

10.1017/s0370164600025827 ◽

1928 ◽

Vol 47 ◽

pp. 230-242 ◽

Cited By ~ 10

Author(s):

W. L. Ferrar

Keyword(s):

Cardinal Function ◽

Image Position

In a previous paper on the Cardinal Function it was proved that, callingthe cardinal function of the table of values αr at the point α + rw], “the cardinal function of the table of values [C(rw′) at the point rw′] is a function C1(x) which coincides with C(x) provided that 0<w′<w, and Σ | αn/n |, Σ |α−n/n| are convergent.”

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The cardinal series in Hilbert space

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100024944 ◽

1949 ◽

Vol 45 (3) ◽

pp. 335-341 ◽

Cited By ~ 3

Author(s):

J. D. Weston

Keyword(s):

Hilbert Space ◽

Point Of View ◽

Mean Square ◽

Interpolation Theory ◽

Smooth Approximation ◽

Cardinal Function ◽

Image Position ◽

Orthogonal Set

The name ‘cardinal function’ was given toby E. T. Whittaker (1), who considered it as a ‘smooth’ approximation to a function f(x), having the same values as f(x) at the points a + rw (r = 0, ± 1, ± 2, …). It has since been extensively studied (2), mainly from the point of view of interpolation theory. Hardy (3), however, observed that the functions νr(t) defined byform a normal orthogonal set on the interval (−∞, ∞), for r = 0, ± 1, ± 2, …. This fact suggests a discussion of the cardinal series from the point of view of mean-square approximation.

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